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  },
  {
    "path": "README.md",
    "content": "# Backpropagation through the Void: Optimizing control variates for black-box gradient estimation\nhttps://arxiv.org/abs/1711.00123\n\nby Will Grathwohl, Dami Choi, Yuhuai Wu, Geoffrey Roeder, David Duvenaud\n\nWe introduce a general framework for learning low-variance, unbiased gradient estimators for black-box functions of random variables, based on gradients of a learned function.\nThese estimators can be jointly trained with model parameters or policies, and are applicable in both discrete and continuous settings.\nWe give unbiased, adaptive analogs of state-of-the-art reinforcement learning methods such as advantage actor-critic.\nWe also demonstrate this framework for training discrete latent-variable models.\n\nCode for VAE Experiments lives here. The Discrete RL experiments can be found at: https://github.com/wgrathwohl/BackpropThroughTheVoidRL. \n\nA simplified, pure-python implementation is in [/relax-autograd/relax.py](/relax-autograd/relax.py)\n\nIf you have any questions about the code or paper please contact Will Grathwohl (wgrathwohl@cs.toronto.edu). The code is in \"research-state\" at the moment and I will be updating it periodically. If you have questions feel free to email me and I will do my best to respond. -Will\n"
  },
  {
    "path": "binary_vae_multilayer_per_layer.py",
    "content": "from tensorflow.examples.tutorials.mnist import input_data\nimport tensorflow as tf\nimport numpy as np\nimport time\nimport os\nimport datasets\n\nimport argparse\n\n\n\"\"\" Helper Functions \"\"\"\ndef safe_log_prob(x, eps=1e-8):\n    return tf.log(tf.clip_by_value(x, eps, 1.0))\n\n\ndef safe_clip(x, eps=1e-8):\n    return tf.clip_by_value(x, eps, 1.0)\n\n\ndef gs(x):\n    return x.get_shape().as_list()\n\n\ndef softplus(x):\n    '''\n    lovingly copied from https://github.com/tensorflow/models/blob/master/research/rebar/utils.py\n    Let m = max(0, x), then,\n\n    sofplus(x) = log(1 + e(x)) = log(e(0) + e(x)) = log(e(m)(e(-m) + e(x-m)))\n                         = m + log(e(-m) + e(x - m))\n\n    The term inside of the log is guaranteed to be between 1 and 2.\n    '''\n    m = tf.maximum(tf.zeros_like(x), x)\n    return m + tf.log(tf.exp(-m) + tf.exp(x - m))\n\n\ndef bernoulli_loglikelihood(b, log_alpha):\n    return b * (-softplus(-log_alpha)) + (1 - b) * (-log_alpha - softplus(-log_alpha))\n\n\ndef bernoulli_loglikelihood_derivitive(b, log_alpha):\n    assert gs(b) == gs(log_alpha)\n    sna = tf.sigmoid(-log_alpha)\n    return b * sna - (1-b) * (1 - sna)\n\n\ndef v_from_u(u, log_alpha, force_same=True):\n    # Lovingly copied from https://github.com/tensorflow/models/blob/master/research/rebar/rebar.py\n    u_prime = tf.nn.sigmoid(-log_alpha)\n    v_1 = (u - u_prime) / safe_clip(1 - u_prime)\n    v_1 = tf.clip_by_value(v_1, 0, 1)\n    v_1 = tf.stop_gradient(v_1)\n    v_1 = v_1 * (1 - u_prime) + u_prime\n    v_0 = u / safe_clip(u_prime)\n    v_0 = tf.clip_by_value(v_0, 0, 1)\n    v_0 = tf.stop_gradient(v_0)\n    v_0 = v_0 * u_prime\n\n    v = tf.where(u > u_prime, v_1, v_0)\n    v = tf.check_numerics(v, 'v sampling is not numerically stable.')\n    if force_same:\n        v = v + tf.stop_gradient(-v + u)  # v and u are the same up to numerical errors\n    return v\n\n\ndef reparameterize(log_alpha, noise, name=None):\n    return tf.identity(log_alpha + safe_log_prob(noise) - safe_log_prob(1 - noise), name=name)\n\n\ndef concrete_relaxation(log_alpha, noise, temp, name):\n    z = log_alpha + safe_log_prob(noise) - safe_log_prob(1 - noise)\n    return tf.sigmoid(z / temp, name=name)\n\n\ndef neg_elbo(x, samples, log_alphas_inf, log_alphas_gen, prior, log=False):\n    assert len(samples) == len(log_alphas_inf) == len(log_alphas_gen)\n    # compute log[q(b1|x)q(b2|b1)...q(bN|bN-1)]\n    log_q_bs = []\n    for b, log_alpha in zip(samples, log_alphas_inf):\n        log_q_cur_given_prev = tf.reduce_sum(bernoulli_loglikelihood(b, log_alpha), axis=1)\n        log_q_bs.append(log_q_cur_given_prev)\n    log_q_b = tf.add_n(log_q_bs)\n    # compute log[p(b1, ..., bN, x)]\n    log_p_x_bs = []\n\n    all_log_alphas_gen = list(reversed(log_alphas_gen)) + [prior]\n    all_samples_gen = [x] + samples\n    for b, log_alpha in zip(all_samples_gen, all_log_alphas_gen):\n        log_p_next_given_cur = tf.reduce_sum(bernoulli_loglikelihood(b, log_alpha), axis=1)\n        log_p_x_bs.append(log_p_next_given_cur)\n    log_p_b_x = tf.add_n(log_p_x_bs)\n\n    if log:\n        for i, log_q in enumerate(log_q_bs):\n            log_p = log_p_x_bs[i+1]\n            kl = tf.reduce_mean(log_q - log_p)\n            tf.summary.scalar(\"kl_{}\".format(i), kl)\n        tf.summary.scalar(\"log_p_x_given_b\", tf.reduce_mean(log_p_x_bs[0]))\n    return -1. * (log_p_b_x - log_q_b), log_q_bs\n\n\n\"\"\" Networks \"\"\"\ndef linear_layer(x, num_latents, name, reuse):\n    with tf.variable_scope(name, reuse=reuse):\n        log_alpha = tf.layers.dense(2. * x - 1., num_latents, name=\"log_alpha\")\n    return log_alpha\n\n\ndef nonlinear_layer(x, num_latents, name, reuse):\n    with tf.variable_scope(name, reuse=reuse):\n        h1 = tf.layers.dense(2. * x - 1., num_latents, activation=tf.tanh, name=\"h1\")\n        h2 = tf.layers.dense(h1, num_latents, activation=tf.tanh, name=\"h2\")\n        log_alpha = tf.layers.dense(h2, num_latents, name=\"log_alpha\")\n    return log_alpha\n\n\ndef inference_network(x, mean, layer, num_layers, num_latents, name, reuse, sampler, samples=[], log_alphas=[]):\n    with tf.variable_scope(name, reuse=reuse):\n        assert len(samples) == len(log_alphas)\n        # copy arrays to avoid them being modified\n        samples = [s for s in samples]\n        log_alphas = [la for la in log_alphas]\n        start = len(samples)\n        for l in range(start, num_layers):\n            if l == 0:\n                inp = ((x - mean) + 1.) / 2.\n            else:\n                inp = samples[-1]\n            log_alpha = layer(inp, num_latents, layer_name(l), reuse)\n            log_alphas.append(log_alpha)\n            sample = sampler.sample(log_alpha, l)\n            samples.append(sample)\n    assert len(log_alphas) == len(samples) == num_layers\n    return log_alphas, samples\n\n\ndef layer_name(l):\n    return \"layer_{}\".format(l)\n\n\ndef Q_name(l):\n    return \"Q_{}\".format(l)\n\ndef generator_network(samples, output_bias, layer, num_layers, num_latents, name, reuse, sampler=None, prior=None):\n    with tf.variable_scope(name, reuse=reuse):\n        log_alphas = []\n        PRODUCE_SAMPLES = False\n        if samples is None:\n            PRODUCE_SAMPLES = True\n            prior_log_alpha = prior\n            samples = [None for l in range(num_layers)]\n            samples[-1] = sampler.sample(prior_log_alpha, num_layers-1)\n        for l in reversed(range(num_layers)):\n            log_alpha = layer(\n                samples[l],\n                784 if l == 0 else num_latents, layer_name(l), reuse\n            )\n            if l == 0:\n                log_alpha = log_alpha + output_bias\n            log_alphas.append(log_alpha)\n            if l > 0 and PRODUCE_SAMPLES:\n                samples[l-1] = sampler.sample(log_alpha, l-1)\n    return log_alphas\n\n\ndef Q_func(x, x_mean, z, bs, name, reuse, depth):\n    inp = tf.concat([x - x_mean, z] + [2. * b - 1 for b in bs], 1)\n    with tf.variable_scope(name, reuse=reuse):\n        h1 = tf.layers.dense(inp, 200, tf.nn.relu, name=\"1\")\n        h2 = tf.layers.dense(h1, 200, tf.nn.relu, name=\"2\")\n        if depth == 2:\n            out = tf.layers.dense(h2, 1, name=\"out\")[:, 0]\n        elif depth == 4:\n            h3 = tf.layers.dense(h2, 200, tf.nn.relu, name=\"3\")\n            h4 = tf.layers.dense(h3, 200, tf.nn.relu, name=\"4\")\n            out = tf.layers.dense(h4, 1, name=\"out\")[:, 0]\n        else:\n            assert False\n    return out\n\n\n\"\"\" Variable Creation \"\"\"\ndef create_log_temp(num):\n    return tf.Variable(\n        [np.log(.5) for i in range(num)],\n        trainable=True,\n        name='log_temperature',\n        dtype=tf.float32\n    )\n\n\ndef create_eta(num):\n    return tf.Variable(\n        [1.0 for i in range(num)],\n        trainable=True,\n        name='eta',\n        dtype=tf.float32\n    )\n\n\nclass BSampler:\n    def __init__(self, u, name):\n        self.u = u\n        self.name = name\n    def sample(self, log_alpha, l):\n        z = reparameterize(log_alpha, self.u[l])\n        b = tf.to_float(tf.stop_gradient(z > 0), name=\"{}_{}\".format(self.name, l))\n        return b\n\n\nclass ZSampler:\n    def __init__(self, u, name):\n        self.u = u\n        self.name = name\n    def sample(self, log_alpha, l):\n        z = reparameterize(log_alpha, self.u[l], name=\"{}_{}\".format(self.name, l))\n        return z\n\n\nclass SIGZSampler:\n    def __init__(self, u, temp, name):\n        self.u = u\n        self.temp = temp\n        self.name = name\n    def sample(self, log_alpha, l):\n        sig_z = concrete_relaxation(log_alpha, self.u[l], self.temp[l], name=\"{}_{}\".format(self.name, l))\n        return sig_z\n\n\ndef log_image(im_vec, name):\n    # produce reconstruction summary\n    a = tf.exp(im_vec)\n    dec_log_theta = a / (1 + a)\n    dec_log_theta_im = tf.reshape(dec_log_theta, [-1, 28, 28, 1])\n    tf.summary.image(name, dec_log_theta_im)\n\n\ndef get_variables(tag, arr=None):\n    if arr is None:\n        return [v for v in tf.global_variables() if tag in v.name]\n    else:\n        return [v for v in arr if tag in v.name]\n\n\ndef main(relaxation=None, learn_prior=True, max_iters=None,\n         batch_size=24, num_latents=200, model_type=None, lr=None,\n         test_bias=False, train_dir=None, iwae_samples=100, dataset=\"mnist\",\n         logf=None, var_lr_scale=10., Q_wd=.0001, Q_depth=-1, checkpoint_path=None):\n\n    valid_batch_size = 100\n\n    if model_type == \"L1\":\n        num_layers = 1\n        layer_type = linear_layer\n    elif model_type == \"L2\":\n        num_layers = 2\n        layer_type = linear_layer\n    elif model_type == \"NL1\":\n        num_layers = 1\n        layer_type = nonlinear_layer\n    else:\n        assert False, \"bad model type {}\".format(model_type)\n\n    sess = tf.Session()\n    if dataset == \"mnist\":\n        X_tr, X_va, X_te = datasets.load_mnist()\n    elif dataset == \"omni\":\n        X_tr, X_va, X_te = datasets.load_omniglot()\n    else:\n        assert False\n    train_mean = np.mean(X_tr, axis=0, keepdims=True)\n    train_output_bias = -np.log(1. / np.clip(train_mean, 0.001, 0.999) - 1.).astype(np.float32)\n\n    x = tf.placeholder(tf.float32, [None, 784])\n    x_im = tf.reshape(x, [-1, 28, 28, 1])\n    tf.summary.image(\"x_true\", x_im)\n\n    # make prior for top b\n    p_prior = tf.Variable(\n        tf.zeros([num_latents],\n        dtype=tf.float32),\n        trainable=learn_prior,\n        name='p_prior',\n    )\n    # create rebar specific variables temperature and eta\n    log_temperatures = [create_log_temp(1) for l in range(num_layers)]\n    temperatures = [tf.exp(log_temp) for log_temp in log_temperatures]\n    batch_temperatures = [tf.reshape(temp, [1, -1]) for temp in temperatures]\n    etas = [create_eta(1) for l in range(num_layers)]\n    batch_etas = [tf.reshape(eta, [1, -1]) for eta in etas]\n\n    # random uniform samples\n    u = [\n        tf.random_uniform([tf.shape(x)[0], num_latents], dtype=tf.float32)\n        for l in range(num_layers)\n    ]\n    # create binary sampler\n    b_sampler = BSampler(u, \"b_sampler\")\n    gen_b_sampler = BSampler(u, \"gen_b_sampler\")\n    # generate hard forward pass\n    encoder_name = \"encoder\"\n    decoder_name = \"decoder\"\n    inf_la_b, samples_b = inference_network(\n        x, train_mean,\n        layer_type, num_layers,\n        num_latents, encoder_name, False, b_sampler\n    )\n    gen_la_b = generator_network(\n        samples_b, train_output_bias,\n        layer_type, num_layers,\n        num_latents, decoder_name, False\n    )\n    log_image(gen_la_b[-1], \"x_pred\")\n    # produce samples\n    _samples_la_b = generator_network(\n        None, train_output_bias,\n        layer_type, num_layers,\n        num_latents, decoder_name, True, sampler=gen_b_sampler, prior=p_prior\n    )\n    log_image(_samples_la_b[-1], \"x_sample\")\n\n    # hard loss evaluation and log probs\n    f_b, log_q_bs = neg_elbo(x, samples_b, inf_la_b, gen_la_b, p_prior, log=True)\n    batch_f_b = tf.expand_dims(f_b, 1)\n    total_loss = tf.reduce_mean(f_b)\n    tf.summary.scalar(\"fb\", total_loss)\n    # optimizer for model parameters\n    model_opt = tf.train.AdamOptimizer(lr, beta2=.99999)\n    # optimizer for variance reducing parameters\n    variance_opt = tf.train.AdamOptimizer(var_lr_scale * lr, beta2=.99999)\n    # get encoder and decoder variables\n    encoder_params = get_variables(encoder_name)\n    decoder_params = get_variables(decoder_name)\n    if learn_prior:\n        decoder_params.append(p_prior)\n    # compute and store gradients of hard loss with respect to encoder_parameters\n    encoder_loss_grads = {}\n    for g, v in model_opt.compute_gradients(total_loss, var_list=encoder_params):\n        encoder_loss_grads[v.name] = g\n    # get gradients for decoder parameters\n    decoder_gradvars = model_opt.compute_gradients(total_loss, var_list=decoder_params)\n    # will hold all gradvars for the model (non-variance adjusting variables)\n    model_gradvars = [gv for gv in decoder_gradvars]\n\n    # conditional samples\n    v = [v_from_u(_u, log_alpha) for _u, log_alpha in zip(u, inf_la_b)]\n    # need to create soft samplers\n    sig_z_sampler = SIGZSampler(u, batch_temperatures, \"sig_z_sampler\")\n    sig_zt_sampler = SIGZSampler(v, batch_temperatures, \"sig_zt_sampler\")\n\n    z_sampler = ZSampler(u, \"z_sampler\")\n    zt_sampler = ZSampler(v, \"zt_sampler\")\n\n    rebars = []\n    reinforces = []\n    variance_objectives = []\n    # have to produce 2 forward passes for each layer for z and zt samples\n    for l in range(num_layers):\n        cur_la_b = inf_la_b[l]\n\n        # if standard rebar or additive relaxation\n        if relaxation == \"rebar\" or relaxation == \"add\":\n            # compute soft samples and soft passes through model and soft elbos\n            cur_z_sample = sig_z_sampler.sample(cur_la_b, l)\n            prev_samples_z = samples_b[:l] + [cur_z_sample]\n\n            cur_zt_sample = sig_zt_sampler.sample(cur_la_b, l)\n            prev_samples_zt = samples_b[:l] + [cur_zt_sample]\n\n            prev_log_alphas = inf_la_b[:l] + [cur_la_b]\n\n            # soft forward passes\n            inf_la_z, samples_z = inference_network(\n                x, train_mean,\n                layer_type, num_layers,\n                num_latents, encoder_name, True, sig_z_sampler,\n                samples=prev_samples_z, log_alphas=prev_log_alphas\n            )\n            gen_la_z = generator_network(\n                samples_z, train_output_bias,\n                layer_type, num_layers,\n                num_latents, decoder_name, True\n            )\n            inf_la_zt, samples_zt = inference_network(\n                x, train_mean,\n                layer_type, num_layers,\n                num_latents, encoder_name, True, sig_zt_sampler,\n                samples=prev_samples_zt, log_alphas=prev_log_alphas\n            )\n            gen_la_zt = generator_network(\n                samples_zt, train_output_bias,\n                layer_type, num_layers,\n                num_latents, decoder_name, True\n            )\n            # soft loss evaluataions\n            f_z, _ = neg_elbo(x, samples_z, inf_la_z, gen_la_z, p_prior)\n            f_zt, _ = neg_elbo(x, samples_zt, inf_la_zt, gen_la_zt, p_prior)\n\n        if relaxation == \"add\" or relaxation == \"all\":\n            # sample z and zt\n            prev_bs = samples_b[:l]\n            cur_z_sample = z_sampler.sample(cur_la_b, l)\n            cur_zt_sample = zt_sampler.sample(cur_la_b, l)\n\n            q_z = Q_func(x, train_mean, cur_z_sample, prev_bs, Q_name(l), False, depth=Q_depth)\n            q_zt = Q_func(x, train_mean, cur_zt_sample, prev_bs, Q_name(l), True, depth=Q_depth)\n            tf.summary.scalar(\"q_z_{}\".format(l), tf.reduce_mean(q_z))\n            tf.summary.scalar(\"q_zt_{}\".format(l), tf.reduce_mean(q_zt))\n            if relaxation == \"add\":\n                f_z = f_z + q_z\n                f_zt = f_zt + q_zt\n            elif relaxation == \"all\":\n                f_z = q_z\n                f_zt = q_zt\n            else:\n                assert False\n        tf.summary.scalar(\"f_z_{}\".format(l), tf.reduce_mean(f_z))\n        tf.summary.scalar(\"f_zt_{}\".format(l), tf.reduce_mean(f_zt))\n        cur_samples_b = samples_b[l]\n        # get gradient of sample log-likelihood wrt current parameter\n        d_log_q_d_la = bernoulli_loglikelihood_derivitive(cur_samples_b, cur_la_b)\n        # get gradient of soft-losses wrt current parameter\n        d_f_z_d_la = tf.gradients(f_z, cur_la_b)[0]\n        d_f_zt_d_la = tf.gradients(f_zt, cur_la_b)[0]\n        batch_f_zt = tf.expand_dims(f_zt, 1)\n        eta = batch_etas[l]\n        # compute rebar and reinforce\n        tf.summary.histogram(\"der_diff_{}\".format(l), d_f_z_d_la - d_f_zt_d_la)\n        tf.summary.histogram(\"d_log_q_d_la_{}\".format(l), d_log_q_d_la)\n        rebar = ((batch_f_b - eta * batch_f_zt) * d_log_q_d_la + eta * (d_f_z_d_la - d_f_zt_d_la)) / batch_size\n        reinforce = batch_f_b * d_log_q_d_la / batch_size\n        rebars.append(rebar)\n        reinforces.append(reinforce)\n        tf.summary.histogram(\"rebar_{}\".format(l), rebar)\n        tf.summary.histogram(\"reinforce_{}\".format(l), reinforce)\n        # backpropogate rebar to individual layer parameters\n        layer_params = get_variables(layer_name(l), arr=encoder_params)\n        layer_rebar_grads = tf.gradients(cur_la_b, layer_params, grad_ys=rebar)\n        # get direct loss grads for each parameter\n        layer_loss_grads = [encoder_loss_grads[v.name] for v in layer_params]\n        # each param's gradient should be rebar + the direct loss gradient\n        layer_grads = [rg + lg for rg, lg in zip(layer_rebar_grads, layer_loss_grads)]\n        for rg, lg, v in zip(layer_rebar_grads, layer_loss_grads, layer_params):\n            tf.summary.histogram(v.name+\"_grad_rebar\", rg)\n            tf.summary.histogram(v.name+\"_grad_loss\", lg)\n        layer_gradvars = list(zip(layer_grads, layer_params))\n        model_gradvars.extend(layer_gradvars)\n        variance_objective = tf.reduce_mean(tf.square(rebar))\n        variance_objectives.append(variance_objective)\n\n    variance_objective = tf.add_n(variance_objectives)\n    variance_vars = log_temperatures + etas\n    if relaxation != \"rebar\":\n        q_vars = get_variables(\"Q_\")\n        wd = tf.add_n([Q_wd * tf.nn.l2_loss(v) for v in q_vars])\n        tf.summary.scalar(\"Q_weight_decay\", wd)\n        variance_vars = variance_vars + q_vars\n    else:\n        wd = 0.0\n    variance_gradvars = variance_opt.compute_gradients(variance_objective+wd, var_list=variance_vars)\n    variance_train_op = variance_opt.apply_gradients(variance_gradvars)\n    model_train_op = model_opt.apply_gradients(model_gradvars)\n    with tf.control_dependencies([model_train_op, variance_train_op]):\n        train_op = tf.no_op()\n\n    for g, v in model_gradvars + variance_gradvars:\n        print(g, v.name)\n        if g is not None:\n            tf.summary.histogram(v.name, v)\n            tf.summary.histogram(v.name+\"_grad\", g)\n\n    val_loss = tf.Variable(1000, trainable=False, name=\"val_loss\", dtype=tf.float32)\n    train_loss = tf.Variable(1000, trainable=False, name=\"train_loss\", dtype=tf.float32)\n    tf.summary.scalar(\"val_loss\", val_loss)\n    tf.summary.scalar(\"train_loss\", train_loss)\n    summ_op = tf.summary.merge_all()\n    summary_writer = tf.summary.FileWriter(train_dir)\n    sess.run(tf.global_variables_initializer())\n\n    # create savers\n    train_saver = tf.train.Saver(tf.global_variables(), max_to_keep=1)\n    val_saver = tf.train.Saver(tf.global_variables(), max_to_keep=1)\n    iwae_elbo = -(tf.reduce_logsumexp(-f_b) - np.log(valid_batch_size))\n\n    if checkpoint_path is None:\n        iters_per_epoch = X_tr.shape[0] // batch_size\n        print(\"Train set has {} examples\".format(X_tr.shape[0]))\n        if relaxation != \"rebar\":\n            print(\"Pretraining Q network\")\n            for i in range(1000):\n                if i % 100 == 0:\n                    print(i)\n                idx = np.random.randint(0, iters_per_epoch-1)\n                batch_xs = X_tr[idx * batch_size: (idx + 1) * batch_size]\n                sess.run(variance_train_op, feed_dict={x: batch_xs})\n        t = time.time()\n        best_val_loss = np.inf\n        for epoch in range(10000000):\n            train_losses = []\n            for i in range(iters_per_epoch):\n                cur_iter = epoch * iters_per_epoch + i\n                if cur_iter > max_iters:\n                    print(\"Training Completed\")\n                    return\n                batch_xs = X_tr[i*batch_size: (i+1) * batch_size]\n                if i % 1000 == 0:\n                    loss, _, = sess.run([total_loss, train_op], feed_dict={x: batch_xs})\n                    #summary_writer.add_summary(sum_str, cur_iter)\n                    time_taken = time.time() - t\n                    t = time.time()\n                    #print(cur_iter, loss, \"{} / batch\".format(time_taken / 1000))\n                    if test_bias:\n                        rebs = []\n                        refs = []\n                        for _i in range(100000):\n                            if _i % 1000 == 0:\n                                print(_i)\n                            rb, re = sess.run([rebars[3], reinforces[3]], feed_dict={x: batch_xs})\n                            rebs.append(rb[:5])\n                            refs.append(re[:5])\n                        rebs = np.array(rebs)\n                        refs = np.array(refs)\n                        re_var = np.log(refs.var(axis=0))\n                        rb_var = np.log(rebs.var(axis=0))\n                        print(\"rebar variance     = {}\".format(rb_var))\n                        print(\"reinforce variance = {}\".format(re_var))\n                        print(\"rebar     = {}\".format(rebs.mean(axis=0)))\n                        print(\"reinforce = {}\\n\".format(refs.mean(axis=0)))\n                else:\n                    loss, _ = sess.run([total_loss, train_op], feed_dict={x: batch_xs})\n\n                train_losses.append(loss)\n\n            # epoch over, run test data\n            iwaes = []\n            for x_va in X_va:\n                x_va_batch = np.array([x_va for i in range(valid_batch_size)])\n                iwae = sess.run(iwae_elbo, feed_dict={x: x_va_batch})\n                iwaes.append(iwae)\n            trl = np.mean(train_losses)\n            val = np.mean(iwaes)\n            print(\"({}) Epoch = {}, Val loss = {}, Train loss = {}\".format(train_dir, epoch, val, trl))\n            logf.write(\"{}: {} {}\\n\".format(epoch, val, trl))\n            sess.run([val_loss.assign(val), train_loss.assign(trl)])\n            if val < best_val_loss:\n                print(\"saving best model\")\n                best_val_loss = val\n                val_saver.save(sess, '{}/best-model'.format(train_dir), global_step=epoch)\n            np.random.shuffle(X_tr)\n            if epoch % 10 == 0:\n                train_saver.save(sess, '{}/model'.format(train_dir), global_step=epoch)\n\n    # run iwae elbo on test set\n    else:\n        val_saver.restore(sess, checkpoint_path)\n        iwae_elbo = -(tf.reduce_logsumexp(-f_b) - np.log(valid_batch_size))\n        iwaes = []\n        elbos = []\n        for x_te in X_te:\n            x_te_batch = np.array([x_te for i in range(100)])\n            iwae, elbo = sess.run([iwae_elbo, f_b], feed_dict={x: x_te_batch})\n            iwaes.append(iwae)\n            elbos.append(elbo)\n        print(\"MEAN IWAE: {}\".format(np.mean(iwaes)))\n        print(\"MEAN ELBO: {}\".format(np.mean(elbos)))\n\n\n\n\nif __name__ == \"__main__\":\n    parser = argparse.ArgumentParser()\n    parser.add_argument(\"--lr\", type=float, default=None)\n    parser.add_argument(\"--relaxation\", type=str, default=None)\n    parser.add_argument(\"--checkpoint_path\", type=str, default=None)\n    parser.add_argument(\"--train_dir\", type=str, default=\"/tmp/test_RELAX\")\n    parser.add_argument(\"--model\", type=str, default=None)\n    parser.add_argument(\"--max_iters\", type=int, default=None)\n    parser.add_argument(\"--dataset\", type=str, default=None)\n    parser.add_argument(\"--var_lr_scale\", type=float, default=10.)\n    parser.add_argument(\"--Q_depth\", type=int, default=-1)\n    parser.add_argument(\"--Q_wd\", type=float, default=0.0)\n    FLAGS = parser.parse_args()\n\n    td = FLAGS.train_dir\n    print(\"Train Dir is {}\".format(td))\n    if os.path.exists(td):\n        print(\"Deleting existing train dir\")\n        import shutil\n        shutil.rmtree(td)\n    os.makedirs(td)\n    # make params file\n    with open(\"{}/params.txt\".format(td), 'w') as f:\n        f.write(\"{}: {}\\n\".format(\"lr\", FLAGS.lr))\n        f.write(\"{}: {}\\n\".format(\"relaxation\", FLAGS.relaxation))\n        f.write(\"{}: {}\\n\".format(\"model\", FLAGS.model))\n        f.write(\"{}: {}\\n\".format(\"max_iters\", FLAGS.max_iters))\n        f.write(\"{}: {}\\n\".format(\"dataset\", FLAGS.dataset))\n        f.write(\"{}: {}\\n\".format(\"var_lr_scale\", FLAGS.var_lr_scale))\n        if FLAGS.relaxation != \"rebar\":\n            f.write(\"{}: {}\\n\".format(\"Q_depth\", FLAGS.Q_depth))\n            f.write(\"{}: {}\\n\".format(\"Q_wd\", FLAGS.Q_wd))\n\n    with open(\"{}/log.txt\".format(td), 'w') as logf:\n        main(\n            relaxation=FLAGS.relaxation, train_dir=td, dataset=FLAGS.dataset,\n            lr=FLAGS.lr, model_type=FLAGS.model, max_iters=FLAGS.max_iters,\n            logf=logf, var_lr_scale=FLAGS.var_lr_scale,\n            Q_depth=FLAGS.Q_depth, Q_wd=FLAGS.Q_wd, checkpoint_path=FLAGS.checkpoint_path\n        )\n"
  },
  {
    "path": "datasets.py",
    "content": "import numpy as np\nimport cPickle as pickle\nimport scipy.io\n\n\ndef load_mnist(data_file=\"/u/wgrathwohl/relaxed-rebar/data/mnist_salakhutdinov_07-19-2017.pkl\"):\n    with open(data_file, 'r') as f:\n         (tr, _), (va, _), (te, _) = pickle.load(f)\n         return tr, va, te\n\n\ndef load_omniglot(data_file='/u/wgrathwohl/relaxed-rebar/data/omniglot_07-19-2017.mat'):\n  \"\"\"Reads in Omniglot images.\n\n  Args:\n    binarize: whether to use the fixed binarization\n\n  Returns:\n    x_train: training images\n    x_valid: validation images\n    x_test: test images\n\n  \"\"\"\n  n_validation=1345\n\n  def reshape_data(data):\n    return data.reshape((-1, 28, 28)).reshape((-1, 28*28), order='fortran')\n\n  omni_raw = scipy.io.loadmat(data_file)\n\n  train_data = reshape_data(omni_raw['data'].T.astype('float32'))\n  test_data = reshape_data(omni_raw['testdata'].T.astype('float32'))\n\n  # Binarize the data with a fixed seed\n  np.random.seed(5)\n  train_data = (np.random.rand(*train_data.shape) < train_data).astype(float)\n  test_data = (np.random.rand(*test_data.shape) < test_data).astype(float)\n\n  shuffle_seed = 123\n  permutation = np.random.RandomState(seed=shuffle_seed).permutation(train_data.shape[0])\n  train_data = train_data[permutation]\n\n  x_train = train_data[:-n_validation]\n  x_valid = train_data[-n_validation:]\n  x_test = test_data\n\n  return x_train, x_valid, x_test"
  },
  {
    "path": "display_grads.py",
    "content": "import matplotlib.pyplot as plt\nimport matplotlib\nimport numpy as np\nimport pickle\n\nfn = '100_samples.pkl'\nwith open(fn, 'r') as f:\n    results = pickle.load(f)\n    \nreinf_fs, reinf_cs, rep_cs, zlaxs = results\n    \nmin_val = -3#np.min(np.concatenate([reinf_fs, reinf_cs, rep_cs, zlaxs]))\nmax_val = 3#np.max(np.concatenate([reinf_fs, reinf_cs, rep_cs, zlaxs]))\nprint(\"++++++++++++++++++++++++++\")\nprint(np.min(reinf_fs), np.max(reinf_fs))\nprint(np.min(reinf_cs), np.max(reinf_cs))\nprint(np.min(reinf_fs-reinf_cs), np.max(reinf_fs-reinf_cs))\nprint(np.min(rep_cs), np.max(rep_cs))\nprint(np.min(zlaxs), np.max(zlaxs))\nprint(\"reinforce_f variance = {}\".format(np.log(reinf_fs.var())))\nprint(\"reinforce_c variance = {}\".format(np.log(reinf_cs.var())))\nprint(\"reparam_c  variance  = {}\".format(np.log(rep_cs.var())))\nprint(\"zlaxs variance       = {}\".format(np.log(zlaxs.var())))\nprint(\"++++++++++++++++++++++++++\")\n\nmatplotlib.rcParams.update({'font.size': 30})\nmatplotlib.rcParams['text.usetex'] = True\nmatplotlib.rcParams['text.latex.preamble'] = [r'\\usepackage{amsmath}']\n\nplt.figure(1, figsize=(20,20))\n\nplt1 = plt.subplot(4, 4, 2)\nplt.hist(reinf_fs, 50, range=(min_val, max_val), normed=1, facecolor='g', alpha=0.75)\nplt.xlim(min_val, max_val)\nplt.title(r'$\\hat g_{\\text{REINFORCE}}[f]$', y=1.025)\nplt.xlabel('unbiased \\nhigh variance')\nplt.ylabel('=', rotation=0, size=40)\nplt1.yaxis.set_label_coords(-0.13,0.4)\nplt1.axes.yaxis.set_ticks([])\n\nplt2 = plt.subplot(4, 4, 3)\nplt.hist(reinf_cs, 50, range=(min_val, max_val), normed=1, facecolor='g', alpha=0.75)\nplt.xlim(min_val, max_val)\nplt.title(r'$\\hat g_{\\text{REINFORCE}}[c_\\phi]$', y=1.025)\nplt.xlabel('biased \\nhigh variance')\nplt.ylabel(r'-', rotation=0, size=60)\nplt2.yaxis.set_label_coords(-0.13,0.35)\nplt2.axes.yaxis.set_ticks([])\n\nplt3 = plt.subplot(4, 4, 4)\nplt.hist(rep_cs, 50, range=(min_val, max_val), normed=1, facecolor='g', alpha=0.75)\nplt.xlim(min_val, max_val)\nplt.title(r'$\\hat g_{\\text{reparam}}[c_\\phi]$', y=1.025)\nplt3.yaxis.set_label_coords(-0.125,0.4)\nplt.xlabel('biased \\nlow variance')\nplt.ylabel('+', rotation=0, size=40)\n\nplt3.axes.yaxis.set_ticks([])\n\nplt4 = plt.subplot(4, 4, 1)\nplt.hist(zlaxs, 50, range=(min_val, max_val), normed=1, facecolor='g', alpha=0.75)\nplt.xlim(min_val, max_val)\nplt.title(r'$\\hat g_{\\text{LAX}}$', y=1.025)\nplt.xlabel('unbiased \\nlow variance')\nplt4.axes.yaxis.set_ticks([])\n\n\nplt.tight_layout()\nplt.savefig('./10k_mnist_vae_grad_hist.pdf', format=\"pdf\", bbox_inches='tight')\nplt.show()\n\n\n"
  },
  {
    "path": "mnist_vae.py",
    "content": "from tensorflow.examples.tutorials.mnist import input_data\nfrom rebar_tf import *\nimport tensorflow as tf\nimport numpy as np\nimport os\ndef encoder(x):\n    if len(gs(x)) > 2:\n        p = np.prod(gs(x)[1:])\n        x = tf.reshape(x, [-1, p])\n    h1 = tf.layers.dense(2. * x - 1., 200, tf.nn.relu, name=\"encoder_1\")\n    h2 = tf.layers.dense(h1, 200, tf.nn.relu, name=\"encoder_2\")\n    log_alpha = tf.layers.dense(h2, 200, name=\"encoder_out\")\n    return log_alpha\n\ndef decoder(b):\n    h1 = tf.layers.dense(2. * b - 1., 200, tf.nn.relu, name=\"decoder_1\")\n    h2 = tf.layers.dense(h1, 200, tf.nn.relu, name=\"decoder_2\")\n    log_alpha = tf.layers.dense(h2, 784, name=\"decoder_out\")\n    return log_alpha\n\ndef Q_func(z):\n    h1 = tf.layers.dense(2. * z - 1., 50, tf.nn.relu, name=\"q_1\", use_bias=True)\n    out = tf.layers.dense(h1, 1, name=\"q_out\", use_bias=True)\n    scale = tf.get_variable(\n        \"q_scale\", shape=[1], dtype=tf.float32,\n        initializer=tf.constant_initializer(0), trainable=True\n    )\n    return scale[0] * out\n\n\nif __name__ == \"__main__\":\n    TRAIN_DIR = \"./rebar_new_u_and_v\"\n    reinforce = False\n    relaxed = False\n    if os.path.exists(TRAIN_DIR):\n        print(\"Deleting existing train dir\")\n        import shutil\n\n        shutil.rmtree(TRAIN_DIR)\n    os.makedirs(TRAIN_DIR)\n    sess = tf.Session()\n    batch_size = 100\n    lr = .0001\n    dataset = input_data.read_data_sets(\"MNIST_data/\", one_hot=True)\n\n    def to_vec(t):\n        return tf.reshape(t, [-1])\n    def from_vec(t):\n        return tf.reshape(t, [batch_size, -1])\n\n    x = tf.placeholder(tf.float32, [batch_size, 784])\n    x_im = tf.reshape(x, [batch_size, 28, 28, 1])\n    tf.summary.image(\"x_true\", x_im)\n    x_binary = tf.to_float(x > .5)\n    log_alpha = encoder(x_binary)\n    log_alpha_v = tf.reshape(log_alpha, [-1])\n    evals = 0\n    def loss(b):\n        log_q_b_given_x = bernoulli_loglikelihood(b, log_alpha)\n        log_q_b_given_x = tf.reduce_mean(tf.reduce_sum(log_q_b_given_x, axis=1))\n\n        log_p_b = bernoulli_loglikelihood(b, tf.zeros_like(log_alpha))\n        log_p_b = tf.reduce_mean(tf.reduce_sum(log_p_b, axis=1))\n\n        with tf.variable_scope(\"decoder\", reuse=evals>0):\n            log_alpha_x_batch = decoder(b)\n        log_p_x_given_b = bernoulli_loglikelihood(x_binary, log_alpha_x_batch)\n        log_p_x_given_b = tf.reduce_mean(tf.reduce_sum(log_p_x_given_b, axis=1))\n        # HACKY BS\n        global evals\n        if evals == 0:\n            # if first eval make image summary\n            a = tf.exp(log_alpha_x_batch)\n            log_theta_x = a / (1 + a)\n            log_theta = tf.reshape(log_theta_x, [batch_size, 28, 28, 1])\n            tf.summary.image(\"x_pred\", log_theta)\n        evals += 1\n        return -tf.expand_dims(log_p_x_given_b + log_p_b - log_q_b_given_x, 0)\n    if relaxed:\n        rebar_optimizer = RelaxedREBAROptimizer(sess, loss, Q_func, log_alpha=log_alpha, learning_rate=lr)\n    else:\n        rebar_optimizer = REBAROptimizer(sess, loss, log_alpha=log_alpha, learning_rate=lr)\n    gen_loss = rebar_optimizer.f_b\n    tf.summary.scalar(\"loss\", gen_loss[0])\n    gen_opt = tf.train.AdamOptimizer(lr)\n    gen_vars = [v for v in tf.trainable_variables() if \"decoder\" in v.name]\n    gen_gradvars = gen_opt.compute_gradients(gen_loss, var_list=gen_vars)\n    gen_train_op = gen_opt.apply_gradients(gen_gradvars)\n\n    alpha_grads = rebar_optimizer.reinforce if reinforce else rebar_optimizer.rebar\n    inf_vars = [v for v in tf.trainable_variables() if \"encode\" in v.name]\n    inf_grads = tf.gradients(log_alpha, inf_vars, grad_ys=alpha_grads)\n    inf_gradvars = zip(inf_grads, inf_vars)\n    inf_opt = tf.train.AdamOptimizer(lr)\n    inf_train_op = inf_opt.apply_gradients(inf_gradvars)\n    if relaxed:\n        gradvars = inf_gradvars + gen_gradvars + rebar_optimizer.variance_gradvars + rebar_optimizer.Q_gradvars\n    else:\n        gradvars = inf_gradvars + gen_gradvars + rebar_optimizer.variance_gradvars\n    for g, v in gradvars:\n        tf.summary.histogram(v.name, v)\n        tf.summary.histogram(v.name+\"_grad\", g)\n\n    if reinforce:\n        with tf.control_dependencies([gen_train_op, inf_train_op]):\n            train_op = tf.no_op()\n    else:\n        with tf.control_dependencies([gen_train_op, inf_train_op, rebar_optimizer.variance_reduction_op]):\n            train_op = tf.no_op()\n\n    summ_op = tf.summary.merge_all()\n    summary_writer = tf.summary.FileWriter(TRAIN_DIR)\n    sess.run(tf.global_variables_initializer())\n    for i in range(250000):\n        batch_xs, _ = dataset.train.next_batch(100)\n        if i % 100 == 0:\n            loss, _, sum_str = sess.run([gen_loss, train_op, summ_op], feed_dict={x: batch_xs})\n            summary_writer.add_summary(sum_str, i)\n            print(i, loss[0])\n        else:\n            loss, _ = sess.run([gen_loss, train_op], feed_dict={x: batch_xs})\n\n"
  },
  {
    "path": "paper/bibliography.bib",
    "content": "@article{liu2017sample,\n      title={Sample-efficient Policy Optimization with Stein Control Variate},\n        author={Liu, Hao and Feng, Yihao and Mao, Yi and Zhou, Dengyong and Peng, Jian and Liu, Qiang},\n          journal={arXiv preprint arXiv:1710.11198},\n            year={2017}\n}\n\n@inproceedings{gu2017interpolated,\n      title={Interpolated policy gradient: Merging on-policy and off-policy gradient estimation for deep reinforcement learning},\n        author={Gu, Shixiang and Lillicrap, Tim and Turner, Richard E and Ghahramani, Zoubin and Sch{\\\"o}lkopf, Bernhard and Levine, Sergey},\n          booktitle={Advances in Neural Information Processing Systems},\n            pages={3849--3858},\n              year={2017}\n}\n\n\n\n@article{oates2017control,\n\ttitle={Control functionals for Monte Carlo integration},\n\tauthor={Oates, Chris J and Girolami, Mark and Chopin, Nicolas},\n\tjournal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)},\n\tvolume={79},\n\tnumber={3},\n\tpages={695--718},\n\tyear={2017},\n\tpublisher={Wiley Online Library}\n}\n\n@article{williams1992simple,\n  title={Simple statistical gradient-following algorithms for connectionist reinforcement learning},\n  author={Williams, Ronald J},\n  journal={Machine learning},\n  volume={8},\n  number={3-4},\n  pages={229--256},\n  year={1992},\n  publisher={Springer}\n}\n\n@inproceedings{peters2006policy,\n  title={Policy gradient methods for robotics},\n  author={Peters, Jan and Schaal, Stefan},\n  booktitle={Intelligent Robots and Systems, 2006 IEEE/RSJ International Conference on},\n  pages={2219--2225},\n  year={2006},\n  organization={IEEE}\n}\n\n@article{maddison2016concrete,\n  title={The concrete distribution: A continuous relaxation of discrete random variables},\n  author={Maddison, Chris J and Mnih, Andriy and Teh, Yee Whye},\n  journal={arXiv preprint arXiv:1611.00712},\n  year={2016}\n}\n\n@article{jang2016categorical,\n  title={Categorical reparameterization with gumbel-softmax},\n  author={Jang, Eric and Gu, Shixiang and Poole, Ben},\n  journal={arXiv preprint arXiv:1611.01144},\n  year={2016}\n}\n\n@article{tucker2017rebar,\n  title={REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models},\n  author={Tucker, George and Mnih, Andriy and Maddison, Chris J and Sohl-Dickstein, Jascha},\n  journal={arXiv preprint arXiv:1703.07370},\n  year={2017}\n}\n\n@inproceedings{ruiz2016generalized,\n  title={The generalized reparameterization gradient},\n  author={Ruiz, Francisco R and AUEB, Michalis Titsias RC and Blei, David},\n  booktitle={Advances in Neural Information Processing Systems},\n  pages={460--468},\n  year={2016}\n}\n\n@article{mnih-dqn-2015,\n    Author = {Mnih, Volodymyr and Kavukcuoglu, Koray and Silver, David and Rusu, Andrei A. and Veness, Joel and Bellemare, Marc G. and Graves, Alex and Riedmiller, Martin and Fidjeland, Andreas K. and Ostrovski, Georg and Petersen, Stig and Beattie, Charles and Sadik, Amir and Antonoglou, Ioannis and King, Helen and Kumaran, Dharshan and Wierstra, Daan and Legg, Shane and Hassabis, Demis},\n    Day = {26},\n    Journal = {Nature},\n    Month = {02},\n    Number = {7540},\n    Pages = {529--533},\n    Title = {Human-level control through deep reinforcement learning},\n    Url = {http://dx.doi.org/10.1038/nature14236},\n    Volume = {518},\n    Year = {2015}\n}\n\n@article{Clevert2016ELUs,\n  author    = {Djork-Arn{\\'e} Clevert, Thomas Unterthiner, and Sepp Hochreiter},\n  title     = {Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)},\n  journal   = {International Conference on Learning Representations},\n  year      = {2016}\n}\n\n@article{kingma2013autoencoding,\n\tauthor    = {Diederik P. Kingma and Max Welling},\n\ttitle     = {Auto-Encoding Variational {B}ayes},\n\tjournal   = {International Conference on Learning Representations},\n\tyear      = {2014}\n}\n\n@inproceedings{rezende2014stochastic,\n  title={Stochastic Backpropagation and Approximate Inference in Deep Generative Models},\n  author={Rezende, Danilo J and Mohamed, Shakir and Wierstra, Daan},\n  booktitle={Proceedings of the 31st International Conference on Machine Learning},\n  pages={1278--1286},\n  year={2014}\n}\n\n@article{rumelhart1986learning,\n  title={Learning representations by back-propagating errors},\n  author={Rumelhart, David E and Hinton, Geoffrey E},\n  journal={Nature},\n  volume={323},\n  pages={9},\n  year={1986}\n}\n\n@inproceedings{kingma2015adam,\n  title={{Adam}: A Method for Stochastic Optimization},\n  year={2015},\n  author={Kingma, Diederik and Ba, Jimmy},\n  booktitle={International Conference on Learning Representations},\n}\n\n@article{kusner2017grammar,\n  title={Grammar Variational Autoencoder},\n  author={Kusner, Matt J and Paige, Brooks and Hern{\\'a}ndez-Lobato, Jos{\\'e} Miguel},\n  journal={arXiv preprint arXiv:1703.01925},\n  year={2017}\n}\n\n@inproceedings{goodfellow2014generative,\n  title={Generative adversarial nets},\n  author={Goodfellow, Ian and Pouget-Abadie, Jean and Mirza, Mehdi and Xu, Bing and Warde-Farley, David and Ozair, Sherjil and Courville, Aaron and Bengio, Yoshua},\n  booktitle={Advances in neural information processing systems},\n  pages={2672--2680},\n  year={2014}\n}\n\n@article{hoffman2013stochastic,\n  title={Stochastic variational inference},\n  author={Hoffman, Matthew D and Blei, David M and Wang, Chong and Paisley, John},\n  journal={The Journal of Machine Learning Research},\n  volume={14},\n  number={1},\n  pages={1303--1347},\n  year={2013}\n}\n\n@article{lillicrap2015continuous,\n  title={Continuous control with deep reinforcement learning},\n  author={Lillicrap, Timothy P and Hunt, Jonathan J and Pritzel, 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title={Gradient estimation using stochastic computation graphs},\n  author={Schulman, John and Heess, Nicolas and Weber, Theophane and Abbeel, Pieter},\n  booktitle={Advances in Neural Information Processing Systems},\n  pages={3528--3536},\n  year={2015}\n}\n\n@inproceedings{mnih2016asynchronous,\n  title={Asynchronous methods for deep reinforcement learning},\n  author={Mnih, Volodymyr and Badia, Adria Puigdomenech and Mirza, Mehdi and Graves, Alex and Lillicrap, Timothy and Harley, Tim and Silver, David and Kavukcuoglu, Koray},\n  booktitle={International Conference on Machine Learning},\n  pages={1928--1937},\n  year={2016}\n}\n\n@phdthesis{speelpenning1980compiling,\n  title={Compiling Fast Partial Derivatives of Functions Given by Algorithms},\n  author={Speelpenning, Bert},\n  year={1980},\n  school={University of Illinois at Urbana-Champaign}\n}\n\n@article{rall1981automatic,\n  title={Automatic differentiation: Techniques and applications},\n  author={Rall, Louis B},\n  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arXiv:1702.08165},\n  year={2017}\n}\n\n@article{gu2016q,\n  title={Q-prop: Sample-efficient policy gradient with an off-policy critic},\n  author={Gu, Shixiang and Lillicrap, Timothy and Ghahramani, Zoubin and Turner, Richard E and Levine, Sergey},\n  journal={arXiv preprint arXiv:1611.02247},\n  year={2016}\n}\n\n@inproceedings{ruiz2016overdispersed,\n  title={Overdispersed black-box variational inference},\n  author={Ruiz, Francisco J.R. and Titsias, Michalis K and Blei, David M},\n  booktitle={Uuncertainty in Artificial Intelligence},\n  year={2016}\n}\n\n@misc{Tieleman2012,\n  title={{Lecture 6.5---RmsProp: Divide the gradient by a running average of its recent magnitude}},\n  author={Tieleman, T. and Hinton, G.},\n  howpublished={COURSERA: Neural Networks for Machine Learning},\n  year={2012}\n}\n\n@article{kimura2000analysis,\n  title={An analysis of actor-critic algorithms using eligibility traces: reinforcement learning with imperfect value functions},\n  author={Kimura, 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  {
    "path": "paper/fancyhdr.sty",
    "content": "% fancyhdr.sty version 3.2\n% Fancy headers and footers for LaTeX.\n% Piet van Oostrum, \n% Dept of Computer and Information Sciences, University of Utrecht,\n% Padualaan 14, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands\n% Telephone: +31 30 2532180. Email: piet@cs.uu.nl\n% ========================================================================\n% LICENCE:\n% This file may be distributed under the terms of the LaTeX Project Public\n% License, as described in lppl.txt in the base LaTeX distribution.\n% Either version 1 or, at your option, any later version.\n% ========================================================================\n% MODIFICATION HISTORY:\n% Sep 16, 1994\n% version 1.4: Correction for use with \\reversemargin\n% Sep 29, 1994:\n% version 1.5: Added the \\iftopfloat, \\ifbotfloat and \\iffloatpage commands\n% Oct 4, 1994:\n% version 1.6: Reset single spacing in headers/footers for use with\n% setspace.sty or doublespace.sty\n% Oct 4, 1994:\n% version 1.7: changed \\let\\@mkboth\\markboth to\n% \\def\\@mkboth{\\protect\\markboth} to make it more robust\n% Dec 5, 1994:\n% version 1.8: corrections for amsbook/amsart: define \\@chapapp and (more\n% importantly) use the \\chapter/sectionmark definitions from ps@headings if\n% they exist (which should be true for all standard classes).\n% May 31, 1995:\n% version 1.9: The proposed \\renewcommand{\\headrulewidth}{\\iffloatpage...\n% construction in the doc did not work properly with the fancyplain style. \n% June 1, 1995:\n% version 1.91: The definition of \\@mkboth wasn't restored on subsequent\n% \\pagestyle{fancy}'s.\n% June 1, 1995:\n% version 1.92: The sequence \\pagestyle{fancyplain} \\pagestyle{plain}\n% \\pagestyle{fancy} would erroneously select the plain version.\n% June 1, 1995:\n% version 1.93: \\fancypagestyle command added.\n% Dec 11, 1995:\n% version 1.94: suggested by Conrad Hughes <chughes@maths.tcd.ie>\n% CJCH, Dec 11, 1995: added \\footruleskip to allow control over footrule\n% position (old hardcoded value of .3\\normalbaselineskip is far too high\n% when used with very small footer fonts).\n% Jan 31, 1996:\n% version 1.95: call \\@normalsize in the reset code if that is defined,\n% otherwise \\normalsize.\n% this is to solve a problem with ucthesis.cls, as this doesn't\n% define \\@currsize. Unfortunately for latex209 calling \\normalsize doesn't\n% work as this is optimized to do very little, so there \\@normalsize should\n% be called. Hopefully this code works for all versions of LaTeX known to\n% mankind.  \n% April 25, 1996:\n% version 1.96: initialize \\headwidth to a magic (negative) value to catch\n% most common cases that people change it before calling \\pagestyle{fancy}.\n% Note it can't be initialized when reading in this file, because\n% \\textwidth could be changed afterwards. This is quite probable.\n% We also switch to \\MakeUppercase rather than \\uppercase and introduce a\n% \\nouppercase command for use in headers. and footers.\n% May 3, 1996:\n% version 1.97: Two changes:\n% 1. Undo the change in version 1.8 (using the pagestyle{headings} defaults\n% for the chapter and section marks. The current version of amsbook and\n% amsart classes don't seem to need them anymore. Moreover the standard\n% latex classes don't use \\markboth if twoside isn't selected, and this is\n% confusing as \\leftmark doesn't work as expected.\n% 2. include a call to \\ps@empty in ps@@fancy. This is to solve a problem\n% in the amsbook and amsart classes, that make global changes to \\topskip,\n% which are reset in \\ps@empty. Hopefully this doesn't break other things.\n% May 7, 1996:\n% version 1.98:\n% Added % after the line  \\def\\nouppercase\n% May 7, 1996:\n% version 1.99: This is the alpha version of fancyhdr 2.0\n% Introduced the new commands \\fancyhead, \\fancyfoot, and \\fancyhf.\n% Changed \\headrulewidth, \\footrulewidth, \\footruleskip to\n% macros rather than length parameters, In this way they can be\n% conditionalized and they don't consume length registers. There is no need\n% to have them as length registers unless you want to do calculations with\n% them, which is unlikely. Note that this may make some uses of them\n% incompatible (i.e. if you have a file that uses \\setlength or \\xxxx=)\n% May 10, 1996:\n% version 1.99a:\n% Added a few more % signs\n% May 10, 1996:\n% version 1.99b:\n% Changed the syntax of \\f@nfor to be resistent to catcode changes of :=\n% Removed the [1] from the defs of \\lhead etc. because the parameter is\n% consumed by the \\@[xy]lhead etc. macros.\n% June 24, 1997:\n% version 1.99c:\n% corrected \\nouppercase to also include the protected form of \\MakeUppercase\n% \\global added to manipulation of \\headwidth.\n% \\iffootnote command added.\n% Some comments added about \\@fancyhead and \\@fancyfoot.\n% Aug 24, 1998\n% version 1.99d\n% Changed the default \\ps@empty to \\ps@@empty in order to allow\n% \\fancypagestyle{empty} redefinition.\n% Oct 11, 2000\n% version 2.0\n% Added LPPL license clause.\n%\n% A check for \\headheight is added. An errormessage is given (once) if the\n% header is too large. Empty headers don't generate the error even if\n% \\headheight is very small or even 0pt. \n% Warning added for the use of 'E' option when twoside option is not used.\n% In this case the 'E' fields will never be used.\n%\n% Mar 10, 2002\n% version 2.1beta\n% New command: \\fancyhfoffset[place]{length}\n% defines offsets to be applied to the header/footer to let it stick into\n% the margins (if length > 0).\n% place is like in fancyhead, except that only E,O,L,R can be used.\n% This replaces the old calculation based on \\headwidth and the marginpar\n% area.\n% \\headwidth will be dynamically calculated in the headers/footers when\n% this is used.\n%\n% Mar 26, 2002\n% version 2.1beta2\n% \\fancyhfoffset now also takes h,f as possible letters in the argument to\n% allow the header and footer widths to be different.\n% New commands \\fancyheadoffset and \\fancyfootoffset added comparable to\n% \\fancyhead and \\fancyfoot.\n% Errormessages and warnings have been made more informative.\n%\n% Dec 9, 2002\n% version 2.1\n% The defaults for \\footrulewidth, \\plainheadrulewidth and\n% \\plainfootrulewidth are changed from \\z@skip to 0pt. In this way when\n% someone inadvertantly uses \\setlength to change any of these, the value\n% of \\z@skip will not be changed, rather an errormessage will be given.\n\n% March 3, 2004\n% Release of version 3.0\n\n% Oct 7, 2004\n% version 3.1\n% Added '\\endlinechar=13' to \\fancy@reset to prevent problems with\n% includegraphics in header when verbatiminput is active.\n\n% March 22, 2005\n% version 3.2\n% reset \\everypar (the real one) in \\fancy@reset because spanish.ldf does\n% strange things with \\everypar between << and >>.\n\n\\def\\ifancy@mpty#1{\\def\\temp@a{#1}\\ifx\\temp@a\\@empty}\n\n\\def\\fancy@def#1#2{\\ifancy@mpty{#2}\\fancy@gbl\\def#1{\\leavevmode}\\else\n                                   \\fancy@gbl\\def#1{#2\\strut}\\fi}\n\n\\let\\fancy@gbl\\global\n\n\\def\\@fancyerrmsg#1{%\n        \\ifx\\PackageError\\undefined\n        \\errmessage{#1}\\else\n        \\PackageError{Fancyhdr}{#1}{}\\fi}\n\\def\\@fancywarning#1{%\n        \\ifx\\PackageWarning\\undefined\n        \\errmessage{#1}\\else\n        \\PackageWarning{Fancyhdr}{#1}{}\\fi}\n\n% Usage: \\@forc \\var{charstring}{command to be executed for each char}\n% This is similar to LaTeX's \\@tfor, but expands the charstring.\n\n\\def\\@forc#1#2#3{\\expandafter\\f@rc\\expandafter#1\\expandafter{#2}{#3}}\n\\def\\f@rc#1#2#3{\\def\\temp@ty{#2}\\ifx\\@empty\\temp@ty\\else\n                                    \\f@@rc#1#2\\f@@rc{#3}\\fi}\n\\def\\f@@rc#1#2#3\\f@@rc#4{\\def#1{#2}#4\\f@rc#1{#3}{#4}}\n\n% Usage: \\f@nfor\\name:=list\\do{body}\n% Like LaTeX's \\@for but an empty list is treated as a list with an empty\n% element\n\n\\newcommand{\\f@nfor}[3]{\\edef\\@fortmp{#2}%\n    \\expandafter\\@forloop#2,\\@nil,\\@nil\\@@#1{#3}}\n\n% Usage: \\def@ult \\cs{defaults}{argument}\n% sets \\cs to the characters from defaults appearing in argument\n% or defaults if it would be empty. All characters are lowercased.\n\n\\newcommand\\def@ult[3]{%\n    \\edef\\temp@a{\\lowercase{\\edef\\noexpand\\temp@a{#3}}}\\temp@a\n    \\def#1{}%\n    \\@forc\\tmpf@ra{#2}%\n        {\\expandafter\\if@in\\tmpf@ra\\temp@a{\\edef#1{#1\\tmpf@ra}}{}}%\n    \\ifx\\@empty#1\\def#1{#2}\\fi}\n% \n% \\if@in <char><set><truecase><falsecase>\n%\n\\newcommand{\\if@in}[4]{%\n    \\edef\\temp@a{#2}\\def\\temp@b##1#1##2\\temp@b{\\def\\temp@b{##1}}%\n    \\expandafter\\temp@b#2#1\\temp@b\\ifx\\temp@a\\temp@b #4\\else #3\\fi}\n\n\\newcommand{\\fancyhead}{\\@ifnextchar[{\\f@ncyhf\\fancyhead h}%\n                                     {\\f@ncyhf\\fancyhead h[]}}\n\\newcommand{\\fancyfoot}{\\@ifnextchar[{\\f@ncyhf\\fancyfoot f}%\n                                     {\\f@ncyhf\\fancyfoot f[]}}\n\\newcommand{\\fancyhf}{\\@ifnextchar[{\\f@ncyhf\\fancyhf{}}%\n                                   {\\f@ncyhf\\fancyhf{}[]}}\n\n% New commands for offsets added\n\n\\newcommand{\\fancyheadoffset}{\\@ifnextchar[{\\f@ncyhfoffs\\fancyheadoffset h}%\n                                           {\\f@ncyhfoffs\\fancyheadoffset h[]}}\n\\newcommand{\\fancyfootoffset}{\\@ifnextchar[{\\f@ncyhfoffs\\fancyfootoffset f}%\n                                           {\\f@ncyhfoffs\\fancyfootoffset f[]}}\n\\newcommand{\\fancyhfoffset}{\\@ifnextchar[{\\f@ncyhfoffs\\fancyhfoffset{}}%\n                                         {\\f@ncyhfoffs\\fancyhfoffset{}[]}}\n\n% The header and footer fields are stored in command sequences with\n% names of the form: \\f@ncy<x><y><z> with <x> for [eo], <y> from [lcr]\n% and <z> from [hf].\n\n\\def\\f@ncyhf#1#2[#3]#4{%\n    \\def\\temp@c{}%\n    \\@forc\\tmpf@ra{#3}%\n        {\\expandafter\\if@in\\tmpf@ra{eolcrhf,EOLCRHF}%\n            {}{\\edef\\temp@c{\\temp@c\\tmpf@ra}}}%\n    \\ifx\\@empty\\temp@c\\else\n        \\@fancyerrmsg{Illegal char `\\temp@c' in \\string#1 argument:\n          [#3]}%\n    \\fi\n    \\f@nfor\\temp@c{#3}%\n        {\\def@ult\\f@@@eo{eo}\\temp@c\n         \\if@twoside\\else\n           \\if\\f@@@eo e\\@fancywarning\n             {\\string#1's `E' option without twoside option is useless}\\fi\\fi\n         \\def@ult\\f@@@lcr{lcr}\\temp@c\n         \\def@ult\\f@@@hf{hf}{#2\\temp@c}%\n         \\@forc\\f@@eo\\f@@@eo\n             {\\@forc\\f@@lcr\\f@@@lcr\n                 {\\@forc\\f@@hf\\f@@@hf\n                     {\\expandafter\\fancy@def\\csname\n                      f@ncy\\f@@eo\\f@@lcr\\f@@hf\\endcsname\n                      {#4}}}}}}\n\n\\def\\f@ncyhfoffs#1#2[#3]#4{%\n    \\def\\temp@c{}%\n    \\@forc\\tmpf@ra{#3}%\n        {\\expandafter\\if@in\\tmpf@ra{eolrhf,EOLRHF}%\n            {}{\\edef\\temp@c{\\temp@c\\tmpf@ra}}}%\n    \\ifx\\@empty\\temp@c\\else\n        \\@fancyerrmsg{Illegal char `\\temp@c' in \\string#1 argument:\n          [#3]}%\n    \\fi\n    \\f@nfor\\temp@c{#3}%\n        {\\def@ult\\f@@@eo{eo}\\temp@c\n         \\if@twoside\\else\n           \\if\\f@@@eo e\\@fancywarning\n             {\\string#1's `E' option without twoside option is useless}\\fi\\fi\n         \\def@ult\\f@@@lcr{lr}\\temp@c\n         \\def@ult\\f@@@hf{hf}{#2\\temp@c}%\n         \\@forc\\f@@eo\\f@@@eo\n             {\\@forc\\f@@lcr\\f@@@lcr\n                 {\\@forc\\f@@hf\\f@@@hf\n                     {\\expandafter\\setlength\\csname\n                      f@ncyO@\\f@@eo\\f@@lcr\\f@@hf\\endcsname\n                      {#4}}}}}%\n     \\fancy@setoffs}\n\n% Fancyheadings version 1 commands. These are more or less deprecated,\n% but they continue to work.\n\n\\newcommand{\\lhead}{\\@ifnextchar[{\\@xlhead}{\\@ylhead}}\n\\def\\@xlhead[#1]#2{\\fancy@def\\f@ncyelh{#1}\\fancy@def\\f@ncyolh{#2}}\n\\def\\@ylhead#1{\\fancy@def\\f@ncyelh{#1}\\fancy@def\\f@ncyolh{#1}}\n\n\\newcommand{\\chead}{\\@ifnextchar[{\\@xchead}{\\@ychead}}\n\\def\\@xchead[#1]#2{\\fancy@def\\f@ncyech{#1}\\fancy@def\\f@ncyoch{#2}}\n\\def\\@ychead#1{\\fancy@def\\f@ncyech{#1}\\fancy@def\\f@ncyoch{#1}}\n\n\\newcommand{\\rhead}{\\@ifnextchar[{\\@xrhead}{\\@yrhead}}\n\\def\\@xrhead[#1]#2{\\fancy@def\\f@ncyerh{#1}\\fancy@def\\f@ncyorh{#2}}\n\\def\\@yrhead#1{\\fancy@def\\f@ncyerh{#1}\\fancy@def\\f@ncyorh{#1}}\n\n\\newcommand{\\lfoot}{\\@ifnextchar[{\\@xlfoot}{\\@ylfoot}}\n\\def\\@xlfoot[#1]#2{\\fancy@def\\f@ncyelf{#1}\\fancy@def\\f@ncyolf{#2}}\n\\def\\@ylfoot#1{\\fancy@def\\f@ncyelf{#1}\\fancy@def\\f@ncyolf{#1}}\n\n\\newcommand{\\cfoot}{\\@ifnextchar[{\\@xcfoot}{\\@ycfoot}}\n\\def\\@xcfoot[#1]#2{\\fancy@def\\f@ncyecf{#1}\\fancy@def\\f@ncyocf{#2}}\n\\def\\@ycfoot#1{\\fancy@def\\f@ncyecf{#1}\\fancy@def\\f@ncyocf{#1}}\n\n\\newcommand{\\rfoot}{\\@ifnextchar[{\\@xrfoot}{\\@yrfoot}}\n\\def\\@xrfoot[#1]#2{\\fancy@def\\f@ncyerf{#1}\\fancy@def\\f@ncyorf{#2}}\n\\def\\@yrfoot#1{\\fancy@def\\f@ncyerf{#1}\\fancy@def\\f@ncyorf{#1}}\n\n\\newlength{\\fancy@headwidth}\n\\let\\headwidth\\fancy@headwidth\n\\newlength{\\f@ncyO@elh}\n\\newlength{\\f@ncyO@erh}\n\\newlength{\\f@ncyO@olh}\n\\newlength{\\f@ncyO@orh}\n\\newlength{\\f@ncyO@elf}\n\\newlength{\\f@ncyO@erf}\n\\newlength{\\f@ncyO@olf}\n\\newlength{\\f@ncyO@orf}\n\\newcommand{\\headrulewidth}{0.4pt}\n\\newcommand{\\footrulewidth}{0pt}\n\\newcommand{\\footruleskip}{.3\\normalbaselineskip}\n\n% Fancyplain stuff shouldn't be used anymore (rather\n% \\fancypagestyle{plain} should be used), but it must be present for\n% compatibility reasons.\n\n\\newcommand{\\plainheadrulewidth}{0pt}\n\\newcommand{\\plainfootrulewidth}{0pt}\n\\newif\\if@fancyplain \\@fancyplainfalse\n\\def\\fancyplain#1#2{\\if@fancyplain#1\\else#2\\fi}\n\n\\headwidth=-123456789sp %magic constant\n\n% Command to reset various things in the headers:\n% a.o.  single spacing (taken from setspace.sty)\n% and the catcode of ^^M (so that epsf files in the header work if a\n% verbatim crosses a page boundary)\n% It also defines a \\nouppercase command that disables \\uppercase and\n% \\Makeuppercase. It can only be used in the headers and footers.\n\\let\\fnch@everypar\\everypar% save real \\everypar because of spanish.ldf\n\\def\\fancy@reset{\\fnch@everypar{}\\restorecr\\endlinechar=13\n \\def\\baselinestretch{1}%\n \\def\\nouppercase##1{{\\let\\uppercase\\relax\\let\\MakeUppercase\\relax\n     \\expandafter\\let\\csname MakeUppercase \\endcsname\\relax##1}}%\n \\ifx\\undefined\\@newbaseline% NFSS not present; 2.09 or 2e\n   \\ifx\\@normalsize\\undefined \\normalsize % for ucthesis.cls\n   \\else \\@normalsize \\fi\n \\else% NFSS (2.09) present\n  \\@newbaseline%\n \\fi}\n\n% Initialization of the head and foot text.\n\n% The default values still contain \\fancyplain for compatibility.\n\\fancyhf{} % clear all\n% lefthead empty on ``plain'' pages, \\rightmark on even, \\leftmark on odd pages\n% evenhead empty on ``plain'' pages, \\leftmark on even, \\rightmark on odd pages\n\\if@twoside\n  \\fancyhead[el,or]{\\fancyplain{}{\\sl\\rightmark}}\n  \\fancyhead[er,ol]{\\fancyplain{}{\\sl\\leftmark}}\n\\else\n  \\fancyhead[l]{\\fancyplain{}{\\sl\\rightmark}}\n  \\fancyhead[r]{\\fancyplain{}{\\sl\\leftmark}}\n\\fi\n\\fancyfoot[c]{\\rm\\thepage} % page number\n\n% Use box 0 as a temp box and dimen 0 as temp dimen. \n% This can be done, because this code will always\n% be used inside another box, and therefore the changes are local.\n\n\\def\\@fancyvbox#1#2{\\setbox0\\vbox{#2}\\ifdim\\ht0>#1\\@fancywarning\n  {\\string#1 is too small (\\the#1): ^^J Make it at least \\the\\ht0.^^J\n    We now make it that large for the rest of the document.^^J\n    This may cause the page layout to be inconsistent, however\\@gobble}%\n  \\dimen0=#1\\global\\setlength{#1}{\\ht0}\\ht0=\\dimen0\\fi\n  \\box0}\n\n% Put together a header or footer given the left, center and\n% right text, fillers at left and right and a rule.\n% The \\lap commands put the text into an hbox of zero size,\n% so overlapping text does not generate an errormessage.\n% These macros have 5 parameters:\n% 1. LEFTSIDE BEARING % This determines at which side the header will stick\n%    out. When \\fancyhfoffset is used this calculates \\headwidth, otherwise\n%    it is \\hss or \\relax (after expansion).\n% 2. \\f@ncyolh, \\f@ncyelh, \\f@ncyolf or \\f@ncyelf. This is the left component.\n% 3. \\f@ncyoch, \\f@ncyech, \\f@ncyocf or \\f@ncyecf. This is the middle comp.\n% 4. \\f@ncyorh, \\f@ncyerh, \\f@ncyorf or \\f@ncyerf. This is the right component.\n% 5. RIGHTSIDE BEARING. This is always \\relax or \\hss (after expansion).\n\n\\def\\@fancyhead#1#2#3#4#5{#1\\hbox to\\headwidth{\\fancy@reset\n  \\@fancyvbox\\headheight{\\hbox\n    {\\rlap{\\parbox[b]{\\headwidth}{\\raggedright#2}}\\hfill\n      \\parbox[b]{\\headwidth}{\\centering#3}\\hfill\n      \\llap{\\parbox[b]{\\headwidth}{\\raggedleft#4}}}\\headrule}}#5}\n\n\\def\\@fancyfoot#1#2#3#4#5{#1\\hbox to\\headwidth{\\fancy@reset\n    \\@fancyvbox\\footskip{\\footrule\n      \\hbox{\\rlap{\\parbox[t]{\\headwidth}{\\raggedright#2}}\\hfill\n        \\parbox[t]{\\headwidth}{\\centering#3}\\hfill\n        \\llap{\\parbox[t]{\\headwidth}{\\raggedleft#4}}}}}#5}\n\n\\def\\headrule{{\\if@fancyplain\\let\\headrulewidth\\plainheadrulewidth\\fi\n    \\hrule\\@height\\headrulewidth\\@width\\headwidth \\vskip-\\headrulewidth}}\n\n\\def\\footrule{{\\if@fancyplain\\let\\footrulewidth\\plainfootrulewidth\\fi\n    \\vskip-\\footruleskip\\vskip-\\footrulewidth\n    \\hrule\\@width\\headwidth\\@height\\footrulewidth\\vskip\\footruleskip}}\n\n\\def\\ps@fancy{%\n\\@ifundefined{@chapapp}{\\let\\@chapapp\\chaptername}{}%for amsbook\n%\n% Define \\MakeUppercase for old LaTeXen.\n% Note: we used \\def rather than \\let, so that \\let\\uppercase\\relax (from\n% the version 1 documentation) will still work.\n%\n\\@ifundefined{MakeUppercase}{\\def\\MakeUppercase{\\uppercase}}{}%\n\\@ifundefined{chapter}{\\def\\sectionmark##1{\\markboth\n{\\MakeUppercase{\\ifnum \\c@secnumdepth>\\z@\n \\thesection\\hskip 1em\\relax \\fi ##1}}{}}%\n\\def\\subsectionmark##1{\\markright {\\ifnum \\c@secnumdepth >\\@ne\n \\thesubsection\\hskip 1em\\relax \\fi ##1}}}%\n{\\def\\chaptermark##1{\\markboth {\\MakeUppercase{\\ifnum \\c@secnumdepth>\\m@ne\n \\@chapapp\\ \\thechapter. \\ \\fi ##1}}{}}%\n\\def\\sectionmark##1{\\markright{\\MakeUppercase{\\ifnum \\c@secnumdepth >\\z@\n \\thesection. \\ \\fi ##1}}}}%\n%\\csname ps@headings\\endcsname % use \\ps@headings defaults if they exist\n\\ps@@fancy\n\\gdef\\ps@fancy{\\@fancyplainfalse\\ps@@fancy}%\n% Initialize \\headwidth if the user didn't\n%\n\\ifdim\\headwidth<0sp\n%\n% This catches the case that \\headwidth hasn't been initialized and the\n% case that the user added something to \\headwidth in the expectation that\n% it was initialized to \\textwidth. We compensate this now. This loses if\n% the user intended to multiply it by a factor. But that case is more\n% likely done by saying something like \\headwidth=1.2\\textwidth. \n% The doc says you have to change \\headwidth after the first call to\n% \\pagestyle{fancy}. This code is just to catch the most common cases were\n% that requirement is violated.\n%\n    \\global\\advance\\headwidth123456789sp\\global\\advance\\headwidth\\textwidth\n\\fi}\n\\def\\ps@fancyplain{\\ps@fancy \\let\\ps@plain\\ps@plain@fancy}\n\\def\\ps@plain@fancy{\\@fancyplaintrue\\ps@@fancy}\n\\let\\ps@@empty\\ps@empty\n\\def\\ps@@fancy{%\n\\ps@@empty % This is for amsbook/amsart, which do strange things with \\topskip\n\\def\\@mkboth{\\protect\\markboth}%\n\\def\\@oddhead{\\@fancyhead\\fancy@Oolh\\f@ncyolh\\f@ncyoch\\f@ncyorh\\fancy@Oorh}%\n\\def\\@oddfoot{\\@fancyfoot\\fancy@Oolf\\f@ncyolf\\f@ncyocf\\f@ncyorf\\fancy@Oorf}%\n\\def\\@evenhead{\\@fancyhead\\fancy@Oelh\\f@ncyelh\\f@ncyech\\f@ncyerh\\fancy@Oerh}%\n\\def\\@evenfoot{\\@fancyfoot\\fancy@Oelf\\f@ncyelf\\f@ncyecf\\f@ncyerf\\fancy@Oerf}%\n}\n% Default definitions for compatibility mode:\n% These cause the header/footer to take the defined \\headwidth as width\n% And to shift in the direction of the marginpar area\n\n\\def\\fancy@Oolh{\\if@reversemargin\\hss\\else\\relax\\fi}\n\\def\\fancy@Oorh{\\if@reversemargin\\relax\\else\\hss\\fi}\n\\let\\fancy@Oelh\\fancy@Oorh\n\\let\\fancy@Oerh\\fancy@Oolh\n\n\\let\\fancy@Oolf\\fancy@Oolh\n\\let\\fancy@Oorf\\fancy@Oorh\n\\let\\fancy@Oelf\\fancy@Oelh\n\\let\\fancy@Oerf\\fancy@Oerh\n\n% New definitions for the use of \\fancyhfoffset\n% These calculate the \\headwidth from \\textwidth and the specified offsets.\n\n\\def\\fancy@offsolh{\\headwidth=\\textwidth\\advance\\headwidth\\f@ncyO@olh\n                   \\advance\\headwidth\\f@ncyO@orh\\hskip-\\f@ncyO@olh}\n\\def\\fancy@offselh{\\headwidth=\\textwidth\\advance\\headwidth\\f@ncyO@elh\n                   \\advance\\headwidth\\f@ncyO@erh\\hskip-\\f@ncyO@elh}\n\n\\def\\fancy@offsolf{\\headwidth=\\textwidth\\advance\\headwidth\\f@ncyO@olf\n                   \\advance\\headwidth\\f@ncyO@orf\\hskip-\\f@ncyO@olf}\n\\def\\fancy@offself{\\headwidth=\\textwidth\\advance\\headwidth\\f@ncyO@elf\n                   \\advance\\headwidth\\f@ncyO@erf\\hskip-\\f@ncyO@elf}\n\n\\def\\fancy@setoffs{%\n% Just in case \\let\\headwidth\\textwidth was used\n  \\fancy@gbl\\let\\headwidth\\fancy@headwidth\n  \\fancy@gbl\\let\\fancy@Oolh\\fancy@offsolh\n  \\fancy@gbl\\let\\fancy@Oelh\\fancy@offselh\n  \\fancy@gbl\\let\\fancy@Oorh\\hss\n  \\fancy@gbl\\let\\fancy@Oerh\\hss\n  \\fancy@gbl\\let\\fancy@Oolf\\fancy@offsolf\n  \\fancy@gbl\\let\\fancy@Oelf\\fancy@offself\n  \\fancy@gbl\\let\\fancy@Oorf\\hss\n  \\fancy@gbl\\let\\fancy@Oerf\\hss}\n\n\\newif\\iffootnote\n\\let\\latex@makecol\\@makecol\n\\def\\@makecol{\\ifvoid\\footins\\footnotetrue\\else\\footnotefalse\\fi\n\\let\\topfloat\\@toplist\\let\\botfloat\\@botlist\\latex@makecol}\n\\def\\iftopfloat#1#2{\\ifx\\topfloat\\empty #2\\else #1\\fi}\n\\def\\ifbotfloat#1#2{\\ifx\\botfloat\\empty #2\\else #1\\fi}\n\\def\\iffloatpage#1#2{\\if@fcolmade #1\\else #2\\fi}\n\n\\newcommand{\\fancypagestyle}[2]{%\n  \\@namedef{ps@#1}{\\let\\fancy@gbl\\relax#2\\relax\\ps@fancy}}\n"
  },
  {
    "path": "paper/generalized_rebar.tex",
    "content": "\\documentclass{article}\n\\usepackage{iclr2018_conference, times}\n\\usepackage{hyperref}       % hyperlinks\n\\usepackage{url}            % simple URL typesetting\n\\usepackage{booktabs}       % professional-quality tables\n\\usepackage{amsfonts}       % blackboard math symbols\n\\usepackage{nicefrac}       % compact symbols for 1/2, etc.\n\\usepackage{microtype}      % microtypography\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{hyperref}\n\\usepackage{graphicx}\n\\usepackage{algorithm}\n\\usepackage{algpseudocode}\n\\usepackage{algorithm}\n\\usepackage{subcaption}\n\\usepackage{algpseudocode}\n\\usepackage{wrapfig} % for side-by\n\\usepackage{bm}\n% rename algorithmic labels\n\\renewcommand{\\algorithmicensure}{\\textbf{return}}\n\\renewcommand{\\algorithmicfunction}{\\textbf{def}}\n\n\\definecolor{mydarkblue}{rgb}{0,0.08,0.45}\n\\hypersetup{colorlinks=true,\n    linkcolor=mydarkblue,\n    citecolor=mydarkblue,\n    filecolor=mydarkblue,\n    urlcolor=mydarkblue}\n\n\\newcommand{\\vtheta}{\\mathbf{\\theta}}\n\\newcommand{\\relaxed}{r}\n\\newcommand{\\controlf}{c}  % Control variate for functions\n\\newcommand{\\controlg}{\\hat g}  % Control variate for gradients\n\\newcommand{\\discreteDist}{p(b|\\theta)}\n\\newcommand{\\loss}{f(b)}\n\\newcommand{\\lossGrad}{\\loss{} \\frac{\\partial}{\\partial \\theta }\\log \\discreteDist{}}\n\\newcommand{\\mcGrad}{\\hat{g}}\n\\newcommand{\\expectedLoss}{\\mathbb{E}_{\\discreteDist{}} \\! \\left[ \\, \\loss{} \\right]}\n\\newcommand{\\expectedLossLogTrick}{\\mathbb{E}_{\\discreteDist{}} \\! \\left[ \\, \\lossGrad{} \\right]}\n\\newcommand{\\var}{\\mathbb{V}}\n\\newcommand{\\vu}{\\mathbf{u}}\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\LL}[1]{\\frac{\\partial \\log \\pi(a_{#1}| s_{#1}, \\theta)}{\\partial \\theta}}\n\\newcommand{\\PT}{\\frac{\\partial}{\\partial \\theta}}\n\\newcommand{\\PP}[1]{\\frac{\\partial}{\\partial #1}}\n\\newcommand{\\PPH}{\\frac{\\partial}{\\partial \\phi}}\n\\newcommand{\\LP}[1]{\\PT \\log p(#1)}\n\\newcommand{\\LZ}[1]{\\frac{\\log \\pi(z_{#1}| s_{#1}, \\theta)}{\\partial \\theta}}\n\n\\newcommand{\\YW}[1]{{\\color{red} \\bf [[YW: #1]]}}\n\\newcommand{\\LAX}{{\\textnormal{LAX}}}\n\\newcommand{\\DLAX}{{\\textnormal{DLAX}}}\n\\newcommand{\\RL}{{\\textnormal{RL}}}\n\\newcommand{\\RELAX}{{\\textnormal{RELAX}}}\n\\newcommand{\\BAR}{{\\textnormal{BAR}}}\n\\newcommand{\\REBAR}{{\\textnormal{REBAR}}}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{exercise}[theorem]{Exercise}\n\\newtheorem{problem}[theorem]{Problem}\n\\newtheorem{question}[theorem]{Question}\n\\newtheorem{observation}[theorem]{Observation}\n\n\n\\title{Backpropagation through the Void:\\\\\nOptimizing control variates for \\\\ black-box gradient estimation}\n% Optimizing gradient control variates\\\\ for black-box expectations}\n\n\\author{Will Grathwohl, Dami Choi, Yuhuai Wu, Geoffrey Roeder, David Duvenaud \\\\\nUniversity of Toronto and Vector Institute\\\\\n\\texttt{\\{wgrathwohl, choidami, ywu, roeder, duvenaud\\}@cs.toronto.edu}\n}\n\n\n\\iclrfinalcopy \n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nGradient-based optimization is the foundation of deep learning and reinforcement learning, but is difficult to apply when the mechanism being optimized is unknown or not differentiable.\n%Even when the mechanism being optimized is unknown or not differentiable, optimization using high-variance or biased gradient estimates is still often the best strategy.\nWe introduce a general framework for learning low-variance, unbiased gradient estimators, applicable to black-box functions of discrete or continuous random variables.\nOur method uses gradients of a surrogate neural network to construct a control variate, which is optimized jointly with the original parameters.\n%Our method applies to both continuous and discrete variables.\nWe demonstrate this framework for training discrete latent-variable models.\nWe also give an unbiased, action-conditional extension of the advantage actor-critic reinforcement learning algorithm.\n\\end{abstract}\n\n\n\\section{Introduction}\nGradient-based optimization has been key to most recent advances in machine learning and reinforcement learning.\nThe back-propagation algorithm \\citep{rumelhart1986learning}, also known as reverse-mode automatic differentiation~\\citep{speelpenning1980compiling, rall1981automatic} computes exact gradients of deterministic, differentiable objective functions.\nThe reparameterization trick \\citep{williams1992simple, kingma2013autoencoding, rezende2014stochastic} allows backpropagation to give unbiased, low-variance estimates of gradients of expectations of continuous random variables.\nThis has allowed effective stochastic optimization of large probabilistic latent-variable models.\n\n%Unfortunately, backpropagation cannot be easily applied to problems involving discrete random variables, or when the function being optimized is a black box~\\citep{schulman2015gradient}.\nUnfortunately, there are many objective functions relevant to the machine learning community for which backpropagation cannot be applied. In reinforcement learning, for example, the function being optimized is unknown to the agent and is treated as a black box~\\citep{schulman2015gradient}. Similarly, when fitting probabilistic models with discrete latent variables, discrete sampling operations create discontinuities giving the objective function zero gradient with respect to its parameters.\n%This is the case in most reinforcement learning settings, or when fitting probabilistic models with discrete latent variables.\nMuch recent work has been devoted to constructing gradient estimators for these situations.\nIn reinforcement learning, advantage actor-critic methods~\\citep{sutton2000policy} give unbiased gradient estimates with reduced variance obtained by jointly optimizing the policy parameters with an estimate of the value function.\nIn discrete latent-variable models, low-variance but biased gradient estimates can be given by continuous relaxations of discrete variables~\\citep{maddison2016concrete, jang2016categorical}.\n\nA recent advance by \\citet{tucker2017rebar} used a continuous relaxation of discrete random variables to build an unbiased and lower-variance gradient estimator,\n%Low-variance estimates of the expectation of the control variate can be computed using the reparameterization trick to produce an unbiased estimator with lower variance than previous methods.\n%Furthermore, \\citet{tucker2017rebar} \nand showed how to tune the free parameters of these relaxations to minimize the estimator's variance during training.\n%\n%\\paragraph{Contributions}\nWe generalize the method of \\citet{tucker2017rebar} to learn a free-form control variate parameterized by a neural network.\nThis gives a lower-variance, unbiased gradient estimator which can be applied to a wider variety of problems.\nMost notably, our method is applicable even when no continuous relaxation is available, as in reinforcement learning or black-box function optimization.\n\n%\\paragraph{Contributions}\n%\\begin{itemize}\n%\\item we keep conditional reparam from rebar while exploring and expanding on gradient signal for minimizing variance of estimator\n%\\item we generalize the concrete and muprop relaxations for a learnable family of functions that can be optimized via gradient decent to minimize the variance of the estimator\n%\\item We show that concrete relax is not optimal, and that control variate based on the discrete objective at continuous input is is not the optimal control variate.\n%\\item we combine these insights into a new family of estimators that generalizes rebar\n%\\end{itemize}\n\n\\begin{figure}[h]\n\\hspace{-1em}\n\\centering\n\\begin{tabular}{c c}\n\\includegraphics[width=.48\\textwidth]{figures/toy_losses_10000_0_499}\n&\n\\includegraphics[width=.48\\textwidth]{figures/variance_100_t_499}\n\\end{tabular}\n\\vspace{-1em}\n%\\includegraphics[width=.325\\textwidth]{figures/relaxations_t_499_which_2}\n\\caption{\n\\emph{Left:} Training curves comparing different gradient estimators on a toy problem: ${\\mathcal{L}(\\theta) = \\E_{p(b|\\theta)} [ (b - 0.499)^2 ]}$\n\\emph{Right:} Log-variance of each estimator's gradient.\n}\n\\label{first figure}\n\\end{figure}\n\n\n\\section{Background: Gradient estimators}\nHow can we choose the parameters of a distribution to maximize an expectation?\nThis problem comes up in reinforcement learning, where we must choose the parameters $\\theta$ of a policy distribution $\\pi(a|s, \\theta)$ to maximize the expected reward $\\mathbb{E}_{\\tau \\sim \\pi} \\left[ R \\right]$ over state-action trajectories $\\tau$.\nIt also comes up in fitting latent-variable models, when we wish to maximize the marginal probability ${p(x|\\theta) = \\sum_z p(x|z) p(z|\\theta) = \\mathbb{E}_{p(z|\\theta)} \\left[ p(x|z) \\right]}$.\nIn this paper, we'll consider the general problem of optimizing\n%\n\\begin{align}\n\\mathcal{L}(\\theta) = \\expectedLoss{}.\n\\end{align}\n\nWhen the parameters $\\theta$ are high-dimensional, gradient-based optimization is appealing because it provides information about how to adjust each parameter individually.\nStochastic optimization is essential for scalablility, but is only guaranteed to converge to a fixed point of the objective when the stochastic gradients $\\hat g$ are unbiased, i.e. ${\\mathbb{E} \\left[ \\hat g \\right] = \\PT \\mathcal{L}(\\theta)}$~\\citep{robbins1951stochastic}.\n\nHow can we build unbiased, stochastic gradient estimators?\nThere are several standard methods:\n\n\\paragraph{The score-function gradient estimator}\nOne of the most generally-applicable gradient estimators is known as the score-function estimator, or REINFORCE~\\citep{williams1992simple}:\n%\n\\begin{align}\n\\hat g_\\textnormal{REINFORCE}[f] =  f \\left( b \\right) \\PT \\log p(b | \\theta), \\qquad b \\sim p(b | \\theta)\n\\end{align}\n%\nThis estimator is unbiased, but in general has high variance.\nIntuitively, this estimator is limited by the fact that it doesn't use any information about how $f$ depends on $b$, only on the final outcome $f(b)$.\n\n\\paragraph{The reparameterization trick}\nWhen $f$ is continuous and differentiable, and the latent variables $b$ can be written as a deterministic, differentiable function of a random draw from a fixed distribution, the reparameterization trick \\citep{williams1992simple, kingma2013autoencoding, rezende2014stochastic} creates a low-variance, unbiased gradient estimator by making the dependence of $b$ on $\\theta$ explicit through a reparameterization function $b=T(\\theta, \\epsilon)$:\n%\n\\begin{align}\n%\\hat g_\\textnormal{reparam} = \\frac{\\partial f \\left( b(\\theta, \\epsilon) \\right)}{\\partial \\theta}, \\qquad \\epsilon \\sim p(\\epsilon)\n\\hat g_\\textnormal{reparam}[f]\n= \\PT f \\left( b \\right)\n= \\frac{\\partial f}{\\partial T}\\frac{\\partial T}{\\partial \\theta} , \n\\qquad \\epsilon \\sim p(\\epsilon) \n\\end{align}\n%\nThis gradient estimator is often used when training high-dimensional, continuous latent-variable models, such as variational autoencoders.\n% or GANs \\citep{goodfellow2014generative}.\nOne intuition for why this gradient estimator is preferable to REINFORCE is that it depends on ${\\partial f} / {\\partial b}$, which exposes the dependence of $f$ on $b$.\n% NOTE for revision: last sentence is confusing. What is the advantage of exposing the dependence of f on b for a gradient estimator?\n\n%The reparameterization trick, while useful, is only applicable when $f$ is a known, differentiable function of a continuous random variable.\n%\\paragraph{Concrete relaxation}\n%For functions of discrete variables, gradient estimators based on continuous relaxations such as the concrete, or Gumbel-softmax relaxations~\\citep{maddison2016concrete, jang2016categorical} are applicable only when $f$ is known.\n%Moreover, they require that $f$ be computable at and behave predictably at inputs outside of the domain of the function.\n%These assumptions greatly restrict the space of functions for which we can apply these methods. \n\n%\\paragraph{Concrete relaxation}\n%When $b$ is discrete and $f$ is known and differentiable, one can obtain biased but low-variance gradients using a continuous relaxation of the discrete random variable $b$.\n%For example, if $b$ is a categorical random variable with parameters $\\theta$,% and  developed a differentiable relaxation of the categorical distribution, called the concrete distribution:\n%\n%\\begin{align}\n%\\hat g_\\textnormal{concrete} = \\PT f \\left( \\texttt{softmax}_\\lambda ( \\log \\theta - \\log(-\\log \\vu)) \\right), \\qquad \\vu \\sim \\textnormal{uniform}[0, 1] \n%\\end{align}\n%\n%where $\\sigma_\\lambda$ is the \\texttt{softmax} function with temperature $\\lambda$.\n%This is called the Gumbel-softmax trick~\\citep{jang2016categorical}, or the concrete distribution~\\citep{maddison2016concrete}\n\n\n\\paragraph{Control variates}\nControl variates are a general method for reducing the variance of a stochastic estimator.\nA control variate is a function $\\controlf(b)$ with a known mean $\\mathbb{E}_{p(b)} [ \\controlf(b) ]$.\nGiven an estimator $\\hat g(b)$, subtracting the control variate from this estimator and adding its mean gives us a new estimator:\n%\n\\begin{align}\n\\hat g_\\textnormal{new}(b) = \\hat g(b) - \\controlf(b) + \\mathbb{E}_{p(b)}[\\controlf(b)]\n\\end{align}\n%\nThis new estimator has the same expectation as the old one, \n%\n%\\begin{align}\n%\\mathbb{E}_{p(b)}\\left[\\hat g_\\textnormal{new}(b) \\right] \n%= \\mathbb{E}_{p(b)}\\left[\\hat g(b) - \\controlf(b) + \\mathbb{E}_{p(b)} \\left[ \\controlf(b) \\right] \\right]\n%= \\mathbb{E}_{p(b)}\\left[ \\hat g(b) \\right]\n%\\end{align}\n%\nbut has lower variance if $\\controlf(b)$ is positively correlated with $\\hat g(b)$.\n% NOTE for revision: \n% I think it's important to stick to the standard definition of a control variate here, which includes the scalar constant eta\n% c(b) can be positively or negatively correlated with f(b) and still reduce variance if the control variate has a learned scalar like eta\n% Eta can be positive or negative. When you learn eta optimally (Cov(g,c) / Var(c), see for example Tucker et al. 2017 sec. 7.1), eta will flip sign so that even if c(b) is negatively correlated, the new estimator will have lower variance .\n\n\\section{Constructing and optimizing a differentiable surrogate}\n\\label{lax section}\nIn this section, we introduce a gradient estimator for the expectation of a function $\\PT \\E_{p(b|\\theta)}[f(b)]$ that can be applied even when $f$ is unknown, or not differentiable, or when $b$ is discrete.\nOur estimator combines the score function estimator, the reparameterization trick, and control variates.\n%In the continuous case, we obtain an unbiased estimator whose variance can potentially be as low as the reparameterization-trick estimator.%, even when $f$ is not differentiable or not computable.\n\nFirst, we consider the case where $b$ is continuous, but that $f$ cannot be differentiated.\nInstead of differentiating through $f$, we build a surrogate of $f$ using a neural network $c_\\phi$, and differentiate $c_\\phi$ instead.\nSince the score-function estimator and reparameterization estimator have the same expectation,\nwe can simply subtract the score-function estimator for $c_\\phi$ and add back its reparameterization estimator.\nThis gives a gradient estimator which we call LAX:\n%\\begin{center}\n%\\begin{tabular}{c|c|c|c}\n%a & b & c & d\n%\\end{tabular}\n%\\end{center}\n%\n\\begin{align}\n\\label{eq:cont_est}\n\\hat g_\\LAX &= \n\\hat{g}_\\textnormal{REINFORCE}[f] - \\hat{g}_\\textnormal{REINFORCE}[c_\\phi] + \\hat{g}_\\textnormal{reparam}[c_\\phi] \\nonumber\\\\\n&= \\left[ f(b) -c_\\phi(b) \\right] \\PT \\log p(b|\\theta) + \\PT c_\\phi(b) \\qquad b = T(\\theta, \\epsilon), \\epsilon \\sim p(\\epsilon).\n\\end{align}\n%\nThis estimator is unbiased for any choice of $c_\\phi$.\nWhen $c_\\phi = f$, then \\LAX{} becomes the reparameterization estimator for $f$.\nThus \\LAX{} can have variance at least as low as the reparameterization estimator. An example of the relative bias and variance of each term in this estimator can be seen below.%in Figure \\ref{fig:grad hist}.\n\\begin{figure}[h!]\n\\includegraphics[width=\\columnwidth]{figures/grad_hist.pdf}\n\\caption{Histograms of samples from the gradient estimators that create LAX. Samples generated from our one-layer VAE experiments (Section \\ref{vae section}).}\n\\label{fig:grad hist}\n\\end{figure}\n\n%\\subsection{Optimizing the gradient control variate with gradients}\n\\subsection{Gradient-based optimization of the control variate}\nSince $\\hat g_\\LAX$ is unbiased for any choice of the surrogate $c_\\phi$, the only remaining problem is to choose a $c_\\phi$ that gives low variance to $\\hat g_\\LAX$.\nHow can we find a $\\phi$ which gives our estimator low variance?\nWe simply optimize $c_\\phi$ using stochastic gradient descent, at the same time as we optimize the parameters $\\theta$ of our model or policy.\n\nTo optimize $c_\\phi$, we require the gradient of the variance of our estimator.\nTo estimate these gradients, we could simply differentiate through the empirical variance over each mini-batch.\nOr, following \\cite{ruiz2016overdispersed} and \\cite{tucker2017rebar}, we can construct an unbiased, single-sample estimator using the fact that our gradient estimator is unbiased.\nFor any unbiased gradient estimator $\\hat g$ with parameters $\\phi$:\n%\n\\begin{align}\n\\PPH \\text{Variance}(\\hat g)\n= \\PPH \\E[\\hat g^2] - \\PPH \\E[\\hat g]^2\n%= \\PPH \\E[\\hat g^2] - \\PPH \\E_{p(b|\\theta)}[f(b)]\n= \\PPH \\E[\\hat g^2]\n= \\E \\left[ \\PPH \\hat g^2 \\right].\n%= \\E \\left[ 2 \\hat g \\frac{\\partial \\hat g}{\\partial \\phi} \\right].\n\\label{eq:vargrad}\n\\end{align}  % Do we need hats on these gs?  Or get rid of them elsewhere.\n%\nThus, an unbiased single-sample estimate of the gradient of the variance of $\\hat g$ is given by $\\partial \\hat g^2 / \\partial \\phi$.\n%{$2 \\hat g \\frac{\\partial \\hat g}{\\partial \\phi}$}.\n\nThis method of directly minimizing the variance of the gradient estimator stands in contrast to other methods such as Q-Prop~\\citep{gu2016q} and advantage actor-critic~\\citep{sutton2000policy}, which train the control variate to minimize the squared error $(f(b) - c_\\phi(b))^2$.\nOur algorithm, which jointly optimizes the parameters $\\theta$ and the surrogate $c_\\phi$ is given in Algorithm~\\ref{lax}.\n%[Todo: talk about the variance of the gradient estimator of the variance of the gradient estimators?]\n\n\\subsubsection{Optimal surrogate}\nWhat is the form of the variance-minimizing $c_\\phi$?\nInspecting the square of \\eqref{eq:cont_est}, we can see that this loss encourages $c_\\phi(b)$ to approximate $f(b)$, but with a weighting based on $\\PT\\log p(b|\\theta)$.  % Todo: reword this awkward sentence.\nMoreover, as $c_\\phi \\rightarrow f$ then $\\hat g_\\textnormal{\\LAX} \\rightarrow \\PT c_\\phi$.\nThus, this objective encourages a balance between the variance of the reparameterization estimator and the variance of the REINFORCE estimator.\nFigure~\\ref{learned-relaxations} shows the learned surrogate on a toy problem.\n\n\n\\begin{algorithm}[h]\n\\begin{algorithmic}\n\\Require $f(\\cdot)$, $\\log p(b|\\theta)$, reparameterized sampler $b = T(\\theta, \\epsilon)$, neural network $c_\\phi(\\cdot)$, \\\\ \\qquad \\quad~step sizes $\\alpha_1, \\alpha_2$ \n\\While {not converged} \n\t\\State $\\epsilon \\sim p(\\epsilon)$ \\Comment Sample noise\n\t\\State $b \\leftarrow T(\\epsilon, \\theta)$ \\Comment Compute input\n\t\\State  $\\hat g_\\theta \\leftarrow \\left[f(b) - c_{\\phi}(b) \\right] \\nabla_\\theta \\log p(b|\\theta) + \\nabla_\\theta c_\\phi(b)$ \\Comment Estimate gradient of objective\n\t\\State  $\\hat g_\\phi \\leftarrow \\partial \\hat g_\\theta^2 / \\partial \\phi$ \\Comment Estimate gradient of variance of gradient\n\t\\State $\\theta \\leftarrow \\theta - \\alpha_1 \\hat{g}_\\theta$ \\Comment Update parameters\n\t\\State $\\phi \\leftarrow \\phi - \\alpha_2 \\hat{g}_\\phi$ \\Comment Update control variate\n\\EndWhile\n\\State \\textbf{return} $\\theta$ \n\\end{algorithmic}\n\\caption{\\LAX{}: Optimizing parameters and a gradient control variate simultaneously.}\n\\label{lax}\n\\end{algorithm}\n\n\\subsection{Discrete random variables and conditional reparameterization}\nWe can adapt the \\LAX{} estimator to the case where $b$ is a discrete random variable by introducing a ``relaxed'' continuous variable $z$.\nWe require a continuous, reparameterizable distribution $p(z|\\theta)$ and a deterministic mapping $H(z)$ such that $H(z) = b \\sim p(b|\\theta)$ when $z \\sim p(z|\\theta)$.\nIn our implementation, we use the Gumbel-softmax trick, the details of which can be found in appendix~\\ref{resample}.\n\nThe discrete version of the \\LAX{} estimator is given by:\n%\n\\begin{align}\n\\label{eq:discrete lax}\n\\hat g_\\DLAX = f(b) \\PT \\log p(b|\\theta) - c_\\phi(z) \\PT \\log p(z|\\theta) + \\PT c_\\phi(z), \\qquad b = H(z), z \\sim p(z|\\theta).\n\\end{align}\n%\nThis estimator is simple to implement and general.\nHowever, if we were able to replace the $\\PT \\log p(z|\\theta)$ in the control variate with $\\PT \\log p(b|\\theta)$ we should be able to achieve a more correlated control variate, and therefore a lower variance estimator. This is the motivation behind our next estimator, which we call RELAX.\n\n%when $f = c_\\phi$ we do not recover the reparameterization estimator as we do with LAX. To achieve this, we must be able to replace the $\\PT \\log p(z|\\theta)$ in the control variate with $\\PT \\log p(b|\\theta)$. This is the motivation behind our next estimator, which we call RELAX.\n\nTo construct a more powerful gradient estimator, we incorporate a further refinement due to~\\cite{tucker2017rebar}.\nSpecifically, we evaluate our control variate both at a relaxed input $z \\sim p(z|\\theta)$, and also at a relaxed input \\emph{conditioned on the discrete variable $b$}, denoted $\\tilde z \\sim p(z|b, \\theta)$. \nDoing so gives us:\n%\n\\begin{align}\n\\hat g_\\textnormal{RELAX} = \\left[ f(b) - c_\\phi(\\tilde{z}) \\right] \\PT \\log p(b|\\theta) + \\PT c_\\phi(z) - \\PT c_\\phi (\\tilde{z}) \\\\\n\\qquad b = H(z), z \\sim p(z|\\theta), \\tilde{z} \\sim p(z|b, \\theta) \\nonumber\n\\end{align}\n%\nThis estimator is unbiased for any $c_\\phi$.\nA proof and a detailed algorithm can be found in appendix~\\ref{relax proof}.\n%\nWe note that the distribution $p(z|b,\\theta)$ must also be reparameterizable.\nWe demonstrate how to perform this conditional reparameterization for Bernoulli and categorical random variables in appendix~\\ref{resample}.\n\n\\subsection{Choosing the control variate architecture}\nThe variance-reduction objective introduced above allows us to use any differentiable, parametric function as our control variate $c_\\phi$. \nHow should we choose the architecture of $c_\\phi$?\nIdeally, we will take advantage of any known structure in $f$.\n\nIn the discrete setting, if $f$ is known and happens to be differentiable, we can use the concrete relaxation~\\citep{jang2016categorical, maddison2016concrete} and let $c_\\phi(z) = f(\\sigma_\\lambda(z))$.\nIn this special case, our estimator is exactly the REBAR estimator.\nWe are also free to add a learned component to the concrete relaxation and let $c_\\phi(z) = f(\\sigma_\\lambda(z)) + {r}_\\rho(z)$ where ${r}_\\rho$ is a neural network with parameters~$\\rho$ making $\\phi = \\{\\rho, \\lambda\\}$.\nWe took this approach in our experiments training discrete variational auto-encoders.\nIf $f$ is unknown, we can simply let $c_\\phi$ be a generic function approximator such as a neural network.\nWe took this simpler approach in our reinforcement learning experiments.\n\n\n\\subsection{Reinforcement learning}\nWe now describe how we apply the \\LAX{} estimator in the reinforcement learning (RL) setting.\nBy reinforcement learning, we refer to the problem of optimizing the parameters $\\theta$ of a policy distribution $\\pi(a | s, \\theta)$ to maximize the sum of rewards.\nIn this setting, the random variable being integrated over is $\\tau$, which denotes a series of $T$ actions and states $[(s_1, a_1), (s_2, a_2), ..., (s_T, a_T)]$.\nThe function whose expectation is being optimized, $R$, maps $\\tau$ to the sum of rewards ${R(\\tau) = \\sum_{t=1}^{T} r_t(s_t, a_t)}$.\n\nAgain, we want to estimate the gradient of an expectation of a black-box function: $ \\PT \\E_{p(\\tau|\\theta)}[R(\\tau)]$.\nThe \\emph{de facto} standard approach is the advantage actor-critic estimator (A2C)~\\citep{sutton2000policy}:\n%We can view the sum of future rewards as a black-box function of state-action trajectories $\\tau$ sampled from our policy.\n%Instead, we typically compute\n%\n%\\begin{align}\n%\\frac{\\partial \\E_\\tau[R]}{\\partial \\theta} = \\E_\\tau \\left[ \\sum_{t=1}^{\\infty} \\LL{t} \\left[ \\sum_{t'=t}^{\\infty} r_{t'} - b(s_t) \\right] \\right]\n%\\end{align}\n%\n%\n\\begin{align}\n\\hat g_{\\textnormal{A2C}} = \\sum_{t=1}^{T} \\LL{t} \\left[ \\sum_{t'=t}^{T} r_{t'} - c_\\phi(s_t) \\right], \\qquad a_t \\sim \\pi(a_t | s_t, \\theta)\n\\label{eq:rl_a2c}\n\\end{align}\n%\nWhere $c_\\phi(s_t)$ is an estimate of the state-value function, $c_\\phi(s) \\approx V^\\pi(s) = \\E_{\\tau}[R|s_1=s].$\nThis estimator is unbiased when $c_\\phi$ does not depend on $a_t$.\nThe main limitations of A2C are that $c_\\phi$ does not depend on $a_t$, and that it's not obvious how to optimize $c_\\phi$.\nUsing the \\LAX{} estimator addresses both of these problems.\n\nFirst, we assume $\\pi(a_t|s_t, \\theta)$ is reparameterizable, meaning that we can write $a_t = a(\\epsilon_t, s_t, \\theta)$, where $\\epsilon_t$ does not depend on $\\theta$.\nWe again introduce a differentiable surrogate $c_\\phi(a,s)$.\nCrucially, this surrogate is a function of the action as well as the state.\n\nThe extension of LAX to Markov decision processes is: \n%\n\\begin{align}\n\\hat g_\\LAX^{\\RL} = \\sum_{t=1}^{T} \\LL{t} \\left[ \\sum_{t'=t}^{T} r_{t'} - c_\\phi(a_t,s_t) \\right] +\\frac{\\partial}{\\partial\\theta} c_\\phi(a_t, s_t), \\\\\na_t = a(\\epsilon_t,s_t, \\theta) \\qquad \\epsilon_t \\sim p(\\epsilon_t)\\nonumber.\n\\label{eq:rl_est}\n\\end{align}\n%\nThis estimator is unbiased if the true dynamics of the system are Markovian w.r.t. the state $s_t$.\nWhen $T = 1$, we recover the special case ${\\hat g_\\LAX^{\\RL} = \\hat g_\\LAX}$.\nComparing $\\hat g_\\LAX^{\\RL}$ to the standard advantage actor-critic estimator in~\\eqref{eq:rl_a2c}, the main difference is that our baseline $c_\\phi(a_t, s_t)$ is action-dependent while still remaining unbiased.\n\nTo optimize the parameters $\\phi$ of our control variate $c_\\phi(a_t, s_t)$, we can again use the single-sample estimator of the gradient of our estimator's variance given in~\\eqref{eq:vargrad}.\nThis approach avoids unstable training dynamics, and doesn't require storage and replay of previous rollouts.\n\nDetails of this derivation, as well as the discrete and conditionally reparameterized version of this estimator can be found in appendix~\\ref{rl appendix}.\n\n\n\\section{Scope and Limitations}\n\\label{limitations}\nThe work most related to ours is the recently-developed REBAR method~\\citep{tucker2017rebar}, which greatly inspired our work.\nThe REBAR estimator is a special case of the \\RELAX{} estimator, when the surrogate is set to ${c_\\phi(z) = \\eta \\cdot f(\\texttt{softmax}_\\lambda(z))}$.\nThe only free parameters of the REBAR estimator are the scaling factor $\\eta$, and the temperature $\\lambda$, which gives limited scope to optimize the surrogate.\nREBAR can only be applied when $f$ is known and differentiable.\nFurthermore, it depends on essentially undefined behavior of the function being optimized, since it evaluates the discrete loss function at continuous inputs.\n\nBecause \\LAX{} and \\RELAX{} can construct a surrogate from scratch, they can be used for optimizing black-box functions, as in reinforcement learning settings where the reward is an unknown function of the environment.\n\\LAX{} and \\RELAX{} only require that we can query the function being optimized, and can sample from and differentiate $p(b|\\theta)$.\n\n%In principle one could use \\RELAX{} to optimize deterministic black-box functions, but only by introducing stochasticity to the inputs.\n%Thus, \\RELAX{} is most suitable for problems where one is already optimizing a distribution over inputs, such as in inference or reinforcement learning.\n% Note that most optimal policies are deterministic?\n\n\\paragraph{Direct dependence on parameters}\nAbove, we assumed that the function $f$ being optimized does not depend directly on $\\theta$, which is usually the case in black-box optimization settings.\nHowever, a dependence on $\\theta$ can occur when training probabilistic models, or when we add a regularizer. % to the parameters. %a black-box optimization problem.\nIn both these settings, if the dependence on $\\theta$ is known and differentiable, we can use the fact that\n%\n\\begin{align}\n\\PT \\E_{p(b|\\theta)}[f(b, \\theta)] = \\E_{p(b|\\theta)}\\left[\\PT f(b, \\theta) + f(b, \\theta)\\PT \\log p(b|\\theta) \\right]\n\\end{align}\n%\nand simply add $\\PT f(b, \\theta)$ to any of the gradient estimators above to recover an unbiased estimator.\n\n\n\\section{Related work}\n%There has been a great deal of other recent work in the area of gradient estimation.\n\\citet{miller2017reducing} reduce the variance of reparameterization gradients in an orthogonal way to ours by approximating the gradient-generating procedure with a simple model and using that model as a control variate.\nNVIL~\\citep{mnih2014neural} and VIMCO~\\citep{mnih2016variational} provide reduced variance gradient estimation in the special case of discrete latent variable models and discrete latent variable models with Monte Carlo objectives.\n\\citet{salimans2017evolution} estimate gradients using a form of finite differences, evaluating hundreds of different parameter values in parallel to construct a gradient estimate.\nIn contrast, our method is a single-sample estimator.\n\n%As gradient estimators become more complex, checking their unbiasedness numerically becomes difficult.\n%The automatic theorem-proving-based unbiasedness checker developed by \\citet{selsam2017developing} may become relevant to this line of research.\n\n%Also: \\citep{levine2016end}\n\n\\citet{staines2012variational} address the general problem of developing gradient estimators for deterministic black-box functions or discrete optimization.\nThey introduce a sampling distribution, and optimize an objective similar to ours.\n\\citet{wierstra2014natural} also introduce a sampling distribution to build a gradient estimator, and consider optimizing the sampling distribution.\nIn the context of general Monte Carlo integration, \\citet{oates2017control} introduce a non-parametric control variate that also leverages gradient information to reduce the variance of an estimator.\n\nIn parallel to our work, there has been a string of recent developments on action-dependent baselines for policy-gradient methods in reinforcement learning.  Such works include \\citet{gu2016q} and \\citet{gu2017interpolated} which train an action-dependent baseline which incorporates off-policy data. \\cite{liu2017sample} independently develop a method similar to LAX applied to continuous control. \\citet{Wu2018Factorized} exploit per-dimension independence of the action distribution in continuous control tasks to produce an action-dependent unbiased baseline. \n\n%In the reinforcement learning setting, the work most similar to ours is $Q$-prop \\citep{gu2016q}.\n%Like our method, $Q$-prop reduces the variance of the policy gradient with a learned, action-dependent control variate whose expectation is approximated via a Monte Carlo sample from a Taylor series expansion of the control variate.\n%Unlike our method, their control variate is trained off-policy.\n%While our method is applicable in both the continuous and discrete action domain, $Q$-prop is only applicable to continuous actions.\n%We are interested in the potential of training our control variate off-policy, but we leave that for further work. \n\n%\\citet{asadi2017mean} reduce the variance of actor-critic gradient estimates by simply summing over all possible actions.\n\n%\\par{Generalized Reparameterization Gradients}\n%REBAR and the generalization in this paper uses a mixture of score function and reparameterization gradients.\n%A recent paper by \\cite{ruiz2016generalized} unifies these two gradient estimators as the generalized reparameterization gradient (GRG).\n%This framework can help disentangle the various components of generalized REBAR.\n\n%REBAR innovation as further decomposition the correction term into secondary reparameterization components\n%note this is a recursive application of the principles of GRG\n%observe that the GRG suggests this recursive application to components of an estimator\n%propose that other estimators could be similarly recursively decomposed?\n\n\n\\section{Applications}\n\\label{Applications}\n%\n\\begin{wrapfigure}[21]{R}{0.50\\textwidth}\n\\centering\n\\vspace{-10mm}\n\\begin{tabular}{c}\nREINFORCE\\\\\n\\hspace{-3mm}\\includegraphics[width=0.50\\columnwidth, clip, trim=0.5cm 8cm 0.8cm 0.5cm]{figures/relaxations_t_499_which_2}\\\\\nREBAR\\\\\n\\hspace{-3mm}\\includegraphics[width=0.50\\columnwidth, clip, trim=0.5cm 5cm 0.8cm 3.9cm]{figures/relaxations_t_499_which_2}\\\\\nRELAX\\\\\n\\hspace{-3mm}\\includegraphics[width=0.50\\columnwidth, clip, trim=0.5cm 0cm 0.8cm 7.2cm]{figures/relaxations_t_499_which_2}\n\\end{tabular}\n\\vspace*{-6mm}\n\\caption{The optimal relaxation for a toy loss function, using different gradient estimators.\nBecause REBAR uses the concrete relaxation of $f$, which happens to be implemented as a quadratic function, the optimal relaxation is constrained to be a warped quadratic.\nIn contrast, RELAX can choose a free-form relaxation.}\n% ${\\mathcal{L}(\\theta) = \\E_{p(b|\\theta)} [ (b - 0.499)^2 ]}$ using REINFORCE (blue), REBAR (green), and RELAX (red)} \n\\label{learned-relaxations}\n\\end{wrapfigure}\n%\nWe demonstrate the effectiveness of our estimator on a number of challenging optimization problems. Following~\\citet{tucker2017rebar} we begin with a simple toy example to illuminate the potential of our method and then continue to the more relevant problems of optimizing binary VAE's and reinforcement learning.\n\n\\subsection{Toy experiment}\nAs a simple example, we follow \\citet{tucker2017rebar} in minimizing $\\mathbb{E}_{p(b|\\theta)}[(b - t)^2]$ as a function of the parameter $\\theta$ where {$p(b|\\theta) = \\textnormal{Bernoulli}(b|\\theta)$}.\n\\citet{tucker2017rebar} set the target $t = .45$.\nWe focus on the more challenging case where $t = .499$.\nFigures~\\ref{first figure}a and \\ref{first figure}b show the relative performance and gradient log-variance of REINFORCE, REBAR, and RELAX.\n\n%With this setting of the target, REBAR and competing methods suffer from high variance and are unable to discover the optimal solution $\\theta = 0$.\n%The fixed concrete relaxation of REBAR is unable to produce a gradient whose signal outweighs the sample noise and is therefore unable to solve this problem noticeably faster than REINFORCE.\n\nFigure~\\ref{learned-relaxations} plots the learned surrogate $c_\\phi$ for a fixed value of $\\theta$. We can see that $c_\\phi$ is near $f$ for all $z$, keeping the variance of the REINFORCE part of the estimator small. Moreover the derivative of $c_\\phi$ is positive for all $z$ meaning that the reparameterization part of the estimator will produce gradients pointing in the correct direction to optimize the expectation. Conversely, the concrete relaxation of REBAR is close to $f$ only near $0$ and $1$ and its gradient points in the correct direction only for values of $z > \\log (\\frac{1-t}{t})$. These factors together result in the RELAX estimator achieving the best performance. \n\n\\subsection{Discrete variational autoencoder}\n\\label{vae section}\nNext, we evaluate the \\RELAX{} estimator on the task of training a variational autoencoder~\\citep{kingma2013autoencoding, rezende2014stochastic} with Bernoulli latent variables.\nWe reproduced the variational autoencoder experiments from \\citet{tucker2017rebar}, training models with one or two layers of 200 Bernoulli random variables with linear or nonlinear mappings between them, on both  the MNIST and Omniglot~\\citep{lake2015human} datasets.\nDetails of these models and our experimental procedure can be found in Appendix~\\ref{app_disc_vae}.\n\nTo take advantage of the available structure in the loss function, we choose the form of our control variate to be $c_\\phi(z) = f(\\sigma_\\lambda(z))+  \\hat{r}_\\rho(z)$ where $\\hat{r}_\\rho$ is a neural network with parameters $\\rho$ and $f(\\sigma_\\lambda(z))$ is the discrete loss function, the evidence lower-bound (ELBO), evaluated at continuously relaxed inputs as in REBAR.  \n%\nIn all experiments, the learned control variate improved the training performance, over the state-of-the-art baseline of REBAR. In both linear models, we achieved improved validation performance as well increased convergence speed. We believe the decrease in validation performance for the nonlinear models was due to overfitting caused by improved optimization of an under-regularized model. We leave exploring this phenomenon to further work. \n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{r l | c c c c c} \nDataset & Model & Concrete & NVIL & MuProp  & REBAR & RELAX\\\\\\midrule \n               & Nonlinear & $-102.2$ & $-101.5$ & -101.1  &  -81.01 &  \\textbf{-78.13} \\\\\n\\textbf{MNIST} & linear one-layer  &-111.3 & $-112.5$ & $-111.7$  & -111.6 & \\textbf{-111.20} \\\\ \n               & linear two-layer  &-99.62 & $-99.6$ & $-99.07$   & -98.22 & \\textbf{-98.00} \\\\\n\\midrule\n               & Nonlinear  & $-110.4$  & $-109.58$ & -108.72  & -56.76 & \\textbf{-56.12} \\\\\n\\textbf{Omniglot} & linear one-layer &-117.23 & $-117.44$ & $-117.09$   & -116.63 & \\textbf{-116.57} \\\\ \n                  & linear two-layer &-109.95 & $-109.98$ & $-109.55$  & -108.71 & \\textbf{-108.54}\n\\end{tabular}\n\\caption{Highest training ELBO for discrete variational autoencoders.}\n\\label{tab:vae tr}\n\\end{table}\n\n\n\n\n%In \\citep{tucker2017rebar}, a separate REBAR estimator was used to estimate the gradients of each model parameter (each weight matrix and bias vector).\n%To apply our estimator to this formulation, we would need to learn a separate relaxation for each model parameter.\n%To get around this, we use our gradient estimator to approximate $g_\\phi \\approx \\PT \\E_{q(b|\\theta)}[f(b)]$ where $x\\cdot W = \\theta$ is the parameters of the Bernoulli latent variables, $W$ is our layer's weight matrix. We then obtain an estimate of $\\PP{W} \\E_{q(b|\\theta)}[f(b)] = g_\\phi\\cdot \\frac{\\partial \\theta}{\\partial W}$. We note this gives us unbiased gradients because \n%\\begin{align}\n%\\E_\\epsilon[g_\\phi(\\epsilon) \\cdot \\frac{\\partial \\theta}{\\partial W}] = \\E_\\epsilon[g_\\phi(\\epsilon)] \\cdot \\frac{\\partial \\theta}{\\partial W} =  \\PT \\E_{q(b|\\theta)}[f(b)] \\cdot \\frac{\\partial \\theta}{\\partial W} = \n%\\frac{\\partial}{\\partial W} \\E_{q(b|\\theta)}[f(b)]\n%\\end{align} \n\n%To provide a fair comparison, we re-implemented REBAR in this way (denoted REBAR-ours in table~\\ref{tab:vae}).\n%We believe this explains the large difference in performance between our implementation and that of \\citep{tucker2017rebar} for the nonlinear models since there are 3 layers of parameters that all share the same gradient estimator.\n%In the linear models, each layer has its own gradient estimator making our implementation closer to that of \\citep{tucker2017rebar}.\nTo obtain training curves we created our own implementation of REBAR, which gave identical or slightly improved performance compared to the implementation of \\citet{tucker2017rebar}.\n\nWhile we obtained a modest improvement in training and validation scores (tables~\\ref{tab:vae tr} and \\ref{tab:vae val}), the most notable improvement provided by \\RELAX{} is in its rate of convergence.\nTraining curves for all models can be seen in Figure~\\ref{fig:vae_curves} and in Appendix~\\ref{extra vae results}.\nIn Table~\\ref{tab:vae epochs} we compare the number of training epochs that are required to match the best validation score of REBAR.\nIn both linear models, RELAX provides an increase in rate of convergence. \n\n\\begin{figure}\n\\centering\n\\hspace*{-.5in}\n\\setlength{\\tabcolsep}{10pt}\n\\renewcommand{\\arraystretch}{0}\n\\begin{tabular}{ccc}\n&MNIST & Omniglot \\\\\n\\rotatebox{90}{\\qquad \\qquad \\qquad \\small -ELBO} & \n\\includegraphics[width=.33\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/MNIST_L1} &\n%\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 3mm]{figures/MNIST_L2} &\n\\includegraphics[width=.35\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/OMNIGLOT_L1}\n%\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 3mm]{figures/OMNIGLOT_L2}\\\\\n\\end{tabular}\n\\caption{Training curves for the VAE Experiments with the one-layer linear model.\n%L1 represents the one-layer linear model and L2 represents the two-layer linear model.\nThe horizontal dashed line indicates the lowest validation error obtained by REBAR.}\n\\label{fig:vae_curves}\n\\end{figure}\n\n%\\begin{figure}[h]\n%\\centering\n%\\includegraphics[width=.4\\textwidth]{figures/OMNIGLOT_L1}\n%\\includegraphics[width=.4\\textwidth]{figures/OMNIGLOT_L2}\n%\\caption{Training curves on OMNIGLOT. The horizontal dashed line indicates the lowest validation score obtained by REBAR.}\n%\\label{fig:vae omni}\n%\\end{figure}\n\n\n\n\n\\subsection{Reinforcement learning}\n\\label{experiments section}\n\nWe apply our gradient estimator to a few simple reinforcement learning environments with discrete and continuous actions.\nWe use the \\RELAX{} and \\LAX{} estimators for discrete and continuous actions, respectively. We compare with the advantage actor-critic algorithm (A2C)~\\citep{sutton2000policy} as a baseline. \n\nAs our control variate does not have the same interpretation as the value function of A2C, it was not directly clear how to add reward bootstrapping and other variance reduction techniques common in RL into our model. For instance, to do reward bootstrapping, we would need to use the state-value function. In the discrete experiments, due to the simplicity of the tasks, we chose not to use reward bootstrapping, and therefore omitted the use of state-value function. However, with the more complicated continuous tasks, we chose to use the value function to enable bootstrapping. In this case, the control variate takes the form: $c_\\phi(a,s) = V(s) + \\hat{c}(a,s)$, where $V(s)$ is trained as it would be in A2C. Full details of our experiments can be found in Appendix~\\ref{experiment appendix}.\n\n% <Dami: i took it out because it doesn't match with the continuous case, and we dont' have variance plots yet>We are aware that better results could be obtained with bootstrapping and larger batch sizes but we wanted to work in the highest possible variance setting to demonstrate the variance reduction capabilities of our approach.\n\n%We test our alrgorithm on the Cart-Pole and Lunar-Lander environments from the OpenAI Gym~\\citep{1606.01540}.\n%We run the Cart-Pole and Lunar-Lander environments for 250 and 1000 episodes, respectively and plot reward and the log-variance of the policy gradients in figure~X.\n\nIn the discrete action setting, we test our approach on the Cart Pole and Lunar Lander environments as provided by the OpenAI gym~\\citep{1606.01540}.\nIn the continuous action setting, we test on the MuJoCo-simulated~\\citep{todorov2012mujoco} environment Inverted Pendulum also found in the OpenAI gym.\n% and Inverted Double Pendulum also found in the OpenAI gym.\nIn all tested environments we observe improved performance and sample efficiency using our method.\nThe results of our experiments can be seen in Figure~\\ref{fig:rl_results}, and Table~\\ref{tab:rl_results}.\n\nWe found that our estimator produced policy gradients with drastically reduced variance (see Figure~\\ref{fig:rl_results}) allowing for larger learning rates to be used while maintaining stable training.\nIn both discrete environments our estimator achieved greater than a 2-times speedup in convergence over the baseline.\n\n\\newcommand{\\rlfig}[1]{{\\includegraphics[height=0.24\\linewidth, clip, trim=3mm 3mm 3mm 3mm]{#1}}}%\n\\newcommand{\\rlfigg}[1]{{\\includegraphics[height=0.24\\linewidth, clip, trim=2mm 2mm 2mm 2mm]{#1}}}%\n\\begin{figure}%\n\\centering\n\\hspace*{-.1in}\n\\setlength{\\tabcolsep}{0pt}\n\\begin{tabular}{cccc}%c}%\n%\\renewcommand{\\arraystretch}{0}\n& Cart-pole & Lunar lander & Inverted pendulum \\\\%& Inverted double pendulum\\\\\n\\rotatebox{90}{\\qquad \\qquad \\small Reward} & \\rlfig{figures/cp_paper} & \n\\rlfig{figures/ll_paper} &\n\\rlfigg{figures/ip_paper_NEW} \\\\%& \\rlfig{figures/idp_paper}\\\\\n\\rotatebox{90}{\\qquad \\qquad \\small Log-Variance} & \\rlfig{figures/cp_paper_var} & \n\\rlfig{figures/ll_paper_var} &\n\\rlfigg{figures/ip_paper_var_NEW} \\\\%& \\rlfig{figures/idp_paper_var}\n\\end{tabular}\n\\caption{\\emph{Top row:} Reward curves.\n\\emph{Bottom row:} Log-variance of policy gradients.\nIn each curve, the center line indicates the mean reward over 5 random seeds.\nThe opaque bars in the top row indicate the 25th and 75th percentiles.\nThe opaque bars in the bottom row indicate 1 standard deviation. Since the gradient estimator is defined at the end of each episode, we display log-variance per episode.\nAfter every 10th training episode 100 episodes were run and the sample log-variance is reported averaged over all policy parameters. }\n\\label{fig:rl_results}\n\\end{figure}\n\n\\begin{table}%\n\\centering\n\\begin{tabular}{l | c c c }%c }%\n\\textbf{Model} & Cart-pole & Lunar lander & Inverted pendulum \\\\\\midrule%& Inverted double pendulum \\\\\\midrule\nA2C             & $1152 \\pm 90$ & $162374 \\pm 17241$                    & $6243 \\pm 164$ \\\\%& $60186 \\pm 3488$  \\\\\nLAX/RELAX & $\\bm{472 \\pm 114}$ & $\\bm{68712 \\pm 20668}$ & $\\bm{2067 \\pm 412}$ \\\\%& $\\bm{60967 \\pm 1669}$\n\\end{tabular}\n\\caption{Mean episodes to solve tasks.\nDefinitions of solving each task can be found in Appendix~\\ref{experiment appendix}.}\n\\label{tab:rl_results}\n\\end{table}\n\n%Code for all experiments can be found at \\href{https://github.com/duvenaud/relax}{github.com/duvenaud/relax}.\n\n\n\\section{Conclusions and future work}\n\\label{conclusion}\nIn this work we synthesized and generalized several standard approaches for constructing gradient estimators.\nWe proposed a generic gradient estimator that can be applied to expectations of known or black-box functions of discrete or continuous random variables, and adds little computational overhead.\nWe also derived a simple extension to reinforcement learning in both discrete and continuous-action domains. \n%This approach is relatively simple to implement and adds little computational overhead. \n\n%The foundation of our approach is the score function gradient estimator with a control variate whose expectation can be estimated with low variance using the reparameterization trick.\n%This control variate is neural network which is trained directly to minimize the variance of the estimated gradients.\n%The central result of this paper is that learning the function in the control variate leads to even better convergence properties and lower variance gradient estimates. \n\n%The generality of this method opens up new possibilities for training non-differentiable models. % which can now be trained via gradient decent.\nFuture applications of this method could include training models with hard attention or memory indexing~\\citep{zaremba2015reinforcement}.\nOne could also apply our estimators to continuous latent-variable models whose likelihood is non-differentiable, such as a 3D rendering engine.\nExtensions to the reparameterization gradient estimator~\\citep{ruiz2016generalized, naesseth2017reparameterization} could also be applied to increase the scope of distributions that can be modeled. \n%There is also room to explore other architecture choices for the control variate.% and to better understand the properties of the optimal control variate. \n%\n\n In the reinforcement learning setting, our method could be combined with other variance-reduction techniques such as generalized advantage estimation~\\citep{kimura2000analysis, schulman2015high}, or other optimization methods, such as KFAC~\\citep{wu2017scalable}.\nOne could also train our control variate off-policy, as in $Q$-prop~\\citep{gu2016q}.\n%We also feel that the relationship between our learned control variate and the action-value function (commonly denoted as $Q$) is worth exploring and understanding in greater detail.\n\n\\subsection*{Acknowledgements}  % Uncomment for arxiv\nWe thank Dougal Maclaurin, Tian Qi Chen, Elliot Creager, and Bowen Xu for helpful discussions.\nWe also thank Christopher Prohm for pointing out an error in one of our derivations. We would also like to thank George Tucker for pointing out a bug in our initially released reinforcement learning code. \n\n\\bibliography{bibliography}\n\\bibliographystyle{iclr2018_conference}\n\n\n\n% ** Questions to answer: \n% (1) is z-tilde a clever way of using Rao-Blackwellization for a part of the reparameterization gradient? This would mean that the reparameterization z-tilde is related to the sufficient statistic of the estimator...\n% (1) also maybe: the Q function has the opportunity to learn an estimator based on the sufficient statistics of the model, which by Rao-Blackwell-Kolmogorov is lower variance\n% (2) What's a better notation to keep the dependence of z on $\\theta$ in view?\n% (3) is the REBAR control variate really using the reparameterization gradient in a meaningful way? Or, is it best viewed as just another f + control variate where control variate is cleverly designed with lower variance? \n\n\n%\\par{Generalizing the reparameterization trick}\n\n%Write sample from distribution $s(\\epsilon)$ as $\\epsilon = \\mathcal{T}^{-1}(\\mathbf{z}; \\mathbf{\\nu})$ for some invertible transform $\\mathcal{T}$ with variational parameters $\\nu$.\n%write out transformed density\n%example: normal with standard normal $s$\n%example: inverse CDF of Gaussian with uniform $s$\n%write out expected gradient under transformation\n%show decomposition of expected gradient into reparameterization and correction terms \n\n%\\par{Applying GRG to REBAR}\n\n%show mapping of terms\n%note denser derivation in REBAR appendix\n\n%\\par{Interpreting REBAR through GRG}\n\n\\clearpage\n\\section*{Appendices}\n\\appendix\n\n\n\n\n\\section{The RELAX Algorithm}\n\\label{relax proof}\n\n\\begin{proof}\n\tWe show that $\\hat g_\\textnormal{RELAX}$ is an unbiased estimator of $\\frac{\\partial}{\\partial \\theta} \\mathbb{E}_{p(b \\vert \\theta)} \\left[ f(b) \\right]$. The estimator is\n\t\n\t\\begin{align*}\n\t\\E_{p(b|\\theta)} \\! \\left[\\left[ f(b) - \\E_{p(\\tilde{z}|b, \\theta)} \\! \\left[c_\\phi(\\tilde{z}) \\right] \\right]\\PT \\log p(b|\\theta)  - \\PT \\E_{p(\\tilde{z}|b, \\theta)} \\! \\left[c_\\phi(\\tilde{z}) \\right] \\right] + \\PT\\E_{p(z|\\theta)} \\! \\left[ c_\\phi(z) \\right]. \\span & \\nonumber \\\\\n\t\\end{align*}\n\tExpanding the expectation for clarity of exposition, we account for each term in the estimator separately:\n\t\\begin{align}\n\t& \\E_{p(b|\\theta)} \\! \\left[ f(b) \\PT \\log p(b|\\theta)  \\right] \\label{one}\\\\ \n\t& - \\E_{p(b|\\theta)} \\left[ \\E_{p(\\tilde{z}|b, \\theta)} \\! \\left[c_\\phi (\\tilde{z}) \\right] \\PT \\log p(b|\\theta)  \\right] \\label{two}\\\\\n\t& - \\E_{p(b|\\theta)}  \\left[ \\PT \\E_{p(\\tilde{z}|b, \\theta)} \\! \\left[c_\\phi(\\tilde{z}) \\right] \\right] \\label{three}\\\\\n\t& + \\PT\\E_{p(z|\\theta)} \\! \\left[ c_\\phi(z) \\right]. \\label{four}\n\t\\end{align}\n\tTerm \\eqref{one} is an unbiased score-function estimator of $\\frac{\\partial}{\\partial \\theta} \\mathbb{E}_{p(b \\vert \\theta)} \\left[ f(b) \\right]$. It remains to show that the other three terms are zero in expectation. Following \\cite{tucker2017rebar} (see the appendices of that paper for a derivation), we rewrite term \\eqref{three} as follows:\n\t\\begin{align}\n\t-\\mathbb{E}_{p(b \\vert \\theta)} \\left[ \\frac{\\partial}{\\partial \\theta} \\mathbb{E}_{p(\\tilde{z} \\vert b, \\theta)} \\left[ c_\\phi (\\tilde{z}) \\right] \\right] =~&\\mathbb{E}_{p(b \\vert \\theta)} \\left[ \\mathbb{E}_{p(\\tilde{z} \\vert b, \\theta)} \\left[ c_\\phi(\\tilde{z}) \\right] \\frac{\\partial}{\\partial \\theta} \\log p(b \\vert \\theta) \\right] \\nonumber \\\\ & -  \\mathbb{E}_{p(z \\vert \\theta)} \\left[ c_\\phi(z) \\frac{\\partial}{\\partial \\theta} \\log p(z) \\right] . \\label{six}\n\t\\end{align}\n\tNote that the first term on the right-hand side of equation \\eqref{six} is equal to term \\eqref{two} with opposite sign. The second term on the right-hand side of equation \\eqref{six} is the score-function estimator of term \\eqref{four}, opposite in sign. The sum of these terms is zero in expectation.\n\t\\\\\n\\end{proof}\n\n\n\\begin{algorithm}[h]\n\t\\begin{algorithmic}\n\t\t\\Require $f(\\cdot)$, $\\log p(b|\\theta)$, reparameterized samplers $b = H(z)$, $z = S(\\epsilon, \\theta)$ and $\\tilde{z} = S(\\epsilon, \\theta | b)$, \\\\ \n\t\t\\hspace{3em} neural network $c_\\phi(\\cdot)$, step sizes $\\alpha_1, \\alpha_2$  \\While {not converged} \n\t\t\\State $\\epsilon_{i}, \\widetilde{\\epsilon_i} \\sim p(\\epsilon)$ \\Comment Sample noise\n\t\t\\State $z_i \\leftarrow S(\\epsilon_i, \\theta)$ \\Comment Compute unconditional relaxed input\n\t\t\\State $b_i \\leftarrow H(z_i)$ \\Comment Compute input\n\t\t\\State $\\widetilde{z_i} \\leftarrow S(\\widetilde{\\epsilon_i}, \\theta | b_i)$ \\Comment Compute conditional relaxed input\n\t\t\\State  $\\hat{g}_\\theta \\leftarrow \\left[f(b_i) - c_{\\phi}(\\widetilde{z_i}) \\right] \\nabla_\\theta \\log p + \\nabla_\\theta c_\\phi(z_i) - \\nabla_\\theta c_\\phi(\\widetilde{z_i})$ \\Comment Estimate gradient\n\t\t\\State  $\\hat{g}_\\phi \\leftarrow \\partial \\hat{g}_\\theta^2 / \\partial \\phi$ \\Comment Estimate gradient of variance of gradient\n\t\t\\State $\\theta \\leftarrow \\theta - \\alpha_1 \\hat{g}_\\theta$ \\Comment Update parameters\n\t\t\\State $\\phi \\leftarrow \\phi - \\alpha_2 \\hat{g}_\\phi$ \\Comment Update control variate\n\t\t\\EndWhile\n\t\t\\State \\textbf{return} $\\theta$ \n\t\\end{algorithmic}\n\t\\caption{\\RELAX{}: Low-variance control variate optimization for black-box gradient estimation.}\n\t\\label{relax}\n\\end{algorithm}\n\n\n\n\n\n\\section{Conditional Re-sampling for Discrete Random Variables}\n\\label{resample}\nWhen applying the RELAX estimator to a function of discrete random variables $b \\sim p(b|\\theta)$, we require that there exists a distribution $p(z|\\theta)$ and a deterministic mapping $H(z)$ such that if $z \\sim p(z|\\theta)$ then $H(z) = b \\sim p(b|\\theta)$. Treating both $b$ and $z$ as random, this procedure defines a probabilistic model $p(b, z | \\theta) = p(b|z)p(z|\\theta)$. The RELAX estimator requires reparameterized samples from $p(z|\\theta)$ and $p(z|b,\\theta)$. We describe how to sample from these distributions in the common cases of $p(b|\\theta) = \\text{Bernoulli}(\\theta)$ and $p(b|\\theta) = \\text{Categorical}(\\theta)$.\n\n\\paragraph{Bernoulli} When $p(b|\\theta)$ is Bernoulli distribution we let $H(z) = \\mathbb{I}(z>0)$ and we sample from $p(z|\\theta)$ with \n$$ z = \\log \\frac{\\theta}{1 - \\theta} + \\log \\frac{u}{1-u}, \\qquad u \\sim \\text{uniform}[0,1].\n$$ \nWe can sample from $p(z|b, \\theta)$ with \n\\[\nv' =    \\left\\{\n\\begin{array}{ll}\n      v\\cdot(1-\\theta) & b = 0 \\\\\n      v\\cdot\\theta + (1 - \\theta) & b = 1 \\\\\n\\end{array} \n\\right.\n\\]\n$$ \\tilde{z} = \\log \\frac{\\theta}{1 - \\theta} + \\log \\frac{v'}{1-v'}, \\qquad v \\sim \\text{uniform}[0, 1].$$\n\n\\paragraph{Categorical} When $p(b|\\theta)$ is a Categorical distribution where $\\theta_i = p(b=i|\\theta)$, we let $H(z) = \\text{argmax}(z)$ and we sample from $p(z|\\theta)$ with \n$$ z = \\log\\theta -\\log(-\\log u), \\qquad u \\sim \\text{uniform}[0,1]^k\n$$ where $k$ is the number of possible outcomes.\n\n\nTo sample from $p(z|b, \\theta)$, we note that the distribution of the largest $\\hat z_b$ is independent of $\\theta$, and can be sampled as $\\hat z_b = -\\log (-\\log v_b)$ where $v_b\\sim \\text{uniform}[0, 1]$.\nThen, the remaining $v_{i\\neq b}$ can be sampled as before but with their underlying noise truncated so $\\hat z_{i \\neq b} < \\hat z_b$. As shown in the appendix of \\cite{tucker2017rebar}, we can then sample from $p(z|b, \\theta)$ with:\n%\n\\begin{align}\n\\hat z_i =    \\left\\{\n\\begin{array}{ll}\n      -\\log(-\\log v_i )& i = b \\\\\n      -\\log \\left( -\\frac{\\log v_i}{\\theta_i} - \\log v_b \\right) & i \\neq b \\\\\n\\end{array} \n\\right.\n\\end{align}\n%\nwhere $v_i \\sim \\text{uniform}[0, 1]$.\n\n\n%To sample from $p(z|b, \\theta)$ we sample a value $v'$ and compute $\\tilde{z} = \\log\\theta -\\log(-\\log v')$. We note that in the unconditional case we would have $v'_b \\sim \\text{uniform}[0, 1]$ but in the conditional case $v'_b \\sim \\text{Beta}\\left[1+\\frac{1 - \\theta_b}{\\theta_b}, 1\\right]$. We first sample $v'_b$ in this way. Then we can sample $v'_{i\\neq b}$ by finding the point in $[0, 1]$ where $z_b = z_{i\\neq b}$ and scaling a uniform random variable $v_i$ to be below that value. \n\n%Intuitively, to sample from $p(z|b, \\theta)$ we should first sample $v\\sim \\text{uniform}[0, 1]^k$, then compute $g_b = \\log\\theta_b -\\log(-\\log(v_b))$. Then we must determine how to scale each $v_{i\\neq b}$ such that $g_{i\\neq b} < g_b$. We can define $v'$ such that\n%Formally,\n%\\[\n%v_i' =    \\left\\{\n%\\begin{array}{ll}\n%      v'_b & i = b \\\\\n %     v_i\\cdot(v'_b)^{\\frac{\\theta_i}{\\theta_b}} & i \\neq b \\\\\n%\\end{array} \n%\\right.\n%\\] \n%$$ v'_b \\sim \\text{Beta}\\left[1+\\frac{1 - \\theta_b}{\\theta_b}, 1\\right], \\qquad v_{i\\neq b} \\sim \\text{uniform}[0, 1]$$ and then $\\tilde{z} = \\log\\theta - \\log(-\\log v')$ which is our sample from $p(z|b, \\theta)$. \n\n%Let $G_{1:k} = -\\log-\\log(U_{i:k})$ be samples from the Gumbel distribution, and learnable parameters $(\\alpha_1, \\dots, \\alpha_k)$ be interpreted as some unnormalized parameterization of the discrete distribution under consideration.\n%Then, consider the following sampling procedure: for each k, find the k that maximizes $\\log \\alpha_k - G_k$, and then set $D_k=1$ and $D_{i \\neq k} = 0$. The Gumbel-Max trick states that sampling from the discrete distribution is equivalent to taking this argmax, that is, $p(D_k = 1) = \\alpha_k / \\sum_{i=1}^n \\alpha_i$.\n\n%Since taking an argmax is still a discontinuous operation, \\cite{maddison2016concrete} and \\cite{jang2016categorical} proposed further relaxing the argmax operator through the softmax function with an additional temperature parameter $\\lambda$:\n%\\begin{equation}\n%x_k = \\frac{\\exp\\{( \\log \\alpha_k+ G_k) / \\lambda\\}}{\\sum_{i=1}^n\\exp\\{( \\log \\alpha_i+ G_i) / \\lambda\\}}\n%\\end{equation}\n%This relaxation allows values within the simplex, but in the low temperature limit, it becomes exactly the discrete argmax.\n%One limitation of the concrete distribution is that it is a biased estimator except in limiting temperature.\n%In other words, a small amount of bias is present for a non-zero temperature.\n\n\n\\section{Derivations of estimators used in Reinforcement learning}\n\\label{rl appendix}\nWe give the derivation of the \\LAX{} estimator used for continuous RL tasks.\n\\begin{theorem}\nThe \\LAX{} estimator,\n\\begin{align}\n\\hat g_\\LAX^{\\RL} = \\sum_{t=1}^{T} \\LL{t} \\left[ \\sum_{t'=t}^{T} r_{t'} - c_\\phi(a_t,s_t) \\right] +\\frac{\\partial}{\\partial\\theta} c_\\phi(a_t, s_t), \\\\\na_t = a_t(\\epsilon_t,s_t,\\theta), \\quad \\epsilon_t \\sim p(\\epsilon_t)\\nonumber,\n\\end{align}\nis unbiased.\n\\end{theorem}\n\\begin{proof}\nNote that by using the score-function estimator, for all $t$, we have \n%\n\\begin{align*}\n\\E_{p(\\tau)}\\Big[\\LL{t} c_\\phi(a_t, s_t)\\Big] = \\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{\\pi(a_t|s_t, \\theta)}\\Big[c_\\phi(a_t, s_t)\\Big]\\Big].\n\\end{align*}\nThen, by adding and subtracting the same term, we have\n\\begin{align*}\n\\PT\\E_{p(\\tau)}[f(\\tau)] &= \\E_{p(\\tau)}\\left[f(\\tau)\\cdot\\LP{\\tau;\\theta}\\right]-\\sum_t\\E_{p(\\tau)}\\Big[\\LL{t} c_\\phi(a_t, s_t)\\Big]+\\\\&\\sum_t \\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{\\pi(a_t|s_t, \\theta)}\\Big[c_\\phi(a_t,s_t)\\Big]\\Big]\\nonumber\\\\\n&= \\E_{p(\\tau)}\\left[ \\sum_{t=1}^{\\infty} \\LL{t}\\left(\\sum_{t'=t}^{\\infty} r_{t'} - c_\\phi(a_t,s_t)\\right)\\right]\\\\\n& \\qquad + \\sum_t \\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\E_{p(\\epsilon_t)}\\Big[\\frac{\\partial}{\\partial\\theta}c_\\phi(a_t(\\epsilon_t,s_t,\\theta), s_t)\\Big]\\Big]\\nonumber\\\\\n&= \\E_{p(\\tau)}\\left[ \\sum_{t=1}^{\\infty} \\LL{t}\\left(\\sum_{t'=t}^{\\infty} r_{t'} - c_\\phi(a_t,s_t)\\right)+\\frac{\\partial}{\\partial\\theta}c_\\phi(a_t(\\epsilon_t,s_t,\\theta), s_t)\\right]\\nonumber\n\\end{align*}\n\\end{proof}\n\nIn the discrete control setting, our policy parameterizes a soft-max distribution which we use to sample actions. We define $z_t\\sim p(z_t|s_t)$, which is equal to $\\sigma (\\log\\pi - \\log(-\\log(u)))$ where $u\\sim \\text{uniform}[0, 1]$, $a_t = \\text{argmax}(z_t)$, $\\sigma$ is the soft-max function. We also define $\\tilde{z_t} \\sim p(z_t|a_t,s_t)$ and uses the same reparametrization trick for sampling $\\tilde{z_t}$ as explicated in Appendix \\ref{resample}.\n\\begin{theorem}\nThe \\RELAX{} estimator,\n\\begin{align}\n\\hat g_\\RELAX^{\\RL} = \\sum_{t=1}^{T} \\LL{t}\\left(\\sum_{t'=t}^{T} r_{t'} - c_\\phi(\\tilde{z}_t, s_t)\\right)-\\frac{\\partial}{\\partial\\theta}c_\\phi(\\tilde{z}_t, s_t)+\\frac{\\partial}{\\partial\\theta}c_\\phi(z_t, s_t), \\label{eq:relaxrlproof}\\\\\n\\tilde{z}_t \\sim p(z_t|a_t,s_t), \\qquad z_t \\sim p(z_t|s_t)\\nonumber, \n\\end{align}\nis unbiased.\n\\end{theorem}\n\\begin{proof}\nNote that by using the score-function estimator, for all $t$, we have \n%\n\\begin{align*}\n& \\E_{p(a_{1:t},s_{1:t})}\\Big[\\LL{t} \\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)]\\Big]\\\\\n &= \\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{\\pi(a_t|s_t, \\theta)}\\Big[\\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)]\\Big]\\Big]\\\\\n&=\\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{p(z_t|s_t)}[c_\\phi(z_t, s_t)]\\Big]\n\\end{align*}\nThen, by adding and subtracting the same term, we have\n\\begin{align*}\n\\PT\\E_{p(\\tau)}[f(\\tau)] &= \\E_{p(\\tau)}\\left[f(\\tau)\\cdot\\LP{\\tau;\\theta}\\right]\\\\\n& \\qquad -\\sum_t\\E_{p(a_{1:t},s_{1:t})}\\Big[\\LL{t} \\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)]\\Big]\\\\\n& \\qquad + \\sum_t\\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{p(z_t|s_t)}[c_\\phi(z_t, s_t)]\\Big]\\nonumber\\\\\n& = \\E_{p(\\tau)}\\left[ \\sum_{t=1}^{\\infty} \\LL{t}\\left(\\sum_{t'=t}^{\\infty} r_{t'} - \\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)]\\right)\\right]\\\\\n& \\qquad + \\sum_t\\E_{p(a_{1:t-1},s_{1:t})}\\Big[\\frac{\\partial}{\\partial\\theta}\\E_{p(z_t|s_t)}[c_\\phi(z_t, s_t)]\\Big]\\nonumber\\\\\n& = \\E_{p(\\tau)}\\Big[ \\sum_{t=1}^{\\infty} \\LL{t}\\left(\\sum_{t'=t}^{\\infty} r_{t'} - \\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)\\right)\\\\\n& \\qquad - \\frac{\\partial}{\\partial\\theta}\\E_{p(z_t|a_t,s_t)}[c_\\phi(z_t, s_t)]+\\frac{\\partial}{\\partial\\theta}\\E_{p(z_t|s_t)}[c_\\phi(z_t, s_t)]\\Big]\\nonumber\n\\end{align*}\nSince $p(z_t|s_t)$ is reparametrizable, we obtain the estimator in Eq.(\\ref{eq:relaxrlproof}).\n\\end{proof}\n\n\\section{Further results on discrete variational autoencoders}\n\\label{extra vae results}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{r l | c c} \n  Dataset & Model & REBAR & RELAX \\\\\\midrule\n & one-layer linear  & -114.32 & \\textbf{-113.62} \\\\ \n\\textbf{MNIST} & two-layer linear  & -101.20 & \\textbf{-100.85}\\\\\n& Nonlinear & \\textbf{-111.12} & 119.19 \\\\ \\midrule\n & one-layer linear & -122.44 & \\textbf{-122.11} \\\\ \n\\textbf{Omniglot}& two-layer linear & -115.83 & \\textbf{-115.42}\\\\\n& Nonlinear& \\textbf{-127.51} & 128.20\n\\end{tabular}\n\\caption{Highest obtained validation ELBO.}\n\\label{tab:vae val}\n\\end{table}\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{r l | c c} \n Dataset & Model  & REBAR & RELAX \\\\\\midrule\n & one-layer  & 857 & \\textbf{531} \\\\ \n\\textbf{MNIST} & two-layer  & 900 & \\textbf{620} \\\\\n& Nonlinear & \\textbf{331} & - \\\\\n\\midrule\n& one-layer & 2086 & \\textbf{566} \\\\ \n\\textbf{Omniglot}  & two-layer & 1027 & \\textbf{673}\\\\\n& Nonlinear & \\textbf{368} & - \n\\end{tabular}\n\\caption{Epochs needed to achieve REBAR's best validation score. ``-'' indicates that the nonlinear RELAX models achieved lower validation scores than REBAR.}\n\\label{tab:vae epochs}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\hspace*{-.5in}\n\\setlength{\\tabcolsep}{10pt}\n\\renewcommand{\\arraystretch}{0}\n\\begin{tabular}{ccc}\n& MNIST & Omniglot \\\\\n\\rotatebox{90}{\\qquad \\qquad \\qquad \\small -ELBO} & \n%\\includegraphics[width=.33\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/MNIST_L1} &\n\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/MNIST_L2} &\n%\\includegraphics[width=.35\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/OMNIGLOT_L1}\n\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/OMNIGLOT_L2}\\\\\n\\end{tabular}\n\\caption{Training curves for the VAE Experiments with the two-layer linear model.\n%L1 represents the one-layer linear model and L2 represents the two-layer linear model.\nThe horizontal dashed line indicates the lowest validation error obtained by REBAR.}\n\\label{fig:vae curves2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\hspace*{-.5in}\n\\setlength{\\tabcolsep}{10pt}\n\\renewcommand{\\arraystretch}{0}\n\\begin{tabular}{ccc}\n& MNIST & Omniglot \\\\\n\\rotatebox{90}{\\qquad \\qquad \\qquad \\small -ELBO} & \n\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/MNIST_NL1} &\n\\includegraphics[width=.31\\textwidth, clip, trim=3mm 3mm 3mm 2mm]{figures/OMNIGLOT_NL1}\\\\\n\\end{tabular}\n\\caption{Training curves for the VAE Experiments with the one-layer nonlinear model.\nThe horizontal dashed line indicates the lowest validation error obtained by REBAR.}\n\\label{fig:vae curves3}\n\\end{figure}\n\n\n\\section{Experimental Details}\n\\label{experiment appendix}\n\n\\subsection{Discrete VAE}\nWe run all models for $2,000,000$ iterations with a batch size of $24$.\nFor the REBAR models, we tested learning rates in $\\{.005, .001, .0005,  .0001, .00005\\}$. \n\n\\RELAX{} adds more hyperparameters.\nThese are the depth of the neural network component of our control variate $r_\\rho$, the weight decay placed on the network, and the scaling on the learning rate for the control variate.\nWe tested neural network models with $l$ layers of 200 units using the ReLU nonlinearity with $l \\in \\{2, 4\\}$.\nWe trained the control variate with weight decay in $\\{.001, .0001\\}$.\nWe trained the control variate with learning rate scaling in $\\{1, 10\\}$.\n\nTo limit the size of hyperparameter search for the RELAX models, we only test the best performing learning rate for the REBAR baseline and the next largest learning rate in our search set.\nIn many cases, we found that RELAX allowed our model to converge at learning rates which made the REBAR estimators diverge.\nWe believe further improvement could be achieved by tuning this parameter. It should be noted that in our experiments, we found the RELAX method to be fairly insensitive to all hyperparameters other than learning rate. In general, we found the larger (4 layer) control variate architecture with weight decay of $.001$ and learning rate scaling of $1$ to work best, but only slightly outperformed other configurations.  \n\nAll presented results are from the models which achieve the highest ELBO on the validation data.\n\\label{app_disc_vae}\n\\subsubsection{One-layer linear model}\nIn the one-layer linear models we optimize the evidence lower bound (ELBO): $$\\log p(x) \\geq \\mathcal{L}(\\theta) = \\E_{q(b|x)}[\\log p(x|b) + \\log p(b) - \\log q(b|x)]$$ where $q(b_1|x) = \\sigma(x\\cdot W_q + \\beta_q)$ and $p(x| b_1) = \\sigma(b_1\\cdot W_p + \\beta_p)$ with weight matrices $W_q,W_p$ and bias vectors $\\beta_q,\\beta_p$.\nThe parameters of the prior $p(b)$ are also learned.\n\n\n\\subsubsection{Two layer linear model}\nIn the two layer linear models we optimize the ELBO $$\\mathcal{L}(\\theta) = \\E_{q(b_2|b_1)q(b_1|x)}[\\log p(x|b_1) + \\log p(b_1|b_2) + \\log p(b_2) - \\log q(b_1|x) - \\log q(b_2|b_1)]$$ where $q(b_1|x) = \\sigma(x\\cdot W_{q_1} + \\beta_{q_1})$, $q(b_2|b_1) = \\sigma(b_1\\cdot W_{q_2} + \\beta_{q_2})$, $p(x| b_1) = \\sigma(b_1\\cdot W_{p_1} + \\beta_{p_1})$, and $p(b_1| b_2) = \\sigma(b_2\\cdot W_{p_2} + \\beta_{p_2})$ with weight matrices $W_{q_1},W_{q_2},W_{p_1},W_{p_2}$ and biases $\\beta_{q_1},\\beta_{q_2},\\beta_{p_1},\\beta_{p_2}$. As in the one-layer model, the prior $p(b_2)$ is also learned.\n\n\\subsubsection{Nonlinear model}\nIn the one-layer nonlinear model, the mappings between random variables consist of 2 deterministic layers with 200 units using the hyperbolic-tangent nonlinearity followed by a linear layer with 200 units. \n\nWe run an identical hyperpameter search in all models. \n\n\n\\subsection{Discrete RL}\nIn both the baseline A2C and RELAX models, the policy and control variate (value function in the baseline model) were two-layer neural networks with 10 units per layer.\nThe ReLU non linearity was used on all layers except for the output layer which was linear.\n\nFor these tasks we estimate the policy gradient with a single Monte Carlo sample.\nWe run one episode of the environment to completion, compute the discounted rewards, and run one iteration of gradient descent.\nWe believe using larger batches will improve performance but would less clearly demonstrate the potential of our method. \n\n%As our control variate does not have the same interpretation as the value function of A2C, it was not directly clear how to add reward bootstrapping and other variance reduction techniques common in RL into our model. We leave the task of incorporating these and other variance reduction techniques to future work.  \n\nBoth models were trained with the RMSProp~\\citep{Tieleman2012} optimizer and a reward discount factor of $.99$ was used. Entropy regularization with a weight of $.01$ was used to  encourage exploration. \n\nBoth models have 2 hyperparameters to tune; the global learning rate and the scaling factor on the learning rate for the control variate (or value function).\nWe complete a grid search for both parameters in $\\{0.01, 0.003, 0.001\\}$ and present the model which ``solves'' the task in the fewest number of episodes averaged over 5 random seeds.\n``Solving'' the tasks was defined by the creators of the OpenAI gym~\\citep{1606.01540}.\nThe Cart Pole task is considered solved if the agent receives an average reward greater than 195 over 100 consecutive episodes.\nThe Lunar Lander task is considered solved if the agent receives an average reward greater than 200 over 100 consecutive episodes. \n\nThe Cart Pole experiments were run for 250,000 frames.\nThe Lunar Lander experiments were run for 5,000,000 frames. \n\nThe results presented for the CartPole and LunarLander environments were obtained using a slightly biased sampler for $p(z|b, \\theta)$.\n\n\\subsection{Continuous RL}\n%The continuous tasks uses both the value function and the control variate to enable bootstrapping, which is needed due to the increased complexity of the problem.\nThe three models- policy, value, and control variate, are two-layer neural networks with 64 hidden units per layer.\nThe value and control variate networks are identical, with the ELU~\\citep{Clevert2016ELUs} nonlinearity in each hidden layer.\nThe policy network has \\texttt{tanh} nonlinearity.\nThe policy network, which parameterizes the Gaussian policy comprises of a network (with the architecture mentioned above) that outputs the mean, and a separate, trainable log standard deviation value that is not input dependent.\nAll three networks have a linear output layer.\nWe selected the batch size to be 2500, meaning for a fixed timestep (2500) we collect multiple rollouts of a task and update the networks' parameters with the batch of episodes.\nPer one policy update, we optimize both the value and control variate network multiple times.\nThe number of times we train  the value network is fixed to 25, while for the control variate, it was chosen to be a hyperparameter. \n%need to add information on whether we build on top of reinforce, or a2c or a2c+GAE\nAll models were trained using ADAM~\\citep{kingma2015adam}, with $\\beta_1=0.9$, $\\beta_2=0.999$, and $\\epsilon=1e-08$. \n\nThe baseline A2C case has 2 hyperparameters to tune: the learning rate for the optimizer for the policy and value network.\nA grid search was done over the set: $\\{0.03, 0.003, 0.0003\\}$.\n\\RELAX{} has 4 hyperparameters to tune: 3 learning rates for the optimizer per network, and the number of training iterations of the control variate per policy gradient update.\nDue to the large number of hyperparameters, we restricted the size of the grid search set to $\\{0.003, 0.0003\\}$ for the learning rates, and $\\{1, 5, 25\\}$ for the control variate training iteration number.\nWe chose the hyperparameter setting that yielded the shortest episode-to-completion time averaged over 5 random seeds.\nAs with the discrete case, we used the definition of completion provided by the OpenAI gym~\\citep{1606.01540} for each task. \n\nThe Inverted Pendulum experiments were run for 1,000,000 frames.\n%The Inverted Double Pendulum experiments were run for 5,000,000 frames.\n\n\\citet{tucker2018mirage} pointed out a bug in our initially released code for the continuous RL experiments. This issue has been fixed in the publicly available code and the results presented in this paper were generated with the corrected code. \n\n\n\\subsubsection{Implementation Considerations}\nFor continuous RL tasks, it is convention to employ a batch of a fixed number of timesteps (here, 2500) in which the number of episodes vary. We follow this convention for the sake of providing a fair comparison to the baseline. However, this causes a complication when calculating the variance loss for the control variate because we must compute the variance averaged over completed episodes, which is difficult to obtain when the number of episodes is not fixed. For this reason, in our implementation we compute the gradients for the control variate outside of the Tensorflow computation graph. However, for practical reasons we recommend using a batch of fixed number of episodes when using our method.\n\\end{document}\n"
  },
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    "path": "paper/iclr2018_conference.bst",
    "content": "%% File: `iclr2017.bst'\n%% A copy of iclm2010.bst, which is a modification of `plainnl.bst' for use with natbib package \n%%\n%% Copyright 2010 Hal Daum\\'e III\n%% Modified by J. Fürnkranz\n%% - Changed labels from (X and Y, 2000) to (X & Y, 2000)\n%%\n%% Copyright 1993-2007 Patrick W Daly\n%% Max-Planck-Institut f\\\"ur Sonnensystemforschung\n%% Max-Planck-Str. 2\n%% D-37191 Katlenburg-Lindau\n%% Germany\n%% E-mail: daly@mps.mpg.de\n%%\n%% This program can be redistributed and/or modified under the terms\n%% of the LaTeX Project Public License Distributed from CTAN\n%% archives in directory macros/latex/base/lppl.txt; either\n%% version 1 of the License, or any later version.\n%%\n % Version and source file information:\n % \\ProvidesFile{icml2010.mbs}[2007/11/26 1.93 (PWD)]\n %\n % BibTeX `plainnat' family\n %   version 0.99b for BibTeX versions 0.99a or later,\n %   for LaTeX versions 2.09 and 2e.\n %\n % For use with the `natbib.sty' package; emulates the corresponding\n %   member of the `plain' family, but with author-year citations.\n %\n % With version 6.0 of `natbib.sty', it may also be used for numerical\n %   citations, while retaining the commands \\citeauthor, \\citefullauthor,\n %   and \\citeyear to print the corresponding information.\n %\n % For version 7.0 of `natbib.sty', the KEY field replaces missing\n %   authors/editors, and the date is left blank in \\bibitem.\n %\n % Includes field EID for the sequence/citation number of electronic journals\n %  which is used instead of page numbers.\n %\n % Includes fields ISBN and ISSN.\n %\n % Includes field URL for Internet addresses.\n %\n % Includes field DOI for Digital Object Idenfifiers.\n %\n % Works best with the url.sty package of Donald Arseneau.\n %\n % Works with identical authors and year are further sorted by\n %   citation key, to preserve any natural sequence.\n %\nENTRY\n  { address\n    author\n    booktitle\n    chapter\n    doi\n    eid\n    edition\n    editor\n    howpublished\n    institution\n    isbn\n    issn\n    journal\n    key\n    month\n    note\n    number\n    organization\n    pages\n    publisher\n    school\n    series\n    title\n    type\n    url\n    volume\n    year\n  }\n  {}\n  { label extra.label sort.label short.list }\n\nINTEGERS { output.state before.all mid.sentence after.sentence after.block }\n\nFUNCTION {init.state.consts}\n{ #0 'before.all :=\n  #1 'mid.sentence :=\n  #2 'after.sentence :=\n  #3 'after.block :=\n}\n\nSTRINGS { s t }\n\nFUNCTION {output.nonnull}\n{ 's :=\n  output.state mid.sentence =\n    { \", \" * write$ }\n    { output.state after.block =\n        { add.period$ write$\n          newline$\n          \"\\newblock \" write$\n        }\n        { output.state before.all =\n            'write$\n            { add.period$ \" \" * write$ }\n          if$\n        }\n      if$\n      mid.sentence 'output.state :=\n    }\n  if$\n  s\n}\n\nFUNCTION {output}\n{ duplicate$ empty$\n    'pop$\n    'output.nonnull\n  if$\n}\n\nFUNCTION {output.check}\n{ 't :=\n  duplicate$ empty$\n    { pop$ \"empty \" t * \" in \" * cite$ * warning$ }\n    'output.nonnull\n  if$\n}\n\nFUNCTION {fin.entry}\n{ add.period$\n  write$\n  newline$\n}\n\nFUNCTION {new.block}\n{ output.state before.all =\n    'skip$\n    { after.block 'output.state := }\n  if$\n}\n\nFUNCTION {new.sentence}\n{ output.state after.block =\n    'skip$\n    { output.state before.all =\n        'skip$\n        { after.sentence 'output.state := }\n      if$\n    }\n  if$\n}\n\nFUNCTION {not}\n{   { #0 }\n    { #1 }\n  if$\n}\n\nFUNCTION {and}\n{   'skip$\n    { pop$ #0 }\n  if$\n}\n\nFUNCTION {or}\n{   { pop$ #1 }\n    'skip$\n  if$\n}\n\nFUNCTION {new.block.checka}\n{ empty$\n    'skip$\n    'new.block\n  if$\n}\n\nFUNCTION {new.block.checkb}\n{ empty$\n  swap$ empty$\n  and\n    'skip$\n    'new.block\n  if$\n}\n\nFUNCTION {new.sentence.checka}\n{ empty$\n    'skip$\n    'new.sentence\n  if$\n}\n\nFUNCTION {new.sentence.checkb}\n{ empty$\n  swap$ empty$\n  and\n    'skip$\n    'new.sentence\n  if$\n}\n\nFUNCTION {field.or.null}\n{ duplicate$ empty$\n    { pop$ \"\" }\n    'skip$\n  if$\n}\n\nFUNCTION {emphasize}\n{ duplicate$ empty$\n    { pop$ \"\" }\n    { \"\\emph{\" swap$ * \"}\" * }\n  if$\n}\n\nINTEGERS { nameptr namesleft numnames }\n\nFUNCTION {format.names}\n{ 's :=\n  #1 'nameptr :=\n  s num.names$ 'numnames :=\n  numnames 'namesleft :=\n    { namesleft #0 > }\n    { s nameptr \"{ff~}{vv~}{ll}{, jj}\" format.name$ 't :=\n      nameptr #1 >\n        { namesleft #1 >\n            { \", \" * t * }\n            { numnames #2 >\n                { \",\" * }\n                'skip$\n              if$\n              t \"others\" =\n                { \" et~al.\" * }\n                { \" and \" * t * }\n              if$\n            }\n          if$\n        }\n        't\n      if$\n      nameptr #1 + 'nameptr :=\n      namesleft #1 - 'namesleft :=\n    }\n  while$\n}\n\nFUNCTION {format.key}\n{ empty$\n    { key field.or.null }\n    { \"\" }\n  if$\n}\n\nFUNCTION {format.authors}\n{ author empty$\n    { \"\" }\n    { author format.names }\n  if$\n}\n\nFUNCTION {format.editors}\n{ editor empty$\n    { \"\" }\n    { editor format.names\n      editor num.names$ #1 >\n        { \" (eds.)\" * }\n        { \" (ed.)\" * }\n      if$\n    }\n  if$\n}\n\nFUNCTION {format.isbn}\n{ isbn empty$\n    { \"\" }\n    { new.block \"ISBN \" isbn * }\n  if$\n}\n\nFUNCTION {format.issn}\n{ issn empty$\n    { \"\" }\n    { new.block \"ISSN \" issn * }\n  if$\n}\n\nFUNCTION {format.url}\n{ url empty$\n    { \"\" }\n    { new.block \"URL \\url{\" url * \"}\" * }\n  if$\n}\n\nFUNCTION {format.doi}\n{ doi empty$\n    { \"\" }\n    { new.block \"\\doi{\" doi * \"}\" * }\n  if$\n}\n\nFUNCTION {format.title}\n{ title empty$\n    { \"\" }\n    { title \"t\" change.case$ }\n  if$\n}\n\nFUNCTION {format.full.names}\n{'s :=\n  #1 'nameptr :=\n  s num.names$ 'numnames :=\n  numnames 'namesleft :=\n    { namesleft #0 > }\n    { s nameptr\n      \"{vv~}{ll}\" format.name$ 't :=\n      nameptr #1 >\n        {\n          namesleft #1 >\n            { \", \" * t * }\n            {\n              numnames #2 >\n                { \",\" * }\n                'skip$\n              if$\n              t \"others\" =\n                { \" et~al.\" * }\n                { \" and \" * t * }\n              if$\n            }\n          if$\n        }\n        't\n      if$\n      nameptr #1 + 'nameptr :=\n      namesleft #1 - 'namesleft :=\n    }\n  while$\n}\n\nFUNCTION {author.editor.full}\n{ author empty$\n    { editor empty$\n        { \"\" }\n        { editor format.full.names }\n      if$\n    }\n    { author format.full.names }\n  if$\n}\n\nFUNCTION {author.full}\n{ author empty$\n    { \"\" }\n    { author format.full.names }\n  if$\n}\n\nFUNCTION {editor.full}\n{ editor empty$\n    { \"\" }\n    { editor format.full.names }\n  if$\n}\n\nFUNCTION {make.full.names}\n{ type$ \"book\" =\n  type$ \"inbook\" =\n  or\n    'author.editor.full\n    { type$ \"proceedings\" =\n        'editor.full\n        'author.full\n      if$\n    }\n  if$\n}\n\nFUNCTION {output.bibitem}\n{ newline$\n  \"\\bibitem[\" write$\n  label write$\n  \")\" make.full.names duplicate$ short.list =\n     { pop$ }\n     { * }\n   if$\n  \"]{\" * write$\n  cite$ write$\n  \"}\" write$\n  newline$\n  \"\"\n  before.all 'output.state :=\n}\n\nFUNCTION {n.dashify}\n{ 't :=\n  \"\"\n    { t empty$ not }\n    { t #1 #1 substring$ \"-\" =\n        { t #1 #2 substring$ \"--\" = not\n            { \"--\" *\n              t #2 global.max$ substring$ 't :=\n            }\n            {   { t #1 #1 substring$ \"-\" = }\n                { \"-\" *\n                  t #2 global.max$ substring$ 't :=\n                }\n              while$\n            }\n          if$\n        }\n        { t #1 #1 substring$ *\n          t #2 global.max$ substring$ 't :=\n        }\n      if$\n    }\n  while$\n}\n\nFUNCTION {format.date}\n{ year duplicate$ empty$\n    { \"empty year in \" cite$ * warning$\n       pop$ \"\" }\n    'skip$\n  if$\n  month empty$\n    'skip$\n    { month\n      \" \" * swap$ *\n    }\n  if$\n  extra.label *\n}\n\nFUNCTION {format.btitle}\n{ title emphasize\n}\n\nFUNCTION {tie.or.space.connect}\n{ duplicate$ text.length$ #3 <\n    { \"~\" }\n    { \" \" }\n  if$\n  swap$ * *\n}\n\nFUNCTION {either.or.check}\n{ empty$\n    'pop$\n    { \"can't use both \" swap$ * \" fields in \" * cite$ * warning$ }\n  if$\n}\n\nFUNCTION {format.bvolume}\n{ volume empty$\n    { \"\" }\n    { \"volume\" volume tie.or.space.connect\n      series empty$\n        'skip$\n        { \" of \" * series emphasize * }\n      if$\n      \"volume and number\" number either.or.check\n    }\n  if$\n}\n\nFUNCTION {format.number.series}\n{ volume empty$\n    { number empty$\n        { series field.or.null }\n        { output.state mid.sentence =\n            { \"number\" }\n            { \"Number\" }\n          if$\n          number tie.or.space.connect\n          series empty$\n            { \"there's a number but no series in \" cite$ * warning$ }\n            { \" in \" * series * }\n          if$\n        }\n      if$\n    }\n    { \"\" }\n  if$\n}\n\nFUNCTION {format.edition}\n{ edition empty$\n    { \"\" }\n    { output.state mid.sentence =\n        { edition \"l\" change.case$ \" edition\" * }\n        { edition \"t\" change.case$ \" edition\" * }\n      if$\n    }\n  if$\n}\n\nINTEGERS { multiresult }\n\nFUNCTION {multi.page.check}\n{ 't :=\n  #0 'multiresult :=\n    { multiresult not\n      t empty$ not\n      and\n    }\n    { t #1 #1 substring$\n      duplicate$ \"-\" =\n      swap$ duplicate$ \",\" =\n      swap$ \"+\" =\n      or or\n        { #1 'multiresult := }\n        { t #2 global.max$ substring$ 't := }\n      if$\n    }\n  while$\n  multiresult\n}\n\nFUNCTION {format.pages}\n{ pages empty$\n    { \"\" }\n    { pages multi.page.check\n        { \"pp.\\ \" pages n.dashify tie.or.space.connect }\n        { \"pp.\\ \" pages tie.or.space.connect }\n      if$\n    }\n  if$\n}\n\nFUNCTION {format.eid}\n{ eid empty$\n    { \"\" }\n    { \"art.\" eid tie.or.space.connect }\n  if$\n}\n\nFUNCTION {format.vol.num.pages}\n{ volume field.or.null\n  number empty$\n    'skip$\n    { \"\\penalty0 (\" number * \")\" * *\n      volume empty$\n        { \"there's a number but no volume in \" cite$ * warning$ }\n        'skip$\n      if$\n    }\n  if$\n  pages empty$\n    'skip$\n    { duplicate$ empty$\n        { pop$ format.pages }\n        { \":\\penalty0 \" * pages n.dashify * }\n      if$\n    }\n  if$\n}\n\nFUNCTION {format.vol.num.eid}\n{ volume field.or.null\n  number empty$\n    'skip$\n    { \"\\penalty0 (\" number * \")\" * *\n      volume empty$\n        { \"there's a number but no volume in \" cite$ * warning$ }\n        'skip$\n      if$\n    }\n  if$\n  eid empty$\n    'skip$\n    { duplicate$ empty$\n        { pop$ format.eid }\n        { \":\\penalty0 \" * eid * }\n      if$\n    }\n  if$\n}\n\nFUNCTION {format.chapter.pages}\n{ chapter empty$\n    'format.pages\n    { type empty$\n        { \"chapter\" }\n        { type \"l\" change.case$ }\n      if$\n      chapter tie.or.space.connect\n      pages empty$\n        'skip$\n        { \", \" * format.pages * }\n      if$\n    }\n  if$\n}\n\nFUNCTION {format.in.ed.booktitle}\n{ booktitle empty$\n    { \"\" }\n    { editor empty$\n        { \"In \" booktitle emphasize * }\n        { \"In \" format.editors * \", \" * booktitle emphasize * }\n      if$\n    }\n  if$\n}\n\nFUNCTION {empty.misc.check}\n{ author empty$ title empty$ howpublished empty$\n  month empty$ year empty$ note empty$\n  and and and and and\n  key empty$ not and\n    { \"all relevant fields are empty in \" cite$ * warning$ }\n    'skip$\n  if$\n}\n\nFUNCTION {format.thesis.type}\n{ type empty$\n    'skip$\n    { pop$\n      type \"t\" change.case$\n    }\n  if$\n}\n\nFUNCTION {format.tr.number}\n{ type empty$\n    { \"Technical Report\" }\n    'type\n  if$\n  number empty$\n    { \"t\" change.case$ }\n    { number tie.or.space.connect }\n  if$\n}\n\nFUNCTION {format.article.crossref}\n{ key empty$\n    { journal empty$\n        { \"need key or journal for \" cite$ * \" to crossref \" * crossref *\n          warning$\n          \"\"\n        }\n        { \"In \\emph{\" journal * \"}\" * }\n      if$\n    }\n    { \"In \" }\n  if$\n  \" \\citet{\" * crossref * \"}\" *\n}\n\nFUNCTION {format.book.crossref}\n{ volume empty$\n    { \"empty volume in \" cite$ * \"'s crossref of \" * crossref * warning$\n      \"In \"\n    }\n    { \"Volume\" volume tie.or.space.connect\n      \" of \" *\n    }\n  if$\n  editor empty$\n  editor field.or.null author field.or.null =\n  or\n    { key empty$\n        { series empty$\n            { \"need editor, key, or series for \" cite$ * \" to crossref \" *\n              crossref * warning$\n              \"\" *\n            }\n            { \"\\emph{\" * series * \"}\" * }\n          if$\n        }\n        'skip$\n      if$\n    }\n    'skip$\n  if$\n  \" \\citet{\" * crossref * \"}\" *\n}\n\nFUNCTION {format.incoll.inproc.crossref}\n{ editor empty$\n  editor field.or.null author field.or.null =\n  or\n    { key empty$\n        { booktitle empty$\n            { \"need editor, key, or booktitle for \" cite$ * \" to crossref \" *\n              crossref * warning$\n              \"\"\n            }\n            { \"In \\emph{\" booktitle * \"}\" * }\n          if$\n        }\n        { \"In \" }\n      if$\n    }\n    { \"In \" }\n  if$\n  \" \\citet{\" * crossref * \"}\" *\n}\n\nFUNCTION {article}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  crossref missing$\n    { journal emphasize \"journal\" output.check\n      eid empty$\n        { format.vol.num.pages output }\n        { format.vol.num.eid output }\n      if$\n      format.date \"year\" output.check\n    }\n    { format.article.crossref output.nonnull\n      eid empty$\n        { format.pages output }\n        { format.eid output }\n      if$\n    }\n  if$\n  format.issn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {book}\n{ output.bibitem\n  author empty$\n    { format.editors \"author and editor\" output.check\n      editor format.key output\n    }\n    { format.authors output.nonnull\n      crossref missing$\n        { \"author and editor\" editor either.or.check }\n        'skip$\n      if$\n    }\n  if$\n  new.block\n  format.btitle \"title\" output.check\n  crossref missing$\n    { format.bvolume output\n      new.block\n      format.number.series output\n      new.sentence\n      publisher \"publisher\" output.check\n      address output\n    }\n    { new.block\n      format.book.crossref output.nonnull\n    }\n  if$\n  format.edition output\n  format.date \"year\" output.check\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {booklet}\n{ output.bibitem\n  format.authors output\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  howpublished address new.block.checkb\n  howpublished output\n  address output\n  format.date output\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {inbook}\n{ output.bibitem\n  author empty$\n    { format.editors \"author and editor\" output.check\n      editor format.key output\n    }\n    { format.authors output.nonnull\n      crossref missing$\n        { \"author and editor\" editor either.or.check }\n        'skip$\n      if$\n    }\n  if$\n  new.block\n  format.btitle \"title\" output.check\n  crossref missing$\n    { format.bvolume output\n      format.chapter.pages \"chapter and pages\" output.check\n      new.block\n      format.number.series output\n      new.sentence\n      publisher \"publisher\" output.check\n      address output\n    }\n    { format.chapter.pages \"chapter and pages\" output.check\n      new.block\n      format.book.crossref output.nonnull\n    }\n  if$\n  format.edition output\n  format.date \"year\" output.check\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {incollection}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  crossref missing$\n    { format.in.ed.booktitle \"booktitle\" output.check\n      format.bvolume output\n      format.number.series output\n      format.chapter.pages output\n      new.sentence\n      publisher \"publisher\" output.check\n      address output\n      format.edition output\n      format.date \"year\" output.check\n    }\n    { format.incoll.inproc.crossref output.nonnull\n      format.chapter.pages output\n    }\n  if$\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {inproceedings}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  crossref missing$\n    { format.in.ed.booktitle \"booktitle\" output.check\n      format.bvolume output\n      format.number.series output\n      format.pages output\n      address empty$\n        { organization publisher new.sentence.checkb\n          organization output\n          publisher output\n          format.date \"year\" output.check\n        }\n        { address output.nonnull\n          format.date \"year\" output.check\n          new.sentence\n          organization output\n          publisher output\n        }\n      if$\n    }\n    { format.incoll.inproc.crossref output.nonnull\n      format.pages output\n    }\n  if$\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {conference} { inproceedings }\n\nFUNCTION {manual}\n{ output.bibitem\n  format.authors output\n  author format.key output\n  new.block\n  format.btitle \"title\" output.check\n  organization address new.block.checkb\n  organization output\n  address output\n  format.edition output\n  format.date output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {mastersthesis}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  \"Master's thesis\" format.thesis.type output.nonnull\n  school \"school\" output.check\n  address output\n  format.date \"year\" output.check\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {misc}\n{ output.bibitem\n  format.authors output\n  author format.key output\n  title howpublished new.block.checkb\n  format.title output\n  howpublished new.block.checka\n  howpublished output\n  format.date output\n  format.issn output\n  format.url output\n  new.block\n  note output\n  fin.entry\n  empty.misc.check\n}\n\nFUNCTION {phdthesis}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.btitle \"title\" output.check\n  new.block\n  \"PhD thesis\" format.thesis.type output.nonnull\n  school \"school\" output.check\n  address output\n  format.date \"year\" output.check\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {proceedings}\n{ output.bibitem\n  format.editors output\n  editor format.key output\n  new.block\n  format.btitle \"title\" output.check\n  format.bvolume output\n  format.number.series output\n  address output\n  format.date \"year\" output.check\n  new.sentence\n  organization output\n  publisher output\n  format.isbn output\n  format.doi output\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {techreport}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  format.tr.number output.nonnull\n  institution \"institution\" output.check\n  address output\n  format.date \"year\" output.check\n  format.url output\n  new.block\n  note output\n  fin.entry\n}\n\nFUNCTION {unpublished}\n{ output.bibitem\n  format.authors \"author\" output.check\n  author format.key output\n  new.block\n  format.title \"title\" output.check\n  new.block\n  note \"note\" output.check\n  format.date output\n  format.url output\n  fin.entry\n}\n\nFUNCTION {default.type} { misc }\n\n\nMACRO {jan} {\"January\"}\n\nMACRO {feb} {\"February\"}\n\nMACRO {mar} {\"March\"}\n\nMACRO {apr} {\"April\"}\n\nMACRO {may} {\"May\"}\n\nMACRO {jun} {\"June\"}\n\nMACRO {jul} {\"July\"}\n\nMACRO {aug} {\"August\"}\n\nMACRO {sep} {\"September\"}\n\nMACRO {oct} {\"October\"}\n\nMACRO {nov} {\"November\"}\n\nMACRO {dec} {\"December\"}\n\n\n\nMACRO {acmcs} {\"ACM Computing Surveys\"}\n\nMACRO {acta} {\"Acta Informatica\"}\n\nMACRO {cacm} {\"Communications of the ACM\"}\n\nMACRO {ibmjrd} {\"IBM Journal of Research and Development\"}\n\nMACRO {ibmsj} {\"IBM Systems Journal\"}\n\nMACRO {ieeese} {\"IEEE Transactions on Software Engineering\"}\n\nMACRO {ieeetc} {\"IEEE Transactions on Computers\"}\n\nMACRO {ieeetcad}\n {\"IEEE Transactions on Computer-Aided Design of Integrated Circuits\"}\n\nMACRO {ipl} {\"Information Processing Letters\"}\n\nMACRO {jacm} {\"Journal of the ACM\"}\n\nMACRO {jcss} {\"Journal of Computer and System Sciences\"}\n\nMACRO {scp} {\"Science of Computer Programming\"}\n\nMACRO {sicomp} {\"SIAM Journal on Computing\"}\n\nMACRO {tocs} {\"ACM Transactions on Computer Systems\"}\n\nMACRO {tods} {\"ACM Transactions on Database Systems\"}\n\nMACRO {tog} {\"ACM Transactions on Graphics\"}\n\nMACRO {toms} {\"ACM Transactions on Mathematical Software\"}\n\nMACRO {toois} {\"ACM Transactions on Office Information Systems\"}\n\nMACRO {toplas} {\"ACM Transactions on Programming Languages and Systems\"}\n\nMACRO {tcs} {\"Theoretical Computer Science\"}\n\n\nREAD\n\nFUNCTION {sortify}\n{ purify$\n  \"l\" change.case$\n}\n\nINTEGERS { len }\n\nFUNCTION {chop.word}\n{ 's :=\n  'len :=\n  s #1 len substring$ =\n    { s len #1 + global.max$ substring$ }\n    's\n  if$\n}\n\nFUNCTION {format.lab.names}\n{ 's :=\n  s #1 \"{vv~}{ll}\" format.name$\n  s num.names$ duplicate$\n  #2 >\n    { pop$ \" et~al.\" * }\n    { #2 <\n        'skip$\n        { s #2 \"{ff }{vv }{ll}{ jj}\" format.name$ \"others\" =\n            { \" et~al.\" * }\n            { \" \\& \" * s #2 \"{vv~}{ll}\" format.name$ * }\n          if$\n        }\n      if$\n    }\n  if$\n}\n\nFUNCTION {author.key.label}\n{ author empty$\n    { key empty$\n        { cite$ #1 #3 substring$ }\n        'key\n      if$\n    }\n    { author format.lab.names }\n  if$\n}\n\nFUNCTION {author.editor.key.label}\n{ author empty$\n    { editor empty$\n        { key empty$\n            { cite$ #1 #3 substring$ }\n            'key\n          if$\n        }\n        { editor format.lab.names }\n      if$\n    }\n    { author format.lab.names }\n  if$\n}\n\nFUNCTION {author.key.organization.label}\n{ author empty$\n    { key empty$\n        { organization empty$\n            { cite$ #1 #3 substring$ }\n            { \"The \" #4 organization chop.word #3 text.prefix$ }\n          if$\n        }\n        'key\n      if$\n    }\n    { author format.lab.names }\n  if$\n}\n\nFUNCTION {editor.key.organization.label}\n{ editor empty$\n    { key empty$\n        { organization empty$\n            { cite$ #1 #3 substring$ }\n            { \"The \" #4 organization chop.word #3 text.prefix$ }\n          if$\n        }\n        'key\n      if$\n    }\n    { editor format.lab.names }\n  if$\n}\n\nFUNCTION {calc.short.authors}\n{ type$ \"book\" =\n  type$ \"inbook\" =\n  or\n    'author.editor.key.label\n    { type$ \"proceedings\" =\n        'editor.key.organization.label\n        { type$ \"manual\" =\n            'author.key.organization.label\n            'author.key.label\n          if$\n        }\n      if$\n    }\n  if$\n  'short.list :=\n}\n\nFUNCTION {calc.label}\n{ calc.short.authors\n  short.list\n  \"(\"\n  *\n  year duplicate$ empty$\n  short.list key field.or.null = or\n     { pop$ \"\" }\n     'skip$\n  if$\n  *\n  'label :=\n}\n\nFUNCTION {sort.format.names}\n{ 's :=\n  #1 'nameptr :=\n  \"\"\n  s num.names$ 'numnames :=\n  numnames 'namesleft :=\n    { namesleft #0 > }\n    {\n      s nameptr \"{vv{ } }{ll{ }}{  ff{ }}{  jj{ }}\" format.name$ 't :=\n      nameptr #1 >\n        {\n          \"   \"  *\n          namesleft #1 = t \"others\" = and\n            { \"zzzzz\" * }\n            { numnames #2 > nameptr #2 = and\n                { \"zz\" * year field.or.null * \"   \" * }\n                'skip$\n              if$\n              t sortify *\n            }\n          if$\n        }\n        { t sortify * }\n      if$\n      nameptr #1 + 'nameptr :=\n      namesleft #1 - 'namesleft :=\n    }\n  while$\n}\n\nFUNCTION {sort.format.title}\n{ 't :=\n  \"A \" #2\n    \"An \" #3\n      \"The \" #4 t chop.word\n    chop.word\n  chop.word\n  sortify\n  #1 global.max$ substring$\n}\n\nFUNCTION {author.sort}\n{ author empty$\n    { key empty$\n        { \"to sort, need author or key in \" cite$ * warning$\n          \"\"\n        }\n        { key sortify }\n      if$\n    }\n    { author sort.format.names }\n  if$\n}\n\nFUNCTION {author.editor.sort}\n{ author empty$\n    { editor empty$\n        { key empty$\n            { \"to sort, need author, editor, or key in \" cite$ * warning$\n              \"\"\n            }\n            { key sortify }\n          if$\n        }\n        { editor sort.format.names }\n      if$\n    }\n    { author sort.format.names }\n  if$\n}\n\nFUNCTION {author.organization.sort}\n{ author empty$\n    { organization empty$\n        { key empty$\n            { \"to sort, need author, organization, or key in \" cite$ * warning$\n              \"\"\n            }\n            { key sortify }\n          if$\n        }\n        { \"The \" #4 organization chop.word sortify }\n      if$\n    }\n    { author sort.format.names }\n  if$\n}\n\nFUNCTION {editor.organization.sort}\n{ editor empty$\n    { organization empty$\n        { key empty$\n            { \"to sort, need editor, organization, or key in \" cite$ * warning$\n              \"\"\n            }\n            { key sortify }\n          if$\n        }\n        { \"The \" #4 organization chop.word sortify }\n      if$\n    }\n    { editor sort.format.names }\n  if$\n}\n\n\nFUNCTION {presort}\n{ calc.label\n  label sortify\n  \"    \"\n  *\n  type$ \"book\" =\n  type$ \"inbook\" =\n  or\n    'author.editor.sort\n    { type$ \"proceedings\" =\n        'editor.organization.sort\n        { type$ \"manual\" =\n            'author.organization.sort\n            'author.sort\n          if$\n        }\n      if$\n    }\n  if$\n  \"    \"\n  *\n  year field.or.null sortify\n  *\n  \"    \"\n  *\n  cite$\n  *\n  #1 entry.max$ substring$\n  'sort.label :=\n  sort.label *\n  #1 entry.max$ substring$\n  'sort.key$ :=\n}\n\nITERATE {presort}\n\nSORT\n\nSTRINGS { longest.label last.label next.extra }\n\nINTEGERS { longest.label.width last.extra.num number.label }\n\nFUNCTION {initialize.longest.label}\n{ \"\" 'longest.label :=\n  #0 int.to.chr$ 'last.label :=\n  \"\" 'next.extra :=\n  #0 'longest.label.width :=\n  #0 'last.extra.num :=\n  #0 'number.label :=\n}\n\nFUNCTION {forward.pass}\n{ last.label label =\n    { last.extra.num #1 + 'last.extra.num :=\n      last.extra.num int.to.chr$ 'extra.label :=\n    }\n    { \"a\" chr.to.int$ 'last.extra.num :=\n      \"\" 'extra.label :=\n      label 'last.label :=\n    }\n  if$\n  number.label #1 + 'number.label :=\n}\n\nFUNCTION {reverse.pass}\n{ next.extra \"b\" =\n    { \"a\" 'extra.label := }\n    'skip$\n  if$\n  extra.label 'next.extra :=\n  extra.label\n  duplicate$ empty$\n    'skip$\n    { \"{\\natexlab{\" swap$ * \"}}\" * }\n  if$\n  'extra.label :=\n  label extra.label * 'label :=\n}\n\nEXECUTE {initialize.longest.label}\n\nITERATE {forward.pass}\n\nREVERSE {reverse.pass}\n\nFUNCTION {bib.sort.order}\n{ sort.label  'sort.key$ :=\n}\n\nITERATE {bib.sort.order}\n\nSORT\n\nFUNCTION {begin.bib}\n{   preamble$ empty$\n    'skip$\n    { preamble$ write$ newline$ }\n  if$\n  \"\\begin{thebibliography}{\" number.label int.to.str$ * \"}\" *\n  write$ newline$\n  \"\\providecommand{\\natexlab}[1]{#1}\"\n  write$ newline$\n  \"\\providecommand{\\url}[1]{\\texttt{#1}}\"\n  write$ newline$\n  \"\\expandafter\\ifx\\csname urlstyle\\endcsname\\relax\"\n  write$ newline$\n  \"  \\providecommand{\\doi}[1]{doi: #1}\\else\"\n  write$ newline$\n  \"  \\providecommand{\\doi}{doi: \\begingroup \\urlstyle{rm}\\Url}\\fi\"\n  write$ newline$\n}\n\nEXECUTE {begin.bib}\n\nEXECUTE {init.state.consts}\n\nITERATE {call.type$}\n\nFUNCTION {end.bib}\n{ newline$\n  \"\\end{thebibliography}\" write$ newline$\n}\n\nEXECUTE {end.bib}\n"
  },
  {
    "path": "paper/iclr2018_conference.sty",
    "content": "%%%% ICLR Macros (LaTex)\n%%%% Adapted by Hugo Larochelle from the NIPS stylefile Macros\n%%%% Style File\n%%%% Dec 12, 1990   Rev Aug 14, 1991; Sept, 1995; April, 1997; April, 1999; October 2014\n\n% This file can be used with Latex2e whether running in main mode, or\n% 2.09 compatibility mode.\n%\n% If using main mode, you need to include the commands\n%             \\documentclass{article}\n%             \\usepackage{iclr14submit_e,times}\n%\n\n% Change the overall width of the page.  If these parameters are\n%       changed, they will require corresponding changes in the\n%       maketitle section.\n%\n\\usepackage{eso-pic} % used by \\AddToShipoutPicture\n\\RequirePackage{fancyhdr}\n\\RequirePackage{natbib}\n\n% modification to natbib citations\n\\setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}}\n\n\\renewcommand{\\topfraction}{0.95}   % let figure take up nearly whole page\n\\renewcommand{\\textfraction}{0.05}  % let figure take up nearly whole page\n\n% Define iclrfinal, set to true if iclrfinalcopy is defined\n\\newif\\ificlrfinal\n\\iclrfinalfalse\n\\def\\iclrfinalcopy{\\iclrfinaltrue}\n\\font\\iclrtenhv  = phvb at 8pt\n\n% Specify the dimensions of each page\n\n\\setlength{\\paperheight}{11in}\n\\setlength{\\paperwidth}{8.5in}\n\n\n\\oddsidemargin .5in    %   Note \\oddsidemargin = \\evensidemargin\n\\evensidemargin .5in\n\\marginparwidth 0.07 true in\n%\\marginparwidth 0.75 true in\n%\\topmargin 0 true pt           % Nominal distance from top of page to top of\n%\\topmargin 0.125in\n\\topmargin -0.625in\n\\addtolength{\\headsep}{0.25in}\n\\textheight 9.0 true in       % Height of text (including footnotes & figures)\n\\textwidth 5.5 true in        % Width of text line.\n\\widowpenalty=10000\n\\clubpenalty=10000\n\n% \\thispagestyle{empty}        \\pagestyle{empty}\n\\flushbottom \\sloppy\n\n% We're never going to need a table of contents, so just flush it to\n% save space --- suggested by drstrip@sandia-2\n\\def\\addcontentsline#1#2#3{}\n\n% Title stuff, taken from deproc.\n\\def\\maketitle{\\par\n\\begingroup\n   \\def\\thefootnote{\\fnsymbol{footnote}}\n   \\def\\@makefnmark{\\hbox to 0pt{$^{\\@thefnmark}$\\hss}} % for perfect author\n                                                        % name centering\n%   The footnote-mark was overlapping the footnote-text,\n%   added the following to fix this problem               (MK)\n   \\long\\def\\@makefntext##1{\\parindent 1em\\noindent\n                            \\hbox to1.8em{\\hss $\\m@th ^{\\@thefnmark}$}##1}\n   \\@maketitle \\@thanks\n\\endgroup\n\\setcounter{footnote}{0}\n\\let\\maketitle\\relax \\let\\@maketitle\\relax\n\\gdef\\@thanks{}\\gdef\\@author{}\\gdef\\@title{}\\let\\thanks\\relax}\n\n% The toptitlebar has been raised to top-justify the first page\n\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyhead{}\n\n% Title (includes both anonimized and non-anonimized versions)\n\\def\\@maketitle{\\vbox{\\hsize\\textwidth\n%\\linewidth\\hsize \\vskip 0.1in \\toptitlebar \\centering\n{\\LARGE\\sc \\@title\\par}\n%\\bottomtitlebar % \\vskip 0.1in %  minus\n\\ificlrfinal\n    \\lhead{Published as a conference paper at ICLR 2018}\n    \\def\\And{\\end{tabular}\\hfil\\linebreak[0]\\hfil\n            \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}\\ignorespaces}%\n  \\def\\AND{\\end{tabular}\\hfil\\linebreak[4]\\hfil\n            \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}\\ignorespaces}%\n    \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}\\@author\\end{tabular}%\n\\else\n       \\lhead{Under review as a conference paper at ICLR 2018}\n   \\def\\And{\\end{tabular}\\hfil\\linebreak[0]\\hfil\n            \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}\\ignorespaces}%\n  \\def\\AND{\\end{tabular}\\hfil\\linebreak[4]\\hfil\n            \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}\\ignorespaces}%\n    \\begin{tabular}[t]{l}\\bf\\rule{\\z@}{24pt}Anonymous authors\\\\Paper under double-blind review\\end{tabular}%\n\\fi\n\\vskip 0.3in minus 0.1in}}\n\n\\renewenvironment{abstract}{\\vskip.075in\\centerline{\\large\\sc\nAbstract}\\vspace{0.5ex}\\begin{quote}}{\\par\\end{quote}\\vskip 1ex}\n\n% sections with less space\n\\def\\section{\\@startsection {section}{1}{\\z@}{-2.0ex plus\n    -0.5ex minus -.2ex}{1.5ex plus 0.3ex\nminus0.2ex}{\\large\\sc\\raggedright}}\n\n\\def\\subsection{\\@startsection{subsection}{2}{\\z@}{-1.8ex plus\n-0.5ex minus -.2ex}{0.8ex plus .2ex}{\\normalsize\\sc\\raggedright}}\n\\def\\subsubsection{\\@startsection{subsubsection}{3}{\\z@}{-1.5ex\nplus      -0.5ex minus -.2ex}{0.5ex plus\n.2ex}{\\normalsize\\sc\\raggedright}}\n\\def\\paragraph{\\@startsection{paragraph}{4}{\\z@}{1.5ex plus\n0.5ex minus .2ex}{-1em}{\\normalsize\\bf}}\n\\def\\subparagraph{\\@startsection{subparagraph}{5}{\\z@}{1.5ex plus\n  0.5ex minus .2ex}{-1em}{\\normalsize\\sc}}\n\\def\\subsubsubsection{\\vskip\n5pt{\\noindent\\normalsize\\rm\\raggedright}}\n\n\n% Footnotes\n\\footnotesep 6.65pt %\n\\skip\\footins 9pt plus 4pt minus 2pt\n\\def\\footnoterule{\\kern-3pt \\hrule width 12pc \\kern 2.6pt }\n\\setcounter{footnote}{0}\n\n% Lists and paragraphs\n\\parindent 0pt\n\\topsep 4pt plus 1pt minus 2pt\n\\partopsep 1pt plus 0.5pt minus 0.5pt\n\\itemsep 2pt plus 1pt minus 0.5pt\n\\parsep 2pt plus 1pt minus 0.5pt\n\\parskip .5pc\n\n\n%\\leftmargin2em\n\\leftmargin3pc\n\\leftmargini\\leftmargin \\leftmarginii 2em\n\\leftmarginiii 1.5em \\leftmarginiv 1.0em \\leftmarginv .5em\n\n%\\labelsep \\labelsep 5pt\n\n\\def\\@listi{\\leftmargin\\leftmargini}\n\\def\\@listii{\\leftmargin\\leftmarginii\n   \\labelwidth\\leftmarginii\\advance\\labelwidth-\\labelsep\n   \\topsep 2pt plus 1pt minus 0.5pt\n   \\parsep 1pt plus 0.5pt minus 0.5pt\n   \\itemsep \\parsep}\n\\def\\@listiii{\\leftmargin\\leftmarginiii\n    \\labelwidth\\leftmarginiii\\advance\\labelwidth-\\labelsep\n    \\topsep 1pt plus 0.5pt minus 0.5pt\n    \\parsep \\z@ \\partopsep 0.5pt plus 0pt minus 0.5pt\n    \\itemsep \\topsep}\n\\def\\@listiv{\\leftmargin\\leftmarginiv\n     \\labelwidth\\leftmarginiv\\advance\\labelwidth-\\labelsep}\n\\def\\@listv{\\leftmargin\\leftmarginv\n     \\labelwidth\\leftmarginv\\advance\\labelwidth-\\labelsep}\n\\def\\@listvi{\\leftmargin\\leftmarginvi\n     \\labelwidth\\leftmarginvi\\advance\\labelwidth-\\labelsep}\n\n\\abovedisplayskip 7pt plus2pt minus5pt%\n\\belowdisplayskip \\abovedisplayskip\n\\abovedisplayshortskip  0pt plus3pt%\n\\belowdisplayshortskip  4pt plus3pt minus3pt%\n\n% Less leading in most fonts (due to the narrow columns)\n% The choices were between 1-pt and 1.5-pt leading\n%\\def\\@normalsize{\\@setsize\\normalsize{11pt}\\xpt\\@xpt} % got rid of @ (MK)\n\\def\\normalsize{\\@setsize\\normalsize{11pt}\\xpt\\@xpt}\n\\def\\small{\\@setsize\\small{10pt}\\ixpt\\@ixpt}\n\\def\\footnotesize{\\@setsize\\footnotesize{10pt}\\ixpt\\@ixpt}\n\\def\\scriptsize{\\@setsize\\scriptsize{8pt}\\viipt\\@viipt}\n\\def\\tiny{\\@setsize\\tiny{7pt}\\vipt\\@vipt}\n\\def\\large{\\@setsize\\large{14pt}\\xiipt\\@xiipt}\n\\def\\Large{\\@setsize\\Large{16pt}\\xivpt\\@xivpt}\n\\def\\LARGE{\\@setsize\\LARGE{20pt}\\xviipt\\@xviipt}\n\\def\\huge{\\@setsize\\huge{23pt}\\xxpt\\@xxpt}\n\\def\\Huge{\\@setsize\\Huge{28pt}\\xxvpt\\@xxvpt}\n\n\\def\\toptitlebar{\\hrule height4pt\\vskip .25in\\vskip-\\parskip}\n\n\\def\\bottomtitlebar{\\vskip .29in\\vskip-\\parskip\\hrule height1pt\\vskip\n.09in} %\n%Reduced second vskip to compensate for adding the strut in \\@author\n\n\n%% % Vertical Ruler\n%% % This code is, largely, from the CVPR 2010 conference style file\n%% % ----- define vruler\n%% \\makeatletter\n%% \\newbox\\iclrrulerbox\n%% \\newcount\\iclrrulercount\n%% \\newdimen\\iclrruleroffset\n%% \\newdimen\\cv@lineheight\n%% \\newdimen\\cv@boxheight\n%% \\newbox\\cv@tmpbox\n%% \\newcount\\cv@refno\n%% \\newcount\\cv@tot\n%% % NUMBER with left flushed zeros  \\fillzeros[<WIDTH>]<NUMBER>\n%% \\newcount\\cv@tmpc@ \\newcount\\cv@tmpc\n%% \\def\\fillzeros[#1]#2{\\cv@tmpc@=#2\\relax\\ifnum\\cv@tmpc@<0\\cv@tmpc@=-\\cv@tmpc@\\fi\n%% \\cv@tmpc=1 %\n%% \\loop\\ifnum\\cv@tmpc@<10 \\else \\divide\\cv@tmpc@ by 10 \\advance\\cv@tmpc by 1 \\fi\n%%    \\ifnum\\cv@tmpc@=10\\relax\\cv@tmpc@=11\\relax\\fi \\ifnum\\cv@tmpc@>10 \\repeat\n%% \\ifnum#2<0\\advance\\cv@tmpc1\\relax-\\fi\n%% \\loop\\ifnum\\cv@tmpc<#1\\relax0\\advance\\cv@tmpc1\\relax\\fi \\ifnum\\cv@tmpc<#1 \\repeat\n%% \\cv@tmpc@=#2\\relax\\ifnum\\cv@tmpc@<0\\cv@tmpc@=-\\cv@tmpc@\\fi \\relax\\the\\cv@tmpc@}%\n%% % \\makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]\n%% \\def\\makevruler[#1][#2][#3][#4][#5]{\\begingroup\\offinterlineskip\n%% \\textheight=#5\\vbadness=10000\\vfuzz=120ex\\overfullrule=0pt%\n%% \\global\\setbox\\iclrrulerbox=\\vbox to \\textheight{%\n%% {\\parskip=0pt\\hfuzz=150em\\cv@boxheight=\\textheight\n%% \\cv@lineheight=#1\\global\\iclrrulercount=#2%\n%% \\cv@tot\\cv@boxheight\\divide\\cv@tot\\cv@lineheight\\advance\\cv@tot2%\n%% \\cv@refno1\\vskip-\\cv@lineheight\\vskip1ex%\n%% \\loop\\setbox\\cv@tmpbox=\\hbox to0cm{{\\iclrtenhv\\hfil\\fillzeros[#4]\\iclrrulercount}}%\n%% \\ht\\cv@tmpbox\\cv@lineheight\\dp\\cv@tmpbox0pt\\box\\cv@tmpbox\\break\n%% \\advance\\cv@refno1\\global\\advance\\iclrrulercount#3\\relax\n%% \\ifnum\\cv@refno<\\cv@tot\\repeat}}\\endgroup}%\n%% \\makeatother\n%% % ----- end of vruler\n\n%% % \\makevruler[<SCALE>][<INITIAL_COUNT>][<STEP>][<DIGITS>][<HEIGHT>]\n%% \\def\\iclrruler#1{\\makevruler[12pt][#1][1][3][0.993\\textheight]\\usebox{\\iclrrulerbox}}\n%% \\AddToShipoutPicture{%\n%% \\ificlrfinal\\else\n%% \\iclrruleroffset=\\textheight\n%% \\advance\\iclrruleroffset by -3.7pt\n%%   \\color[rgb]{.7,.7,.7}\n%%   \\AtTextUpperLeft{%\n%%     \\put(\\LenToUnit{-35pt},\\LenToUnit{-\\iclrruleroffset}){%left ruler\n%%       \\iclrruler{\\iclrrulercount}}\n%%   }\n%% \\fi\n%% }\n%%% To add a vertical bar on the side\n%\\AddToShipoutPicture{\n%\\AtTextLowerLeft{\n%\\hspace*{-1.8cm}\n%\\colorbox[rgb]{0.7,0.7,0.7}{\\small \\parbox[b][\\textheight]{0.1cm}{}}}\n%}\n\n"
  },
  {
    "path": "paper/natbib.sty",
    "content": "%%\n%% This is file `natbib.sty',\n%% generated with the docstrip utility.\n%%\n%% The original source files were:\n%%\n%% natbib.dtx  (with options: `package,all')\n%% =============================================\n%% IMPORTANT NOTICE:\n%% \n%% This program can be redistributed and/or modified under the terms\n%% of the LaTeX Project Public License Distributed from CTAN\n%% archives in directory macros/latex/base/lppl.txt; either\n%% version 1 of the License, or any later version.\n%% \n%% This is a generated file.\n%% It may not be distributed without the original source file natbib.dtx.\n%% \n%% Full documentation can be obtained by LaTeXing that original file.\n%% Only a few abbreviated comments remain here to describe the usage.\n%% =============================================\n%% Copyright 1993-2009 Patrick W Daly\n%% Max-Planck-Institut f\\\"ur Sonnensystemforschung\n%% Max-Planck-Str. 2\n%% D-37191 Katlenburg-Lindau\n%% Germany\n%% E-mail: daly@mps.mpg.de\n\\NeedsTeXFormat{LaTeX2e}[1995/06/01]\n\\ProvidesPackage{natbib}\n        [2009/07/16 8.31 (PWD, AO)]\n\n % This package reimplements the LaTeX \\cite command to be used for various\n % citation styles, both author-year and numerical. It accepts BibTeX\n % output intended for many other packages, and therefore acts as a\n % general, all-purpose citation-style interface.\n %\n % With standard numerical .bst files, only numerical citations are\n % possible. With an author-year .bst file, both numerical and\n % author-year citations are possible.\n %\n % If author-year citations are selected, \\bibitem must have one of the\n %   following forms:\n %   \\bibitem[Jones et al.(1990)]{key}...\n %   \\bibitem[Jones et al.(1990)Jones, Baker, and Williams]{key}...\n %   \\bibitem[Jones et al., 1990]{key}...\n %   \\bibitem[\\protect\\citeauthoryear{Jones, Baker, and Williams}{Jones\n %       et al.}{1990}]{key}...\n %   \\bibitem[\\protect\\citeauthoryear{Jones et al.}{1990}]{key}...\n %   \\bibitem[\\protect\\astroncite{Jones et al.}{1990}]{key}...\n %   \\bibitem[\\protect\\citename{Jones et al., }1990]{key}...\n %   \\harvarditem[Jones et al.]{Jones, Baker, and Williams}{1990}{key}...\n %\n % This is either to be made up manually, or to be generated by an\n % appropriate .bst file with BibTeX.\n %                            Author-year mode     ||   Numerical mode\n % Then, \\citet{key}  ==>>  Jones et al. (1990)    ||   Jones et al. [21]\n %       \\citep{key}  ==>> (Jones et al., 1990)    ||   [21]\n % Multiple citations as normal:\n % \\citep{key1,key2}  ==>> (Jones et al., 1990; Smith, 1989) || [21,24]\n %                           or  (Jones et al., 1990, 1991)  || [21,24]\n %                           or  (Jones et al., 1990a,b)     || [21,24]\n % \\cite{key} is the equivalent of \\citet{key} in author-year mode\n %                         and  of \\citep{key} in numerical mode\n % Full author lists may be forced with \\citet* or \\citep*, e.g.\n %       \\citep*{key}      ==>> (Jones, Baker, and Williams, 1990)\n % Optional notes as:\n %   \\citep[chap. 2]{key}    ==>> (Jones et al., 1990, chap. 2)\n %   \\citep[e.g.,][]{key}    ==>> (e.g., Jones et al., 1990)\n %   \\citep[see][pg. 34]{key}==>> (see Jones et al., 1990, pg. 34)\n %  (Note: in standard LaTeX, only one note is allowed, after the ref.\n %   Here, one note is like the standard, two make pre- and post-notes.)\n %   \\citealt{key}          ==>> Jones et al. 1990\n %   \\citealt*{key}         ==>> Jones, Baker, and Williams 1990\n %   \\citealp{key}          ==>> Jones et al., 1990\n %   \\citealp*{key}         ==>> Jones, Baker, and Williams, 1990\n % Additional citation possibilities (both author-year and numerical modes)\n %   \\citeauthor{key}       ==>> Jones et al.\n %   \\citeauthor*{key}      ==>> Jones, Baker, and Williams\n %   \\citeyear{key}         ==>> 1990\n %   \\citeyearpar{key}      ==>> (1990)\n %   \\citetext{priv. comm.} ==>> (priv. comm.)\n %   \\citenum{key}          ==>> 11 [non-superscripted]\n % Note: full author lists depends on whether the bib style supports them;\n %       if not, the abbreviated list is printed even when full requested.\n %\n % For names like della Robbia at the start of a sentence, use\n %   \\Citet{dRob98}         ==>> Della Robbia (1998)\n %   \\Citep{dRob98}         ==>> (Della Robbia, 1998)\n %   \\Citeauthor{dRob98}    ==>> Della Robbia\n %\n %\n % Citation aliasing is achieved with\n %   \\defcitealias{key}{text}\n %   \\citetalias{key}  ==>> text\n %   \\citepalias{key}  ==>> (text)\n %\n % Defining the citation mode and punctual (citation style)\n %   \\setcitestyle{<comma-separated list of keywords, same\n %     as the package options>}\n % Example: \\setcitestyle{square,semicolon}\n % Alternatively:\n % Use \\bibpunct with 6 mandatory arguments:\n %    1. opening bracket for citation\n %    2. closing bracket\n %    3. citation separator (for multiple citations in one \\cite)\n %    4. the letter n for numerical styles, s for superscripts\n %        else anything for author-year\n %    5. punctuation between authors and date\n %    6. punctuation between years (or numbers) when common authors missing\n % One optional argument is the character coming before post-notes. It\n %   appears in square braces before all other arguments. May be left off.\n % Example (and default) \\bibpunct[, ]{(}{)}{;}{a}{,}{,}\n %\n % To make this automatic for a given bib style, named newbib, say, make\n % a local configuration file, natbib.cfg, with the definition\n %   \\newcommand{\\bibstyle@newbib}{\\bibpunct...}\n % Then the \\bibliographystyle{newbib} will cause \\bibstyle@newbib to\n % be called on THE NEXT LATEX RUN (via the aux file).\n %\n % Such preprogrammed definitions may be invoked anywhere in the text\n %  by calling \\citestyle{newbib}. This is only useful if the style specified\n %  differs from that in \\bibliographystyle.\n %\n % With \\citeindextrue and \\citeindexfalse, one can control whether the\n % \\cite commands make an automatic entry of the citation in the .idx\n % indexing file. For this, \\makeindex must also be given in the preamble.\n %\n % Package Options: (for selecting punctuation)\n %   round  -  round parentheses are used (default)\n %   square -  square brackets are used   [option]\n %   curly  -  curly braces are used      {option}\n %   angle  -  angle brackets are used    <option>\n %   semicolon  -  multiple citations separated by semi-colon (default)\n %   colon  - same as semicolon, an earlier confusion\n %   comma  -  separated by comma\n %   authoryear - selects author-year citations (default)\n %   numbers-  selects numerical citations\n %   super  -  numerical citations as superscripts\n %   sort   -  sorts multiple citations according to order in ref. list\n %   sort&compress   -  like sort, but also compresses numerical citations\n %   compress - compresses without sorting\n %   longnamesfirst  -  makes first citation full author list\n %   sectionbib - puts bibliography in a \\section* instead of \\chapter*\n %   merge - allows the citation key to have a * prefix,\n %           signifying to merge its reference with that of the previous citation.\n %   elide - if references are merged, repeated portions of later ones may be removed.\n %   mcite - recognizes and ignores the * prefix for merging.\n % Punctuation so selected dominates over any predefined ones.\n % Package options are called as, e.g.\n %        \\usepackage[square,comma]{natbib}\n % LaTeX the source file natbib.dtx to obtain more details\n % or the file natnotes.tex for a brief reference sheet.\n %-----------------------------------------------------------\n\\providecommand\\@ifxundefined[1]{%\n \\ifx#1\\@undefined\\expandafter\\@firstoftwo\\else\\expandafter\\@secondoftwo\\fi\n}%\n\\providecommand\\@ifnum[1]{%\n \\ifnum#1\\expandafter\\@firstoftwo\\else\\expandafter\\@secondoftwo\\fi\n}%\n\\providecommand\\@ifx[1]{%\n \\ifx#1\\expandafter\\@firstoftwo\\else\\expandafter\\@secondoftwo\\fi\n}%\n\\providecommand\\appdef[2]{%\n \\toks@\\expandafter{#1}\\@temptokena{#2}%\n \\edef#1{\\the\\toks@\\the\\@temptokena}%\n}%\n\\@ifclassloaded{agu2001}{\\PackageError{natbib}\n  {The agu2001 class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n\\@ifclassloaded{agutex}{\\PackageError{natbib}\n  {The AGUTeX class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n\\@ifclassloaded{aguplus}{\\PackageError{natbib}\n  {The aguplus class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n\\@ifclassloaded{nlinproc}{\\PackageError{natbib}\n  {The nlinproc class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n\\@ifclassloaded{egs}{\\PackageError{natbib}\n  {The egs class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n\\@ifclassloaded{egu}{\\PackageError{natbib}\n  {The egu class already includes natbib coding,\\MessageBreak\n   so you should not add it explicitly}\n  {Type <Return> for now, but then later remove\\MessageBreak\n   the command \\protect\\usepackage{natbib} from the document}\n  \\endinput}{}\n % Define citation punctuation for some author-year styles\n % One may add and delete at this point\n % Or put additions into local configuration file natbib.cfg\n\\newcommand\\bibstyle@chicago{\\bibpunct{(}{)}{;}{a}{,}{,}}\n\\newcommand\\bibstyle@named{\\bibpunct{[}{]}{;}{a}{,}{,}}\n\\newcommand\\bibstyle@agu{\\bibpunct{[}{]}{;}{a}{,}{,~}}%Amer. Geophys. Union\n\\newcommand\\bibstyle@copernicus{\\bibpunct{(}{)}{;}{a}{,}{,}}%Copernicus Publications\n\\let\\bibstyle@egu=\\bibstyle@copernicus\n\\let\\bibstyle@egs=\\bibstyle@copernicus\n\\newcommand\\bibstyle@agsm{\\bibpunct{(}{)}{,}{a}{}{,}\\gdef\\harvardand{\\&}}\n\\newcommand\\bibstyle@kluwer{\\bibpunct{(}{)}{,}{a}{}{,}\\gdef\\harvardand{\\&}}\n\\newcommand\\bibstyle@dcu{\\bibpunct{(}{)}{;}{a}{;}{,}\\gdef\\harvardand{and}}\n\\newcommand\\bibstyle@aa{\\bibpunct{(}{)}{;}{a}{}{,}} %Astronomy & Astrophysics\n\\newcommand\\bibstyle@pass{\\bibpunct{(}{)}{;}{a}{,}{,}}%Planet. & Space Sci\n\\newcommand\\bibstyle@anngeo{\\bibpunct{(}{)}{;}{a}{,}{,}}%Annales Geophysicae\n\\newcommand\\bibstyle@nlinproc{\\bibpunct{(}{)}{;}{a}{,}{,}}%Nonlin.Proc.Geophys.\n % Define citation punctuation for some numerical styles\n\\newcommand\\bibstyle@cospar{\\bibpunct{/}{/}{,}{n}{}{}%\n     \\gdef\\bibnumfmt##1{##1.}}\n\\newcommand\\bibstyle@esa{\\bibpunct{(Ref.~}{)}{,}{n}{}{}%\n     \\gdef\\bibnumfmt##1{##1.\\hspace{1em}}}\n\\newcommand\\bibstyle@nature{\\bibpunct{}{}{,}{s}{}{\\textsuperscript{,}}%\n     \\gdef\\bibnumfmt##1{##1.}}\n % The standard LaTeX styles\n\\newcommand\\bibstyle@plain{\\bibpunct{[}{]}{,}{n}{}{,}}\n\\let\\bibstyle@alpha=\\bibstyle@plain\n\\let\\bibstyle@abbrv=\\bibstyle@plain\n\\let\\bibstyle@unsrt=\\bibstyle@plain\n % The author-year modifications of the standard styles\n\\newcommand\\bibstyle@plainnat{\\bibpunct{[}{]}{,}{a}{,}{,}}\n\\let\\bibstyle@abbrvnat=\\bibstyle@plainnat\n\\let\\bibstyle@unsrtnat=\\bibstyle@plainnat\n\\newif\\ifNAT@numbers \\NAT@numbersfalse\n\\newif\\ifNAT@super \\NAT@superfalse\n\\let\\NAT@merge\\z@\n\\DeclareOption{numbers}{\\NAT@numberstrue\n   \\ExecuteOptions{square,comma,nobibstyle}}\n\\DeclareOption{super}{\\NAT@supertrue\\NAT@numberstrue\n   \\renewcommand\\NAT@open{}\\renewcommand\\NAT@close{}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{authoryear}{\\NAT@numbersfalse\n   \\ExecuteOptions{round,semicolon,bibstyle}}\n\\DeclareOption{round}{%\n      \\renewcommand\\NAT@open{(} \\renewcommand\\NAT@close{)}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{square}{%\n      \\renewcommand\\NAT@open{[} \\renewcommand\\NAT@close{]}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{angle}{%\n      \\renewcommand\\NAT@open{$<$} \\renewcommand\\NAT@close{$>$}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{curly}{%\n      \\renewcommand\\NAT@open{\\{} \\renewcommand\\NAT@close{\\}}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{comma}{\\renewcommand\\NAT@sep{,}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{semicolon}{\\renewcommand\\NAT@sep{;}\n   \\ExecuteOptions{nobibstyle}}\n\\DeclareOption{colon}{\\ExecuteOptions{semicolon}}\n\\DeclareOption{nobibstyle}{\\let\\bibstyle=\\@gobble}\n\\DeclareOption{bibstyle}{\\let\\bibstyle=\\@citestyle}\n\\newif\\ifNAT@openbib \\NAT@openbibfalse\n\\DeclareOption{openbib}{\\NAT@openbibtrue}\n\\DeclareOption{sectionbib}{\\def\\NAT@sectionbib{on}}\n\\def\\NAT@sort{\\z@}\n\\def\\NAT@cmprs{\\z@}\n\\DeclareOption{sort}{\\def\\NAT@sort{\\@ne}}\n\\DeclareOption{compress}{\\def\\NAT@cmprs{\\@ne}}\n\\DeclareOption{sort&compress}{\\def\\NAT@sort{\\@ne}\\def\\NAT@cmprs{\\@ne}}\n\\DeclareOption{mcite}{\\let\\NAT@merge\\@ne}\n\\DeclareOption{merge}{\\@ifnum{\\NAT@merge<\\tw@}{\\let\\NAT@merge\\tw@}{}}\n\\DeclareOption{elide}{\\@ifnum{\\NAT@merge<\\thr@@}{\\let\\NAT@merge\\thr@@}{}}\n\\@ifpackageloaded{cite}{\\PackageWarningNoLine{natbib}\n  {The `cite' package should not be used\\MessageBreak\n   with natbib. Use option `sort' instead}\\ExecuteOptions{sort}}{}\n\\@ifpackageloaded{mcite}{\\PackageWarningNoLine{natbib}\n  {The `mcite' package should not be used\\MessageBreak\n   with natbib. Use option `merge' instead}\\ExecuteOptions{merge}}{}\n\\@ifpackageloaded{citeref}{\\PackageError{natbib}\n  {The `citeref' package must be loaded after natbib}%\n  {Move \\protect\\usepackage{citeref} to after \\string\\usepackage{natbib}}}{}\n\\newif\\ifNAT@longnames\\NAT@longnamesfalse\n\\DeclareOption{longnamesfirst}{\\NAT@longnamestrue}\n\\DeclareOption{nonamebreak}{\\def\\NAT@nmfmt#1{\\mbox{\\NAT@up#1}}}\n\\def\\NAT@nmfmt#1{{\\NAT@up#1}}\n\\renewcommand\\bibstyle[1]{\\csname bibstyle@#1\\endcsname}\n\\AtBeginDocument{\\global\\let\\bibstyle=\\@gobble}\n\\let\\@citestyle\\bibstyle\n\\newcommand\\citestyle[1]{\\@citestyle{#1}\\let\\bibstyle\\@gobble}\n\\newcommand\\bibpunct[7][, ]%\n  {\\gdef\\NAT@open{#2}\\gdef\\NAT@close{#3}\\gdef\n   \\NAT@sep{#4}\\global\\NAT@numbersfalse\n     \\ifx #5n\\global\\NAT@numberstrue\\global\\NAT@superfalse\n   \\else\n     \\ifx #5s\\global\\NAT@numberstrue\\global\\NAT@supertrue\n   \\fi\\fi\n   \\gdef\\NAT@aysep{#6}\\gdef\\NAT@yrsep{#7}%\n   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 }%\n  \\else \\NAT@nm\n  \\fi\n }{%\n  \\if\\relax\\NAT@date\\relax\n   \\begingroup\\reset@font\\bfseries(year?)\\endgroup\n   \\PackageWarning{natbib}{%\n    Year undefined for citation`\\@citeb' \\MessageBreak on page \\thepage%\n   }%\n  \\else \\NAT@date\n  \\fi\n }%\n}%\n\\let\\citenumfont=\\@empty\n\\newcommand\\NAT@citex{}\n\\def\\NAT@citex%\n  [#1][#2]#3{%\n  \\NAT@reset@parser\n  \\NAT@sort@cites{#3}%\n  \\NAT@reset@citea\n  \\@cite{\\let\\NAT@nm\\@empty\\let\\NAT@year\\@empty\n    \\@for\\@citeb:=\\NAT@cite@list\\do\n    {\\@safe@activestrue\n     \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb\\@empty}%\n     \\@safe@activesfalse\n     \\@ifundefined{b@\\@citeb\\@extra@b@citeb}{\\@citea%\n       {\\reset@font\\bfseries ?}\\NAT@citeundefined\n                 \\PackageWarning{natbib}%\n       {Citation `\\@citeb' on page \\thepage \\space undefined}\\def\\NAT@date{}}%\n     {\\let\\NAT@last@nm=\\NAT@nm\\let\\NAT@last@yr=\\NAT@year\n      \\NAT@parse{\\@citeb}%\n      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\\@citea\\NAT@hyper@{%\n             \\NAT@nmfmt{\\NAT@nm}%\n             \\hyper@natlinkbreak{%\n               \\NAT@aysep\\NAT@spacechar}{\\@citeb\\@extra@b@citeb\n             }%\n             \\NAT@date\n           }%\n         \\fi\n       \\fi\n     \\or\\@citea\\NAT@hyper@{\\NAT@nmfmt{\\NAT@nm}}%\n     \\or\\@citea\\NAT@hyper@{\\NAT@date}%\n     \\or\\@citea\\NAT@hyper@{\\NAT@alias}%\n     \\fi \\NAT@def@citea\n     \\else\n       \\ifcase\\NAT@ctype\n        \\if\\relax\\NAT@date\\relax\n          \\@citea\\NAT@hyper@{\\NAT@nmfmt{\\NAT@nm}}%\n        \\else\n         \\ifx\\NAT@last@nm\\NAT@nm\\NAT@yrsep\n            \\ifx\\NAT@last@yr\\NAT@year\n              \\def\\NAT@temp{{?}}%\n              \\ifx\\NAT@temp\\NAT@exlab\\PackageWarningNoLine{natbib}%\n               {Multiple citation on page \\thepage: same authors and\n               year\\MessageBreak without distinguishing extra\n               letter,\\MessageBreak appears as question mark}\\fi\n              \\NAT@hyper@{\\NAT@exlab}%\n            \\else\n              \\unskip\\NAT@spacechar\n              \\NAT@hyper@{\\NAT@date}%\n            \\fi\n         \\else\n           \\@citea\\NAT@hyper@{%\n             \\NAT@nmfmt{\\NAT@nm}%\n             \\hyper@natlinkbreak{\\NAT@spacechar\\NAT@@open\\if*#1*\\else#1\\NAT@spacechar\\fi}%\n               {\\@citeb\\@extra@b@citeb}%\n             \\NAT@date\n           }%\n         \\fi\n        \\fi\n       \\or\\@citea\\NAT@hyper@{\\NAT@nmfmt{\\NAT@nm}}%\n       \\or\\@citea\\NAT@hyper@{\\NAT@date}%\n       \\or\\@citea\\NAT@hyper@{\\NAT@alias}%\n       \\fi\n       \\if\\relax\\NAT@date\\relax\n         \\NAT@def@citea\n       \\else\n         \\NAT@def@citea@close\n       \\fi\n     \\fi\n     }}\\ifNAT@swa\\else\\if*#2*\\else\\NAT@cmt#2\\fi\n     \\if\\relax\\NAT@date\\relax\\else\\NAT@@close\\fi\\fi}{#1}{#2}}\n\\def\\NAT@spacechar{\\ }%\n\\def\\NAT@separator{\\NAT@sep\\NAT@penalty}%\n\\def\\NAT@reset@citea{\\c@NAT@ctr\\@ne\\let\\@citea\\@empty}%\n\\def\\NAT@def@citea{\\def\\@citea{\\NAT@separator\\NAT@space}}%\n\\def\\NAT@def@citea@space{\\def\\@citea{\\NAT@separator\\NAT@spacechar}}%\n\\def\\NAT@def@citea@close{\\def\\@citea{\\NAT@@close\\NAT@separator\\NAT@space}}%\n\\def\\NAT@def@citea@box{\\def\\@citea{\\NAT@mbox{\\NAT@@close}\\NAT@separator\\NAT@spacechar}}%\n\\newif\\ifNAT@par \\NAT@partrue\n\\newcommand\\NAT@@open{\\ifNAT@par\\NAT@open\\fi}\n\\newcommand\\NAT@@close{\\ifNAT@par\\NAT@close\\fi}\n\\newcommand\\NAT@alias{\\@ifundefined{al@\\@citeb\\@extra@b@citeb}{%\n  {\\reset@font\\bfseries(alias?)}\\PackageWarning{natbib}\n  {Alias undefined for citation `\\@citeb'\n  \\MessageBreak on page \\thepage}}{\\@nameuse{al@\\@citeb\\@extra@b@citeb}}}\n\\let\\NAT@up\\relax\n\\newcommand\\NAT@Up[1]{{\\let\\protect\\@unexpandable@protect\\let~\\relax\n  \\expandafter\\NAT@deftemp#1}\\expandafter\\NAT@UP\\NAT@temp}\n\\newcommand\\NAT@deftemp[1]{\\xdef\\NAT@temp{#1}}\n\\newcommand\\NAT@UP[1]{\\let\\@tempa\\NAT@UP\\ifcat a#1\\MakeUppercase{#1}%\n  \\let\\@tempa\\relax\\else#1\\fi\\@tempa}\n\\newcommand\\shortcites[1]{%\n  \\@bsphack\\@for\\@citeb:=#1\\do\n  {\\@safe@activestrue\n   \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb\\@empty}%\n   \\@safe@activesfalse\n   \\global\\@namedef{bv@\\@citeb\\@extra@b@citeb}{}}\\@esphack}\n\\newcommand\\NAT@biblabel[1]{\\hfill}\n\\newcommand\\NAT@biblabelnum[1]{\\bibnumfmt{#1}}\n\\let\\bibnumfmt\\@empty\n\\providecommand\\@biblabel[1]{[#1]}\n\\AtBeginDocument{\\ifx\\bibnumfmt\\@empty\\let\\bibnumfmt\\@biblabel\\fi}\n\\newcommand\\NAT@bibsetnum[1]{\\settowidth\\labelwidth{\\@biblabel{#1}}%\n   \\setlength{\\leftmargin}{\\labelwidth}\\addtolength{\\leftmargin}{\\labelsep}%\n   \\setlength{\\itemsep}{\\bibsep}\\setlength{\\parsep}{\\z@}%\n   \\ifNAT@openbib\n     \\addtolength{\\leftmargin}{\\bibindent}%\n     \\setlength{\\itemindent}{-\\bibindent}%\n     \\setlength{\\listparindent}{\\itemindent}%\n     \\setlength{\\parsep}{0pt}%\n   \\fi\n}\n\\newlength{\\bibhang}\n\\setlength{\\bibhang}{1em}\n\\newlength{\\bibsep}\n {\\@listi \\global\\bibsep\\itemsep \\global\\advance\\bibsep by\\parsep}\n\n\\newcommand\\NAT@bibsetup%\n   [1]{\\setlength{\\leftmargin}{\\bibhang}\\setlength{\\itemindent}{-\\leftmargin}%\n       \\setlength{\\itemsep}{\\bibsep}\\setlength{\\parsep}{\\z@}}\n\\newcommand\\NAT@set@cites{%\n  \\ifNAT@numbers\n    \\ifNAT@super \\let\\@cite\\NAT@citesuper\n       \\def\\NAT@mbox##1{\\unskip\\nobreak\\textsuperscript{##1}}%\n       \\let\\citeyearpar=\\citeyear\n       \\let\\NAT@space\\relax\n       \\def\\NAT@super@kern{\\kern\\p@}%\n    \\else\n       \\let\\NAT@mbox=\\mbox\n       \\let\\@cite\\NAT@citenum\n       \\let\\NAT@space\\NAT@spacechar\n       \\let\\NAT@super@kern\\relax\n    \\fi\n    \\let\\@citex\\NAT@citexnum\n    \\let\\@biblabel\\NAT@biblabelnum\n    \\let\\@bibsetup\\NAT@bibsetnum\n    \\renewcommand\\NAT@idxtxt{\\NAT@name\\NAT@spacechar\\NAT@open\\NAT@num\\NAT@close}%\n    \\def\\natexlab##1{}%\n    \\def\\NAT@penalty{\\penalty\\@m}%\n  \\else\n    \\let\\@cite\\NAT@cite\n    \\let\\@citex\\NAT@citex\n    \\let\\@biblabel\\NAT@biblabel\n    \\let\\@bibsetup\\NAT@bibsetup\n    \\let\\NAT@space\\NAT@spacechar\n    \\let\\NAT@penalty\\@empty\n    \\renewcommand\\NAT@idxtxt{\\NAT@name\\NAT@spacechar\\NAT@open\\NAT@date\\NAT@close}%\n    \\def\\natexlab##1{##1}%\n  \\fi}\n\\AtBeginDocument{\\NAT@set@cites}\n\\AtBeginDocument{\\ifx\\SK@def\\@undefined\\else\n\\ifx\\SK@cite\\@empty\\else\n  \\SK@def\\@citex[#1][#2]#3{\\SK@\\SK@@ref{#3}\\SK@@citex[#1][#2]{#3}}\\fi\n\\ifx\\SK@citeauthor\\@undefined\\def\\HAR@checkdef{}\\else\n  \\let\\citeauthor\\SK@citeauthor\n  \\let\\citefullauthor\\SK@citefullauthor\n  \\let\\citeyear\\SK@citeyear\\fi\n\\fi}\n\\newif\\ifNAT@full\\NAT@fullfalse\n\\newif\\ifNAT@swa\n\\DeclareRobustCommand\\citet\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\z@\\NAT@partrue\n     \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\newcommand\\NAT@citetp{\\@ifnextchar[{\\NAT@@citetp}{\\NAT@@citetp[]}}\n\\newcommand\\NAT@@citetp{}\n\\def\\NAT@@citetp[#1]{\\@ifnextchar[{\\@citex[#1]}{\\@citex[][#1]}}\n\\DeclareRobustCommand\\citep\n   {\\begingroup\\NAT@swatrue\\let\\NAT@ctype\\z@\\NAT@partrue\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\cite\n    {\\begingroup\\let\\NAT@ctype\\z@\\NAT@partrue\\NAT@swatrue\n      \\@ifstar{\\NAT@fulltrue\\NAT@cites}{\\NAT@fullfalse\\NAT@cites}}\n\\newcommand\\NAT@cites{\\@ifnextchar [{\\NAT@@citetp}{%\n     \\ifNAT@numbers\\else\n     \\NAT@swafalse\n     \\fi\n    \\NAT@@citetp[]}}\n\\DeclareRobustCommand\\citealt\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\z@\\NAT@parfalse\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\citealp\n   {\\begingroup\\NAT@swatrue\\let\\NAT@ctype\\z@\\NAT@parfalse\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\citenum\n   {\\begingroup\n     \\NAT@swatrue\\let\\NAT@ctype\\z@\\NAT@parfalse\\let\\textsuperscript\\NAT@spacechar\n     \\NAT@citexnum[][]}\n\\DeclareRobustCommand\\citeauthor\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\@ne\\NAT@parfalse\n    \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\Citet\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\z@\\NAT@partrue\n     \\let\\NAT@up\\NAT@Up\n     \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\Citep\n   {\\begingroup\\NAT@swatrue\\let\\NAT@ctype\\z@\\NAT@partrue\n     \\let\\NAT@up\\NAT@Up\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\Citealt\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\z@\\NAT@parfalse\n     \\let\\NAT@up\\NAT@Up\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\Citealp\n   {\\begingroup\\NAT@swatrue\\let\\NAT@ctype\\z@\\NAT@parfalse\n     \\let\\NAT@up\\NAT@Up\n         \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\Citeauthor\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\@ne\\NAT@parfalse\n     \\let\\NAT@up\\NAT@Up\n    \\@ifstar{\\NAT@fulltrue\\NAT@citetp}{\\NAT@fullfalse\\NAT@citetp}}\n\\DeclareRobustCommand\\citeyear\n   {\\begingroup\\NAT@swafalse\\let\\NAT@ctype\\tw@\\NAT@parfalse\\NAT@citetp}\n\\DeclareRobustCommand\\citeyearpar\n   {\\begingroup\\NAT@swatrue\\let\\NAT@ctype\\tw@\\NAT@partrue\\NAT@citetp}\n\\newcommand\\citetext[1]{\\NAT@open#1\\NAT@close}\n\\DeclareRobustCommand\\citefullauthor\n   {\\citeauthor*}\n\\newcommand\\defcitealias[2]{%\n   \\@ifundefined{al@#1\\@extra@b@citeb}{}\n   {\\PackageWarning{natbib}{Overwriting existing alias for citation #1}}\n   \\@namedef{al@#1\\@extra@b@citeb}{#2}}\n\\DeclareRobustCommand\\citetalias{\\begingroup\n   \\NAT@swafalse\\let\\NAT@ctype\\thr@@\\NAT@parfalse\\NAT@citetp}\n\\DeclareRobustCommand\\citepalias{\\begingroup\n   \\NAT@swatrue\\let\\NAT@ctype\\thr@@\\NAT@partrue\\NAT@citetp}\n\\renewcommand\\nocite[1]{\\@bsphack\n  \\@for\\@citeb:=#1\\do{%\n    \\@safe@activestrue\n    \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb\\@empty}%\n    \\@safe@activesfalse\n    \\if@filesw\\immediate\\write\\@auxout{\\string\\citation{\\@citeb}}\\fi\n    \\if*\\@citeb\\else\n    \\@ifundefined{b@\\@citeb\\@extra@b@citeb}{%\n       \\NAT@citeundefined \\PackageWarning{natbib}%\n       {Citation `\\@citeb' undefined}}{}\\fi}%\n  \\@esphack}\n\\newcommand\\NAT@parse[1]{%\n  \\begingroup\n   \\let\\protect=\\@unexpandable@protect\n   \\let~\\relax\n   \\let\\active@prefix=\\@gobble\n   \\edef\\NAT@temp{\\csname b@#1\\@extra@b@citeb\\endcsname}%\n   \\aftergroup\\NAT@split\n   \\expandafter\n  \\endgroup\n  \\NAT@temp{}{}{}{}{}@@%\n  \\expandafter\\NAT@parse@date\\NAT@date??????@@%\n  \\ifciteindex\\NAT@index\\fi\n}%\n\\def\\NAT@split#1#2#3#4#5@@{%\n  \\gdef\\NAT@num{#1}\\gdef\\NAT@name{#3}\\gdef\\NAT@date{#2}%\n  \\gdef\\NAT@all@names{#4}%\n  \\ifx\\NAT@num\\@empty\\gdef\\NAT@num{0}\\fi\n  \\ifx\\NAT@noname\\NAT@all@names \\gdef\\NAT@all@names{#3}\\fi\n}%\n\\def\\NAT@reset@parser{%\n  \\global\\let\\NAT@num\\@empty\n  \\global\\let\\NAT@name\\@empty\n  \\global\\let\\NAT@date\\@empty\n  \\global\\let\\NAT@all@names\\@empty\n}%\n\\newcommand\\NAT@parse@date{}\n\\def\\NAT@parse@date#1#2#3#4#5#6@@{%\n  \\ifnum\\the\\catcode`#1=11\\def\\NAT@year{}\\def\\NAT@exlab{#1}\\else\n  \\ifnum\\the\\catcode`#2=11\\def\\NAT@year{#1}\\def\\NAT@exlab{#2}\\else\n  \\ifnum\\the\\catcode`#3=11\\def\\NAT@year{#1#2}\\def\\NAT@exlab{#3}\\else\n  \\ifnum\\the\\catcode`#4=11\\def\\NAT@year{#1#2#3}\\def\\NAT@exlab{#4}\\else\n    \\def\\NAT@year{#1#2#3#4}\\def\\NAT@exlab{{#5}}\\fi\\fi\\fi\\fi}\n\\newcommand\\NAT@index{}\n\\let\\NAT@makeindex=\\makeindex\n\\renewcommand\\makeindex{\\NAT@makeindex\n  \\renewcommand\\NAT@index{\\@bsphack\\begingroup\n     \\def~{\\string~}\\@wrindex{\\NAT@idxtxt}}}\n\\newcommand\\NAT@idxtxt{\\NAT@name\\NAT@spacechar\\NAT@open\\NAT@date\\NAT@close}\n\\@ifxundefined\\@indexfile{}{\\let\\NAT@makeindex\\relax\\makeindex}\n\\newif\\ifciteindex \\citeindexfalse\n\\newcommand\\citeindextype{default}\n\\newcommand\\NAT@index@alt{{\\let\\protect=\\noexpand\\let~\\relax\n  \\xdef\\NAT@temp{\\NAT@idxtxt}}\\expandafter\\NAT@exp\\NAT@temp\\@nil}\n\\newcommand\\NAT@exp{}\n\\def\\NAT@exp#1\\@nil{\\index[\\citeindextype]{#1}}\n\n\\AtBeginDocument{%\n\\@ifpackageloaded{index}{\\let\\NAT@index=\\NAT@index@alt}{}}\n\\newcommand\\NAT@ifcmd{\\futurelet\\NAT@temp\\NAT@ifxcmd}\n\\newcommand\\NAT@ifxcmd{\\ifx\\NAT@temp\\relax\\else\\expandafter\\NAT@bare\\fi}\n\\def\\NAT@bare#1(#2)#3(@)#4\\@nil#5{%\n  \\if @#2\n    \\expandafter\\NAT@apalk#1, , \\@nil{#5}%\n  \\else\n  \\NAT@wrout{\\the\\c@NAT@ctr}{#2}{#1}{#3}{#5}%\n\\fi\n}\n\\newcommand\\NAT@wrout[5]{%\n\\if@filesw\n      {\\let\\protect\\noexpand\\let~\\relax\n       \\immediate\n       \\write\\@auxout{\\string\\bibcite{#5}{{#1}{#2}{{#3}}{{#4}}}}}\\fi\n\\ignorespaces}\n\\def\\NAT@noname{{}}\n\\renewcommand\\bibitem{\\@ifnextchar[{\\@lbibitem}{\\@lbibitem[]}}%\n\\let\\NAT@bibitem@first@sw\\@secondoftwo\n\\def\\@lbibitem[#1]#2{%\n  \\if\\relax\\@extra@b@citeb\\relax\\else\n    \\@ifundefined{br@#2\\@extra@b@citeb}{}{%\n     \\@namedef{br@#2}{\\@nameuse{br@#2\\@extra@b@citeb}}%\n    }%\n  \\fi\n  \\@ifundefined{b@#2\\@extra@b@citeb}{%\n   \\def\\NAT@num{}%\n  }{%\n   \\NAT@parse{#2}%\n  }%\n  \\def\\NAT@tmp{#1}%\n  \\expandafter\\let\\expandafter\\bibitemOpen\\csname NAT@b@open@#2\\endcsname\n  \\expandafter\\let\\expandafter\\bibitemShut\\csname NAT@b@shut@#2\\endcsname\n  \\@ifnum{\\NAT@merge>\\@ne}{%\n   \\NAT@bibitem@first@sw{%\n    \\@firstoftwo\n   }{%\n    \\@ifundefined{NAT@b*@#2}{%\n     \\@firstoftwo\n    }{%\n     \\expandafter\\def\\expandafter\\NAT@num\\expandafter{\\the\\c@NAT@ctr}%\n     \\@secondoftwo\n    }%\n   }%\n  }{%\n   \\@firstoftwo\n  }%\n  {%\n   \\global\\advance\\c@NAT@ctr\\@ne\n   \\@ifx{\\NAT@tmp\\@empty}{\\@firstoftwo}{%\n    \\@secondoftwo\n   }%\n   {%\n    \\expandafter\\def\\expandafter\\NAT@num\\expandafter{\\the\\c@NAT@ctr}%\n    \\global\\NAT@stdbsttrue\n   }{}%\n   \\bibitem@fin\n   \\item[\\hfil\\NAT@anchor{#2}{\\NAT@num}]%\n   \\global\\let\\NAT@bibitem@first@sw\\@secondoftwo\n   \\NAT@bibitem@init\n  }%\n  {%\n   \\NAT@anchor{#2}{}%\n   \\NAT@bibitem@cont\n   \\bibitem@fin\n  }%\n  \\@ifx{\\NAT@tmp\\@empty}{%\n    \\NAT@wrout{\\the\\c@NAT@ctr}{}{}{}{#2}%\n  }{%\n    \\expandafter\\NAT@ifcmd\\NAT@tmp(@)(@)\\@nil{#2}%\n  }%\n}%\n\\def\\bibitem@fin{%\n \\@ifxundefined\\@bibstop{}{\\csname bibitem@\\@bibstop\\endcsname}%\n}%\n\\def\\NAT@bibitem@init{%\n \\let\\@bibstop\\@undefined\n}%\n\\def\\NAT@bibitem@cont{%\n \\let\\bibitem@Stop\\bibitemStop\n \\let\\bibitem@NoStop\\bibitemContinue\n}%\n\\def\\BibitemOpen{%\n \\bibitemOpen\n}%\n\\def\\BibitemShut#1{%\n \\bibitemShut\n \\def\\@bibstop{#1}%\n \\let\\bibitem@Stop\\bibitemStop\n \\let\\bibitem@NoStop\\bibitemNoStop\n}%\n\\def\\bibitemStop{}%\n\\def\\bibitemNoStop{.\\spacefactor\\@mmm\\space}%\n\\def\\bibitemContinue{\\spacefactor\\@mmm\\space}%\n\\mathchardef\\@mmm=3000 %\n\\providecommand{\\bibAnnote}[3]{%\n  \\BibitemShut{#1}%\n  \\def\\@tempa{#3}\\@ifx{\\@tempa\\@empty}{}{%\n   \\begin{quotation}\\noindent\n    \\textsc{Key:}\\ #2\\\\\\textsc{Annotation:}\\ \\@tempa\n   \\end{quotation}%\n  }%\n}%\n\\providecommand{\\bibAnnoteFile}[2]{%\n  \\IfFileExists{#2}{%\n    \\bibAnnote{#1}{#2}{\\input{#2}}%\n  }{%\n    \\bibAnnote{#1}{#2}{}%\n  }%\n}%\n\\let\\bibitemOpen\\relax\n\\let\\bibitemShut\\relax\n\\def\\bibfield{\\@ifnum{\\NAT@merge>\\tw@}{\\@bibfield}{\\@secondoftwo}}%\n\\def\\@bibfield#1#2{%\n \\begingroup\n  \\let\\Doi\\@gobble\n  \\let\\bibinfo\\relax\n  \\let\\restore@protect\\@empty\n  \\protected@edef\\@tempa{#2}%\n  \\aftergroup\\def\\aftergroup\\@tempa\n \\expandafter\\endgroup\\expandafter{\\@tempa}%\n \\expandafter\\@ifx\\expandafter{\\csname @bib#1\\endcsname\\@tempa}{%\n  \\expandafter\\let\\expandafter\\@tempa\\csname @bib@X#1\\endcsname\n }{%\n  \\expandafter\\let\\csname @bib#1\\endcsname\\@tempa\n  \\expandafter\\let\\expandafter\\@tempa\\csname @bib@Y#1\\endcsname\n }%\n \\@ifx{\\@tempa\\relax}{\\let\\@tempa\\@firstofone}{}%\n \\@tempa{#2}%\n}%\n\\def\\bibinfo#1{%\n \\expandafter\\let\\expandafter\\@tempa\\csname bibinfo@X@#1\\endcsname\n \\@ifx{\\@tempa\\relax}{\\@firstofone}{\\@tempa}%\n}%\n\\def\\@bib@Xauthor#1{\\let\\@bib@Xjournal\\@gobble}%\n\\def\\@bib@Xjournal#1{\\begingroup\\let\\bibinfo@X@journal\\@bib@Z@journal#1\\endgroup}%\n\\def\\@bibibid@#1{\\textit{ibid}.}%\n\\appdef\\NAT@bibitem@init{%\n \\let\\@bibauthor  \\@empty\n \\let\\@bibjournal \\@empty\n \\let\\@bib@Z@journal\\@bibibid@\n}%\n\\ifx\\SK@lbibitem\\@undefined\\else\n   \\let\\SK@lbibitem\\@lbibitem\n   \\def\\@lbibitem[#1]#2{%\n     \\SK@lbibitem[#1]{#2}\\SK@\\SK@@label{#2}\\ignorespaces}\\fi\n\\newif\\ifNAT@stdbst \\NAT@stdbstfalse\n\n\\AtEndDocument{%\n  \\ifNAT@stdbst\\if@filesw\n   \\immediate\\write\\@auxout{%\n    \\string\\providecommand\\string\\NAT@force@numbers{}%\n    \\string\\NAT@force@numbers\n   }%\n  \\fi\\fi\n }\n\\newcommand\\NAT@force@numbers{%\n  \\ifNAT@numbers\\else\n  \\PackageError{natbib}{Bibliography not compatible with author-year\n  citations.\\MessageBreak\n  Press <return> to continue in numerical citation style}\n  {Check the bibliography entries for non-compliant syntax,\\MessageBreak\n   or select author-year BibTeX style, e.g. plainnat}%\n  \\global\\NAT@numberstrue\\fi}\n\n\\providecommand\\bibcite{}\n\\renewcommand\\bibcite[2]{%\n \\@ifundefined{b@#1\\@extra@binfo}{\\relax}{%\n   \\NAT@citemultiple\n   \\PackageWarningNoLine{natbib}{Citation `#1' multiply defined}%\n }%\n \\global\\@namedef{b@#1\\@extra@binfo}{#2}%\n}%\n\\AtEndDocument{\\NAT@swatrue\\let\\bibcite\\NAT@testdef}\n\\newcommand\\NAT@testdef[2]{%\n  \\def\\NAT@temp{#2}%\n  \\expandafter \\ifx \\csname b@#1\\@extra@binfo\\endcsname\\NAT@temp\n  \\else\n    \\ifNAT@swa \\NAT@swafalse\n      \\PackageWarningNoLine{natbib}{%\n        Citation(s) may have changed.\\MessageBreak\n        Rerun to get citations correct%\n      }%\n    \\fi\n  \\fi\n}%\n\\newcommand\\NAT@apalk{}\n\\def\\NAT@apalk#1, #2, #3\\@nil#4{%\n  \\if\\relax#2\\relax\n    \\global\\NAT@stdbsttrue\n    \\NAT@wrout{#1}{}{}{}{#4}%\n  \\else\n    \\NAT@wrout{\\the\\c@NAT@ctr}{#2}{#1}{}{#4}%\n  \\fi\n}%\n\\newcommand\\citeauthoryear{}\n\\def\\citeauthoryear#1#2#3(@)(@)\\@nil#4{%\n  \\if\\relax#3\\relax\n    \\NAT@wrout{\\the\\c@NAT@ctr}{#2}{#1}{}{#4}%\n  \\else\n    \\NAT@wrout{\\the\\c@NAT@ctr}{#3}{#2}{#1}{#4}%\n  \\fi\n}%\n\\newcommand\\citestarts{\\NAT@open}%\n\\newcommand\\citeends{\\NAT@close}%\n\\newcommand\\betweenauthors{and}%\n\\newcommand\\astroncite{}\n\\def\\astroncite#1#2(@)(@)\\@nil#3{%\n \\NAT@wrout{\\the\\c@NAT@ctr}{#2}{#1}{}{#3}%\n}%\n\\newcommand\\citename{}\n\\def\\citename#1#2(@)(@)\\@nil#3{\\expandafter\\NAT@apalk#1#2, \\@nil{#3}}\n\\newcommand\\harvarditem[4][]{%\n \\if\\relax#1\\relax\n   \\bibitem[#2(#3)]{#4}%\n \\else\n   \\bibitem[#1(#3)#2]{#4}%\n 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\\ifx\\NAT@cite@list\\@empty\\else\n   \\protected@edef\\NAT@cite@list{\\expandafter\\NAT@xcom\\NAT@cite@list @@}%\n  \\fi\n}%\n\\def\\NAT@make@cite@list{%\n  \\advance\\count@\\@ne\n  \\@safe@activestrue\n  \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb\\@empty}%\n  \\@safe@activesfalse\n  \\@ifundefined{b@\\@citeb\\@extra@b@citeb}%\n   {\\def\\NAT@num{A}}%\n   {\\NAT@parse{\\@citeb}}%\n  \\NAT@ifcat@num\\NAT@num\n   {\\@tempcnta\\NAT@num \\relax\n    \\@ifnum{\\@tempcnta<\\@tempcntb}{%\n      \\let\\NAT@@cite@list=\\NAT@cite@list\n      \\let\\NAT@cite@list\\@empty\n      \\begingroup\\let\\@celt=\\NAT@celt\\NAT@num@list\\endgroup\n      \\protected@edef\\NAT@num@list{%\n       \\expandafter\\NAT@num@celt \\NAT@num@list \\@gobble @%\n      }%\n    }{%\n      \\protected@edef\\NAT@num@list{\\NAT@num@list \\@celt{\\NAT@num}}%\n      \\protected@edef\\NAT@cite@list{\\NAT@cite@list\\@citeb,}%\n      \\@tempcntb\\@tempcnta\n    }%\n   }%\n   {\\protected@edef\\NAT@nonsort@list{\\NAT@nonsort@list\\@citeb,}}%\n}%\n\\def\\NAT@celt#1{%\n  \\@ifnum{#1>\\@tempcnta}{%\n    \\xdef\\NAT@cite@list{\\NAT@cite@list\\@citeb,\\NAT@@cite@list}%\n    \\let\\@celt\\@gobble\n  }{%\n    \\expandafter\\def@NAT@cite@lists\\NAT@@cite@list\\@@\n  }%\n}%\n\\def\\NAT@num@celt#1#2{%\n \\ifx#1\\@celt\n  \\@ifnum{#2>\\@tempcnta}{%\n    \\@celt{\\number\\@tempcnta}%\n    \\@celt{#2}%\n  }{%\n    \\@celt{#2}%\n    \\expandafter\\NAT@num@celt\n  }%\n \\fi\n}%\n\\def\\def@NAT@cite@lists#1,#2\\@@{%\n  \\xdef\\NAT@cite@list{\\NAT@cite@list#1,}%\n  \\xdef\\NAT@@cite@list{#2}%\n}%\n\\def\\NAT@nextc#1,#2@@{#1,}\n\\def\\NAT@restc#1,#2{#2}\n\\def\\NAT@xcom#1,@@{#1}\n\\InputIfFileExists{natbib.cfg}\n       {\\typeout{Local config file natbib.cfg used}}{}\n%% \n%% <<<<< End of generated file <<<<<<\n%%\n%% End of file `natbib.sty'.\n"
  },
  {
    "path": "pytorch_test.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\n\nfrom pytorch_toy import *\n\n\ndef _parse_args(args):\n    parser = argparse.ArgumentParser(\n        description='sanity check for pytorch_toy gradient estimators')\n    parser.add_argument('--num-mc-samples', type=int, default=10000)\n    parser.add_argument('--latent-dim', type=int, default=3)\n    parser.add_argument('--param-seed', type=int, default=0)\n    parser.add_argument('--mc-seed', type=int, default=0)\n    return parser.parse_args(args)\n\n\ndef test(args=None):\n    args = _parse_args(args)\n    torch.manual_seed(args.param_seed)\n    theta = torch.rand(args.latent_dim).clamp(1e-5, 1. - 1e-5)\n    logits = torch.log(theta) - torch.log1p(-theta)\n    logits.requires_grad_(True)\n    q_func = QFunc(args.latent_dim, 5)\n\n    def f(samples):\n        return torch.sum(\n            (samples - torch.linspace(0.2, 0.9, args.latent_dim)) ** 2,\n            1\n        )\n\n    def expected_f(logits):\n        # population iterates over all possible configurations of D\n        # bernoulli samples\n        population = torch.stack(\n            [\n                torch.FloatTensor(b)\n                for b in product([0.0, 1.0], repeat=args.latent_dim)\n            ],\n            dim=0,\n        )\n        return torch.sum(\n            f(population) * torch.prod(\n                torch.sigmoid(logits * (population * 2. - 1.)), dim=1))\n\n    def monte_carlo_estimator(logits, estimator, **kwargs):\n        torch.manual_seed(args.mc_seed)\n        u = torch.rand(args.num_mc_samples, args.latent_dim)\n        v = torch.rand(args.num_mc_samples, args.latent_dim)\n        # add extra samples to (new) batch index\n        logits = logits.unsqueeze(0).expand(\n            args.num_mc_samples, args.latent_dim)\n        z = logits + torch.log(u) - torch.log1p(-u)\n        b = z.gt(0.).type_as(z)\n        f_b = f(b)\n        d_logits = estimator(f_b=f_b, b=b, logits=logits, z=z, v=v, **kwargs)\n        return d_logits.mean(0)\n\n    print(\"Gradient estimators:\")\n    print(\"Exact            : {}\".format(\n        torch.autograd.grad([expected_f(logits)], [logits])[0].numpy()\n    ))\n    print(\"Reinforce        : {}\".format(\n        monte_carlo_estimator(logits, reinforce).numpy()))\n    print(\"Rebar, temp = 1  : {}\".format(\n        monte_carlo_estimator(\n            logits, rebar,\n            eta=torch.ones(args.latent_dim) * 0.3,\n            log_temp=torch.ones(args.latent_dim).log(),\n            target=None,\n            loss_func=lambda x, t: f(x)).numpy()))\n    print(\"Rebar, temp = 10 : {}\".format(\n        monte_carlo_estimator(\n            logits, rebar,\n            eta=torch.ones(args.latent_dim) * 0.3,\n            log_temp=(torch.ones(args.latent_dim) * 10.).log(),\n            target=None,\n            loss_func=lambda x, t: f(x)).numpy()))\n    print(\"Rebar, eta = 0   : {}\".format(\n        monte_carlo_estimator(\n            logits, rebar,\n            eta=torch.zeros(args.latent_dim),\n            log_temp=torch.ones(args.latent_dim).log(),\n            target=None,\n            loss_func=lambda x, t: f(x)).numpy()))\n    print(\"Relax            : {}\".format(\n        monte_carlo_estimator(\n            logits, relax,\n            eta=torch.ones(args.latent_dim) * 0.3,\n            log_temp=torch.ones(args.latent_dim).log(),\n            q_func=q_func).numpy()))\n\n\nif __name__ == '__main__':\n    test()\n"
  },
  {
    "path": "pytorch_toy.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\n\nfrom itertools import product\n\nimport argparse\nimport numpy as np\nimport torch\n\n\nclass QFunc(torch.nn.Module):\n    '''Control variate for RELAX'''\n\n    def __init__(self, num_latents, hidden_size=10):\n        super(QFunc, self).__init__()\n        self.h1 = torch.nn.Linear(num_latents, hidden_size)\n        self.nonlin = torch.nn.Tanh()\n        self.out = torch.nn.Linear(hidden_size, 1)\n\n    def forward(self, z):\n        # the multiplication by 2 and subtraction is from toy.py...\n        # it doesn't change the bias of the estimator, I guess\n        z = self.h1(z * 2. - 1.)\n        z = self.nonlin(z)\n        z = self.out(z)\n        return z\n\n\ndef loss_func(b, t):\n    return ((b - t) ** 2).mean(1)\n\n\ndef _parse_args(args):\n    parser = argparse.ArgumentParser(\n        description='Toy experiment from backpropagation throught the void, '\n        'written in pytorch')\n    parser.add_argument(\n        '--estimator', choices=['reinforce', 'relax', 'rebar'],\n        default='reinforce')\n    parser.add_argument('--rand-seed', type=int, default=42)\n    parser.add_argument('--iters', type=int, default=5000)\n    parser.add_argument('--batch-size', type=int, default=1)\n    parser.add_argument('--target', type=float, default=.499)\n    parser.add_argument('--num-latents', type=int, default=1)\n    parser.add_argument('--lr', type=float, default=.01)\n    return parser.parse_args(args)\n\n\ndef reinforce(f_b, b, logits, **kwargs):\n    log_prob = torch.distributions.Bernoulli(logits=logits).log_prob(b)\n    d_log_prob = torch.autograd.grad(\n        [log_prob], [logits], grad_outputs=torch.ones_like(log_prob))[0]\n    d_logits = f_b.unsqueeze(1) * d_log_prob\n    return d_logits\n\n\ndef _get_z_tilde(logits, b, v):\n    theta = torch.sigmoid(logits)\n    v_prime = v * (b - 1.) * (theta - 1.) + b * (v * theta + 1. - theta)\n    z_tilde = logits + torch.log(v_prime) - torch.log1p(-v_prime)\n    return z_tilde\n\n\ndef rebar(\n        f_b, b, logits, z, v, eta, log_temp, target, loss_func=loss_func,\n        **kwargs):\n    z_tilde = _get_z_tilde(logits, b, v)\n    temp = torch.exp(log_temp).unsqueeze(0)\n    sig_z = torch.sigmoid(z / temp)\n    sig_z_tilde = torch.sigmoid(z_tilde / temp)\n    f_z = loss_func(sig_z, target)\n    f_z_tilde = loss_func(sig_z_tilde, target)\n    log_prob = torch.distributions.Bernoulli(logits=logits).log_prob(b)\n    d_log_prob = torch.autograd.grad(\n        [log_prob], [logits], grad_outputs=torch.ones_like(log_prob))[0]\n    d_f_z = torch.autograd.grad(\n        [f_z], [logits], grad_outputs=torch.ones_like(f_z),\n        create_graph=True, retain_graph=True)[0]\n    d_f_z_tilde = torch.autograd.grad(\n        [f_z_tilde], [logits], grad_outputs=torch.ones_like(f_z_tilde),\n        create_graph=True, retain_graph=True)[0]\n    diff = f_b.unsqueeze(1) - eta * f_z_tilde.unsqueeze(1)\n    d_logits = diff * d_log_prob + eta * (d_f_z - d_f_z_tilde)\n    var_loss = (d_logits ** 2).mean()\n    var_loss.backward()\n    return d_logits.detach()\n\n\ndef relax(f_b, b, logits, z, v, log_temp, q_func, **kwargs):\n    z_tilde = _get_z_tilde(logits, b, v)\n    temp = torch.exp(log_temp).unsqueeze(0)\n    sig_z = torch.sigmoid(z / temp)\n    sig_z_tilde = torch.sigmoid(z_tilde / temp)\n    f_z = q_func(sig_z)[:, 0]\n    f_z_tilde = q_func(sig_z_tilde)[:, 0]\n    log_prob = torch.distributions.Bernoulli(logits=logits).log_prob(b)\n    d_log_prob = torch.autograd.grad(\n        [log_prob], [logits], grad_outputs=torch.ones_like(log_prob))[0]\n    d_f_z = torch.autograd.grad(\n        [f_z], [logits], grad_outputs=torch.ones_like(f_z),\n        create_graph=True, retain_graph=True)[0]\n    d_f_z_tilde = torch.autograd.grad(\n        [f_z_tilde], [logits], grad_outputs=torch.ones_like(f_z_tilde),\n        create_graph=True, retain_graph=True)[0]\n    diff = f_b.unsqueeze(1) - f_z_tilde.unsqueeze(1)\n    d_logits = diff * d_log_prob + d_f_z - d_f_z_tilde\n    var_loss = (d_logits.mean(0) ** 2).mean()\n    var_loss.backward()\n    return d_logits.detach()\n\n\ndef run_toy_example(args=None):\n    args = _parse_args(args)\n    print('Target is {}'.format(args.target))\n    target = torch.Tensor(1, args.num_latents)\n    target.fill_(args.target)\n    logits = torch.zeros(args.num_latents, requires_grad=True)\n    eta = torch.ones(args.num_latents, requires_grad=True)\n    log_temp = torch.from_numpy(\n        np.array([.5] * args.num_latents, dtype=np.float32))\n    log_temp.requires_grad_(True)\n    q_func = QFunc(args.num_latents)\n    torch.manual_seed(args.rand_seed)\n    if args.estimator == 'reinforce':\n        estimator = reinforce\n        tunable = []\n    elif args.estimator == 'rebar':\n        estimator = rebar\n        tunable = [eta, log_temp]\n    else:\n        estimator = relax\n        tunable = [log_temp] + list(q_func.parameters())\n    logit_optim = torch.optim.Adam([logits], lr=args.lr)\n    if tunable:\n        tune_optim = torch.optim.Adam(tunable, lr=args.lr)\n    else:\n        tune_optim = None\n    for i in range(args.iters):\n        logit_optim.zero_grad()\n        if tune_optim:\n            tune_optim.zero_grad()\n        u = torch.rand(args.batch_size, args.num_latents)\n        v = torch.rand(args.batch_size, args.num_latents)\n        z = logits + torch.log(u) - torch.log1p(-u)\n        b = z.gt(0.).type_as(z)\n        f_b = loss_func(b, target)\n        d_logits = estimator(\n            f_b=f_b, b=b, u=u, v=v, z=z, target=target, logits=logits,\n            log_temp=log_temp, eta=eta, q_func=q_func,\n        )\n        logits.backward(d_logits.mean(0))  # mean of batch\n        d_logits = d_logits.numpy()\n        logit_optim.step()\n        if tune_optim:\n            tune_optim.step()\n        thetas = torch.sigmoid(logits.detach()).numpy()\n        loss = thetas * (1 - args.target) ** 2\n        loss += (1 - thetas) * args.target ** 2\n        loss = loss.mean()\n        mean = d_logits.mean()\n        std = d_logits.std()\n        print(\n            'Iter: {} Loss: {:.03f} Thetas: {} Mean: {:.03f} Std: {:.03f} '\n            'Temp: {:.03f}'.format(\n                i, loss, thetas, mean, std, torch.exp(log_temp).item())\n        )\n\n\nif __name__ == '__main__':\n    run_toy_example()\n"
  },
  {
    "path": "rebar_baseline/config.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\n\"\"\"Configuration variables.\"\"\"\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nDATA_DIR = 'data'\nMNIST_BINARIZED = 'mnist_salakhutdinov_07-19-2017.pkl'\nMNIST_FLOAT = 'mnist_train_xs_07-19-2017.npy'\nOMNIGLOT = 'omniglot_07-19-2017.mat'\n"
  },
  {
    "path": "rebar_baseline/datasets.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\n\"\"\"Library of datasets for REBAR.\"\"\"\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport random\nimport os\nimport scipy.io\nimport numpy as np\nimport cPickle as pickle\nimport tensorflow as tf\nimport config\ngfile = tf.gfile\n\n\ndef load_data(hparams):\n  # Load data\n  if hparams.task in ['sbn', 'sp']:\n    reader = read_MNIST\n  elif hparams.task == 'omni':\n    reader = read_omniglot\n  x_train, x_valid, x_test = reader(binarize=not hparams.dynamic_b)\n\n  return x_train, x_valid, x_test\n\ndef read_MNIST(binarize=False):\n  \"\"\"Reads in MNIST images.\n\n  Args:\n    binarize: whether to use the fixed binarization\n\n  Returns:\n    x_train: 50k training images\n    x_valid: 10k validation images\n    x_test: 10k test images\n\n  \"\"\"\n  with gfile.FastGFile(os.path.join(config.DATA_DIR, config.MNIST_BINARIZED), 'r') as f:\n    (x_train, _), (x_valid, _), (x_test, _) = pickle.load(f)\n\n  if not binarize:\n    with gfile.FastGFile(os.path.join(config.DATA_DIR, config.MNIST_FLOAT), 'r') as f:\n      x_train = np.load(f).reshape(-1, 784)\n\n  return x_train, x_valid, x_test\n\ndef read_omniglot(binarize=False):\n  \"\"\"Reads in Omniglot images.\n\n  Args:\n    binarize: whether to use the fixed binarization\n\n  Returns:\n    x_train: training images\n    x_valid: validation images\n    x_test: test images\n\n  \"\"\"\n  n_validation=1345\n\n  def reshape_data(data):\n    return data.reshape((-1, 28, 28)).reshape((-1, 28*28), order='fortran')\n\n  omni_raw = scipy.io.loadmat(os.path.join(config.DATA_DIR, config.OMNIGLOT))\n\n  train_data = reshape_data(omni_raw['data'].T.astype('float32'))\n  test_data = reshape_data(omni_raw['testdata'].T.astype('float32'))\n\n  # Binarize the data with a fixed seed\n  if binarize:\n    np.random.seed(5)\n    train_data = (np.random.rand(*train_data.shape) < train_data).astype(float)\n    test_data = (np.random.rand(*test_data.shape) < test_data).astype(float)\n\n  shuffle_seed = 123\n  permutation = np.random.RandomState(seed=shuffle_seed).permutation(train_data.shape[0])\n  train_data = train_data[permutation]\n\n  x_train = train_data[:-n_validation]\n  x_valid = train_data[-n_validation:]\n  x_test = test_data\n\n  return x_train, x_valid, x_test\n\n"
  },
  {
    "path": "rebar_baseline/download_data.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\n\"\"\"Download MNIST, Omniglot datasets for Rebar.\"\"\"\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport urllib\nimport gzip\nimport os\nimport config\nimport struct\nimport numpy as np\nimport cPickle as pickle\nimport datasets\n\nMNIST_URL = 'http://yann.lecun.com/exdb/mnist'\nMNIST_BINARIZED_URL = 'http://www.cs.toronto.edu/~larocheh/public/datasets/binarized_mnist'\nOMNIGLOT_URL = 'https://github.com/yburda/iwae/raw/master/datasets/OMNIGLOT/chardata.mat'\n\nMNIST_FLOAT_TRAIN = 'train-images-idx3-ubyte'\n\n\ndef load_mnist_float(local_filename):\n  with open(local_filename, 'rb') as f:\n    f.seek(4)\n    nimages, rows, cols = struct.unpack('>iii', f.read(12))\n    dim = rows*cols\n\n    images = np.fromfile(f, dtype=np.dtype(np.ubyte))\n    images = (images/255.0).astype('float32').reshape((nimages, dim))\n\n  return images\n\nif __name__ == '__main__':\n  if not os.path.exists(config.DATA_DIR):\n    os.makedirs(config.DATA_DIR)\n\n  # Get MNIST and convert to npy file\n  local_filename = os.path.join(config.DATA_DIR, MNIST_FLOAT_TRAIN)\n  if not os.path.exists(local_filename):\n    urllib.urlretrieve(\"%s/%s.gz\" % (MNIST_URL, MNIST_FLOAT_TRAIN), local_filename+'.gz')\n    with gzip.open(local_filename+'.gz', 'rb') as f:\n      file_content = f.read()\n    with open(local_filename, 'wb') as f:\n      f.write(file_content)\n    os.remove(local_filename+'.gz')\n\n  mnist_float_train = load_mnist_float(local_filename)[:-10000]\n  # save in a nice format\n  np.save(os.path.join(config.DATA_DIR, config.MNIST_FLOAT), mnist_float_train)\n\n  # Get binarized MNIST\n  splits = ['train', 'valid', 'test']\n  mnist_binarized = []\n  for split in splits:\n    filename = 'binarized_mnist_%s.amat' % split\n    url = '%s/binarized_mnist_%s.amat' % (MNIST_BINARIZED_URL, split)\n    local_filename = os.path.join(config.DATA_DIR, filename)\n    if not os.path.exists(local_filename):\n      urllib.urlretrieve(url, local_filename)\n\n    with open(local_filename, 'rb') as f:\n      mnist_binarized.append((np.array([map(int, line.split()) for line in f.readlines()]).astype('float32'), None))\n\n  # save in a nice format\n  with open(os.path.join(config.DATA_DIR, config.MNIST_BINARIZED), 'w') as out:\n    pickle.dump(mnist_binarized, out)\n\n  # Get Omniglot\n  local_filename = os.path.join(config.DATA_DIR, config.OMNIGLOT)\n  if not os.path.exists(local_filename):\n    urllib.urlretrieve(OMNIGLOT_URL,\n                       local_filename)\n\n"
  },
  {
    "path": "rebar_baseline/logger.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\n\"\"\"Logger for REBAR\"\"\"\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nclass Logger:\n  def __init__(self):\n    pass\n\n  def log(self, key, value):\n    pass\n\n  def flush(self):\n    pass\n"
  },
  {
    "path": "rebar_baseline/rebar.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport functools\nimport tensorflow as tf\nimport numpy as np\nfrom scipy.misc import logsumexp\n\nimport tensorflow.contrib.slim as slim\nfrom tensorflow.python.ops import init_ops\nimport utils as U\n\ntry:\n  xrange          # Python 2\nexcept NameError:\n  xrange = range  # Python 3\n\nFLAGS = tf.flags.FLAGS\n\nQ_COLLECTION = \"q_collection\"\nP_COLLECTION = \"p_collection\"\n\nclass SBN(object):  # REINFORCE\n\n  def __init__(self,\n               hparams,\n               activation_func=tf.nn.sigmoid,\n               mean_xs = None,\n               eval_mode=False):\n    self.eval_mode = eval_mode\n    self.hparams = hparams\n    self.mean_xs = mean_xs\n    self.train_bias= -np.log(1./np.clip(mean_xs, 0.001, 0.999)-1.).astype(np.float32)\n    self.activation_func = activation_func\n\n    self.n_samples = tf.placeholder('int32')\n    self.x = tf.placeholder('float', [None, self.hparams.n_input])\n    self._x = tf.tile(self.x, [self.n_samples, 1])\n\n    self.batch_size = tf.shape(self._x)[0]\n\n    self.uniform_samples = dict()\n    self.uniform_samples_v = dict()\n    self.prior = tf.Variable(tf.zeros([self.hparams.n_hidden],\n                                      dtype=tf.float32),\n                             name='p_prior',\n                             collections=[tf.GraphKeys.GLOBAL_VARIABLES, P_COLLECTION])\n\n    self.run_recognition_network = False\n    self.run_generator_network = False\n    self.run_q_func = False\n\n    # Initialize temperature\n    self.pre_temperature_variable = tf.Variable(\n        np.log(self.hparams.temperature),\n        trainable=False,\n        dtype=tf.float32)\n    self.temperature_variable = tf.exp(self.pre_temperature_variable)\n\n    self.global_step = tf.Variable(0, trainable=False)\n    self.baseline_loss = []\n    self.ema = tf.train.ExponentialMovingAverage(decay=0.999)\n    self.maintain_ema_ops = []\n    self.optimizer_class = tf.train.AdamOptimizer(\n        learning_rate=1*self.hparams.learning_rate,\n        beta2=self.hparams.beta2)\n\n    self._generate_randomness()\n    self._create_network()\n\n\n  def initialize(self, sess):\n    self.sess = sess\n\n  def _create_eta(self, shape=[], collection='CV'):\n    return 2 * tf.sigmoid(tf.Variable(tf.zeros(shape), trainable=False,\n                                      collections=[collection, tf.GraphKeys.GLOBAL_VARIABLES, Q_COLLECTION]))\n\n  def _create_baseline(self, n_output=1, n_hidden=100,\n                       is_zero_init=False,\n                       collection='BASELINE'):\n    # center input\n    h = self._x\n    if self.mean_xs is not None:\n      h -= self.mean_xs\n\n    if is_zero_init:\n      initializer = init_ops.zeros_initializer()\n    else:\n      initializer = slim.variance_scaling_initializer()\n\n    with slim.arg_scope([slim.fully_connected],\n                        variables_collections=[collection, Q_COLLECTION],\n                        trainable=False,\n                        weights_initializer=initializer):\n      h = slim.fully_connected(h, n_hidden, activation_fn=tf.nn.tanh)\n      baseline = slim.fully_connected(h, n_output, activation_fn=None)\n\n      if n_output == 1:\n        baseline = tf.reshape(baseline, [-1])  # very important to reshape\n    return baseline\n\n\n  def _create_transformation(self, input, n_output, reuse, scope_prefix):\n    \"\"\"Create the deterministic transformation between stochastic layers.\n\n    If self.hparam.nonlinear:\n        2 x tanh layers\n    Else:\n        1 x linear layer\n    \"\"\"\n    if \"q_func\" in scope_prefix:\n      h = slim.fully_connected(input,\n                               self.hparams.n_hidden,\n                               reuse=reuse,\n                               activation_fn=tf.nn.relu,\n                               scope='%s_nonlinear_1' % scope_prefix)\n      h = slim.fully_connected(h,\n                               n_output,\n                               reuse=reuse,\n                               activation_fn=None,\n                               scope='%s_nonlinear_2' % scope_prefix)\n    if self.hparams.nonlinear:\n      h = slim.fully_connected(input,\n                               self.hparams.n_hidden,\n                               reuse=reuse,\n                               activation_fn=tf.nn.tanh,\n                               scope='%s_nonlinear_1' % scope_prefix)\n      h = slim.fully_connected(h,\n                               self.hparams.n_hidden,\n                               reuse=reuse,\n                               activation_fn=tf.nn.tanh,\n                               scope='%s_nonlinear_2' % scope_prefix)\n      h = slim.fully_connected(h,\n                               n_output,\n                               reuse=reuse,\n                               activation_fn=None,\n                               scope='%s' % scope_prefix)\n    else:\n      h = slim.fully_connected(input,\n                               n_output,\n                               reuse=reuse,\n                               activation_fn=None,\n                               scope='%s' % scope_prefix)\n    return h\n\n  def _recognition_network(self, sampler=None, log_likelihood_func=None):\n    \"\"\"x values -> samples from Q and return log Q(h|x).\"\"\"\n    samples = {}\n    reuse = None if not self.run_recognition_network else True\n\n    # Set defaults\n    if sampler is None:\n      sampler = self._random_sample\n\n    if log_likelihood_func is None:\n      log_likelihood_func = lambda sample, log_params: (\n        U.binary_log_likelihood(sample['activation'], log_params))\n\n    logQ = []\n\n\n    if self.hparams.task in ['sbn', 'omni']:\n      # Initialize the edge case\n      samples[-1] = {'activation': self._x}\n      if self.mean_xs is not None:\n        samples[-1]['activation'] -= self.mean_xs  # center the input\n      samples[-1]['activation'] = (samples[-1]['activation'] + 1)/2.0\n\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[Q_COLLECTION]):\n        for i in xrange(self.hparams.n_layer):\n          # Set up the input to the layer\n          input = 2.0*samples[i-1]['activation'] - 1.0\n\n          # Create the conditional distribution (output is the logits)\n          h = self._create_transformation(input,\n                                          n_output=self.hparams.n_hidden,\n                                          reuse=reuse,\n                                          scope_prefix='q_%d' % i)\n\n          samples[i] = sampler(h, self.uniform_samples[i], i)\n          logQ.append(log_likelihood_func(samples[i], h))\n\n      self.run_recognition_network = True\n      return logQ, samples\n    elif self.hparams.task == 'sp':\n      # Initialize the edge case\n      samples[-1] = {'activation': tf.split(self._x,\n                                            num_or_size_splits=2,\n                                            axis=1)[0]}  # top half of digit\n      if self.mean_xs is not None:\n        samples[-1]['activation'] -= np.split(self.mean_xs, 2, 0)[0]  # center the input\n      samples[-1]['activation'] = (samples[-1]['activation'] + 1)/2.0\n\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[Q_COLLECTION]):\n        for i in xrange(self.hparams.n_layer):\n          # Set up the input to the layer\n          input = 2.0*samples[i-1]['activation'] - 1.0\n\n          # Create the conditional distribution (output is the logits)\n          h = self._create_transformation(input,\n                                          n_output=self.hparams.n_hidden,\n                                          reuse=reuse,\n                                          scope_prefix='q_%d' % i)\n\n          samples[i] = sampler(h, self.uniform_samples[i], i)\n          logQ.append(log_likelihood_func(samples[i], h))\n\n      self.run_recognition_network = True\n      return logQ, samples\n\n  def _generator_network(self, samples, logQ, log_likelihood_func=None):\n    '''Returns learning signal and function.\n\n    This is the implementation for SBNs for the ELBO.\n\n    Args:\n      samples: dictionary of sampled latent variables\n      logQ: list of log q(h_i) terms\n      log_likelihood_func: function used to compute log probs for the latent\n        variables\n\n    Returns:\n      learning_signal: the \"reward\" function\n      function_term: part of the function that depends on the parameters\n        and needs to have the gradient taken through\n    '''\n    reuse=None if not self.run_generator_network else True\n\n    if self.hparams.task in ['sbn', 'omni']:\n      if log_likelihood_func is None:\n        log_likelihood_func = lambda sample, log_params: (\n          U.binary_log_likelihood(sample['activation'], log_params))\n\n      logPPrior = log_likelihood_func(\n          samples[self.hparams.n_layer-1],\n          tf.expand_dims(self.prior, 0))\n\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[P_COLLECTION]):\n\n        for i in reversed(xrange(self.hparams.n_layer)):\n          if i == 0:\n            n_output = self.hparams.n_input\n          else:\n            n_output = self.hparams.n_hidden\n          input = 2.0*samples[i]['activation']-1.0\n\n          h = self._create_transformation(input,\n                                          n_output,\n                                          reuse=reuse,\n                                          scope_prefix='p_%d' % i)\n\n          if i == 0:\n            # Assume output is binary\n            logP = U.binary_log_likelihood(self._x, h + self.train_bias)\n          else:\n            logPPrior += log_likelihood_func(samples[i-1], h)\n\n      self.run_generator_network = True\n      return logP + logPPrior - tf.add_n(logQ), logP + logPPrior\n    elif self.hparams.task == 'sp':\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[P_COLLECTION]):\n        n_output = int(self.hparams.n_input/2)\n        i = self.hparams.n_layer - 1  # use the last layer\n        input = 2.0*samples[i]['activation']-1.0\n\n        h = self._create_transformation(input,\n                                        n_output,\n                                        reuse=reuse,\n                                        scope_prefix='p_%d' % i)\n\n        # Predict on the lower half of the image\n        logP = U.binary_log_likelihood(tf.split(self._x,\n                                              num_or_size_splits=2,\n                                              axis=1)[1],\n                                     h + np.split(self.train_bias, 2, 0)[1])\n\n      self.run_generator_network = True\n      return logP, logP\n  \n  def _q_func(self, samples, collection='Q_FUNC'):\n    '''Returns learning signal and function.\n  \n    This is the implementation for SBNs for the ELBO.\n  \n    Args:\n      samples: dictionary of sampled latent variables\n      logQ: list of log q(h_i) terms\n      log_likelihood_func: function used to compute log probs for the latent\n        variables\n  \n    Returns:\n      learning_signal: the \"reward\" function\n      function_term: part of the function that depends on the parameters\n        and needs to have the gradient taken through\n    '''\n    reuse=None if not self.run_q_func else True\n  \n    if self.hparams.task in ['sbn', 'omni']:\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[collection, tf.GraphKeys.GLOBAL_VARIABLES, Q_COLLECTION]):\n  \n#        for i in reversed(xrange(self.hparams.n_layer)):\n#          if i == 0:\n#            n_output = self.hparams.n_input\n#          else:\n#            n_output = self.hparams.n_hidden\n        n_output = self.hparams.n_input\n        i = self.hparams.n_layer - 1  # use the last layer\n        input = 2.0*samples[i]['activation']-1.0\n  \n        h = self._create_transformation(input,\n                                        n_output,\n                                        reuse=reuse,\n                                        scope_prefix='q_func_%d' % i)\n        h = tf.reduce_sum(h)\n          \n      self.run_q_func = True\n      return h, h\n    elif self.hparams.task == 'sp':\n      with slim.arg_scope([slim.fully_connected],\n                          weights_initializer=slim.variance_scaling_initializer(),\n                          variables_collections=[collection, tf.GraphKeys.GLOBAL_VARIABLES, Q_COLLECTION]):\n        n_output = int(self.hparams.n_input/2)\n        i = self.hparams.n_layer - 1  # use the last layer\n        input = 2.0*samples[i]['activation']-1.0\n  \n        h = self._create_transformation(input,\n                                        n_output,\n                                        reuse=reuse,\n                                        scope_prefix='q_func_%d' % i)\n      self.run_q_func = True\n      return h, h\n\n  def _create_loss(self):\n    # Hard loss\n    logQHard, samples = self._recognition_network()\n    reinforce_learning_signal, reinforce_model_grad = self._generator_network(samples, logQHard)\n    logQHard = tf.add_n(logQHard)\n\n    # REINFORCE\n    learning_signal = tf.stop_gradient(U.center(reinforce_learning_signal))\n    self.optimizerLoss = -(learning_signal*logQHard +\n                           reinforce_model_grad)\n    self.lHat = map(tf.reduce_mean, [\n        reinforce_learning_signal,\n        U.rms(learning_signal),\n    ])\n\n    return reinforce_learning_signal\n\n  def _reshape(self, t):\n    return tf.transpose(tf.reshape(t,\n                      [self.n_samples, -1]))\n\n\n  def compute_tensor_variance(self, t):\n    \"\"\"Compute the mean per component variance.\n\n    Use a moving average to estimate the required moments.\n    \"\"\"\n    t_sq = tf.reduce_mean(tf.square(t))\n    self.maintain_ema_ops.append(self.ema.apply([t, t_sq]))\n\n    # mean per component variance\n    variance_estimator = (self.ema.average(t_sq) -\n                          tf.reduce_mean(\n                              tf.square(self.ema.average(t))))\n\n    return variance_estimator\n\n  def _create_train_op(self, grads_and_vars, extra_grads_and_vars=[]):\n    '''\n    Args:\n      grads_and_vars: gradients to apply and compute running average variance\n      extra_grads_and_vars: gradients to apply (not used to compute average variance)\n    '''\n    # Variance summaries\n    first_moment = U.vectorize(grads_and_vars, skip_none=True)\n    second_moment = tf.square(first_moment)\n    self.maintain_ema_ops.append(self.ema.apply([first_moment, second_moment]))\n\n    # Add baseline losses\n    if len(self.baseline_loss) > 0:\n      mean_baseline_loss = tf.reduce_mean(tf.add_n(self.baseline_loss))\n      extra_grads_and_vars += self.optimizer_class.compute_gradients(\n          mean_baseline_loss,\n          var_list=tf.get_collection('BASELINE'))\n\n    # Ensure that all required tensors are computed before updates are executed\n    extra_optimizer = tf.train.AdamOptimizer(\n        learning_rate=10*self.hparams.learning_rate,\n        beta2=self.hparams.beta2)\n\n    with tf.control_dependencies(\n        [tf.group(*[g for g, _ in (grads_and_vars + extra_grads_and_vars) if g is not None])]):\n\n      # Filter out the P_COLLECTION variables if we're in eval mode\n      if self.eval_mode:\n        grads_and_vars = [(g, v) for g, v in grads_and_vars\n                          if v not in tf.get_collection(P_COLLECTION)]\n\n      train_op = self.optimizer_class.apply_gradients(grads_and_vars,\n                                                      global_step=self.global_step)\n\n      if len(extra_grads_and_vars) > 0:\n        extra_train_op = extra_optimizer.apply_gradients(extra_grads_and_vars)\n      else:\n        extra_train_op = tf.no_op()\n\n      self.optimizer = tf.group(train_op, extra_train_op, *self.maintain_ema_ops)\n\n    # per parameter variance\n    variance_estimator = (self.ema.average(second_moment) -\n        tf.square(self.ema.average(first_moment)))\n    self.grad_variance = tf.reduce_mean(variance_estimator)\n\n  def _create_network(self):\n    logF = self._create_loss()\n    self.optimizerLoss = tf.reduce_mean(self.optimizerLoss)\n\n    # Setup optimizer\n    grads_and_vars = self.optimizer_class.compute_gradients(self.optimizerLoss)\n    self._create_train_op(grads_and_vars)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\n  def partial_fit(self, X, n_samples=1):\n    if hasattr(self, 'grad_variances'):\n      grad_variance_field_to_return = self.grad_variances\n    else:\n      grad_variance_field_to_return = self.grad_variance\n    _, res, grad_variance, step, temperature = self.sess.run(\n        (self.optimizer, self.lHat, grad_variance_field_to_return, self.global_step, self.temperature_variable),\n        feed_dict={self.x: X, self.n_samples: n_samples})\n    return res, grad_variance, step, temperature\n\n  def partial_grad(self, X, n_samples=1):\n    control_variate_grads, step = self.sess.run(\n        (self.control_variate_grads, self.global_step),\n        feed_dict={self.x: X, self.n_samples: n_samples})\n    return control_variate_grads, step\n\n  def partial_eval(self, X, n_samples=5):\n    if n_samples < 1000:\n      res, iwae = self.sess.run(\n          (self.lHat, self.iwae),\n          feed_dict={self.x: X, self.n_samples: n_samples})\n      res = [iwae] + res\n    else:  # special case to handle OOM\n      assert n_samples % 100 == 0, \"When using large # of samples, it must be divisble by 100\"\n      res = []\n      for i in xrange(int(n_samples/100)):\n        logF, = self.sess.run(\n            (self.logF,),\n            feed_dict={self.x: X, self.n_samples: 100})\n        res.append(logsumexp(logF, axis=1))\n      res = [np.mean(logsumexp(res, axis=0) - np.log(n_samples))]\n    return res\n\n\n  # Random samplers\n  def _mean_sample(self, log_alpha, _, layer):\n    \"\"\"Returns mean of random variables parameterized by log_alpha.\"\"\"\n    mu = tf.nn.sigmoid(log_alpha)\n    return {\n        'preactivation': mu,\n        'activation': mu,\n        'log_param': log_alpha,\n    }\n\n  def _generate_randomness(self):\n    for i in xrange(self.hparams.n_layer):\n      self.uniform_samples[i] = tf.stop_gradient(tf.random_uniform(\n          [self.batch_size, self.hparams.n_hidden]))\n\n  def _u_to_v(self, log_alpha, u, eps = 1e-8):\n    \"\"\"Convert u to tied randomness in v.\"\"\"\n    u_prime = tf.nn.sigmoid(-log_alpha)  # g(u') = 0\n\n    v_1 = (u - u_prime) / tf.clip_by_value(1 - u_prime, eps, 1)\n    v_1 = tf.clip_by_value(v_1, 0, 1)\n    v_1 = tf.stop_gradient(v_1)\n    v_1 = v_1*(1 - u_prime) + u_prime\n    v_0 = u / tf.clip_by_value(u_prime, eps, 1)\n    v_0 = tf.clip_by_value(v_0, 0, 1)\n    v_0 = tf.stop_gradient(v_0)\n    v_0 = v_0 * u_prime\n\n    v = tf.where(u > u_prime, v_1, v_0)\n    v = tf.check_numerics(v, 'v sampling is not numerically stable.')\n    v = v + tf.stop_gradient(-v + u)  # v and u are the same up to numerical errors\n\n    return v\n\n  def _random_sample(self, log_alpha, u, layer):\n    \"\"\"Returns sampled random variables parameterized by log_alpha.\"\"\"\n    # Generate tied randomness for later\n    if layer not in self.uniform_samples_v:\n      self.uniform_samples_v[layer] = self._u_to_v(log_alpha, u)\n\n    # Sample random variable underlying softmax/argmax\n    x = log_alpha + U.safe_log_prob(u) - U.safe_log_prob(1 - u)\n    samples = tf.stop_gradient(tf.to_float(x > 0))\n\n    return {\n        'preactivation': x,\n        'activation': samples,\n        'log_param': log_alpha,\n    }\n\n  def _random_sample_soft(self, log_alpha, u, layer, temperature=None):\n    \"\"\"Returns sampled random variables parameterized by log_alpha.\"\"\"\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    # Sample random variable underlying softmax/argmax\n    x = log_alpha + U.safe_log_prob(u) - U.safe_log_prob(1 - u)\n    x /= tf.expand_dims(temperature, -1)\n\n    if self.hparams.muprop_relaxation:\n      y = tf.nn.sigmoid(x + log_alpha * tf.expand_dims(temperature/(temperature + 1), -1))\n    else:\n      y = tf.nn.sigmoid(x)\n\n    return {\n        'preactivation': x,\n        'activation': y,\n        'log_param': log_alpha\n    }\n\n  def _random_sample_soft_v(self, log_alpha, _, layer, temperature=None):\n    \"\"\"Returns sampled random variables parameterized by log_alpha.\"\"\"\n    v = self.uniform_samples_v[layer]\n\n    return self._random_sample_soft(log_alpha, v, layer, temperature)\n\n  def get_gumbel_gradient(self):\n    logQ, softSamples = self._recognition_network(sampler=self._random_sample_soft)\n    logQ = tf.add_n(logQ)\n    logPPrior, logP = self._generator_network(softSamples)\n\n    softELBO = logPPrior + logP - logQ\n    gumbel_gradient = (self.optimizer_class.\n                       compute_gradients(softELBO))\n    debug = {\n        'softELBO': softELBO,\n    }\n\n    return gumbel_gradient, debug\n\n  # samplers used for quadratic version\n  def _random_sample_switch(self, log_alpha, u, layer, switch_layer, temperature=None):\n    \"\"\"Run partial discrete, then continuous path.\n\n       Args:\n        switch_layer: this layer and beyond will be continuous\n    \"\"\"\n    if layer < switch_layer:\n      return self._random_sample(log_alpha, u, layer)\n    else:\n      return self._random_sample_soft(log_alpha, u, layer, temperature)\n\n  def _random_sample_switch_v(self, log_alpha, u, layer, switch_layer, temperature=None):\n    \"\"\"Run partial discrete, then continuous path.\n\n       Args:\n        switch_layer: this layer and beyond will be continuous\n    \"\"\"\n    if layer < switch_layer:\n      return self._random_sample(log_alpha, u, layer)\n    else:\n      return self._random_sample_soft_v(log_alpha, u, layer, temperature)\n\n\n  # #####\n  # Gradient computation\n  # #####\n  def get_nvil_gradient(self):\n    \"\"\"Compute the NVIL gradient.\"\"\"\n    # Hard loss\n    logQHard, samples = self._recognition_network()\n    ELBO, reinforce_model_grad = self._generator_network(samples, logQHard)\n    logQHard = tf.add_n(logQHard)\n\n    # Add baselines (no variance normalization)\n    learning_signal = tf.stop_gradient(ELBO) - self._create_baseline()\n\n    # Set up losses\n    self.baseline_loss.append(tf.square(learning_signal))\n    optimizerLoss = -(tf.stop_gradient(learning_signal)*logQHard +\n                           reinforce_model_grad)\n    optimizerLoss = tf.reduce_mean(optimizerLoss)\n\n    nvil_gradient = self.optimizer_class.compute_gradients(optimizerLoss)\n    debug = {\n        'ELBO': ELBO,\n        'RMS of centered learning signal': U.rms(learning_signal),\n    }\n\n    return nvil_gradient, debug\n\n\n  def get_simple_muprop_gradient(self):\n    \"\"\" Computes the simple muprop gradient.\n\n    This muprop control variate does not include the linear term.\n    \"\"\"\n    # Hard loss\n    logQHard, hardSamples = self._recognition_network()\n    hardELBO, reinforce_model_grad = self._generator_network(hardSamples, logQHard)\n\n    # Soft loss\n    logQ, muSamples = self._recognition_network(sampler=self._mean_sample)\n    muELBO, _  = self._generator_network(muSamples, logQ)\n\n    scaling_baseline = self._create_eta(collection='BASELINE')\n    learning_signal = (hardELBO\n                       - scaling_baseline * muELBO\n                       - self._create_baseline())\n    self.baseline_loss.append(tf.square(learning_signal))\n\n    optimizerLoss = -(tf.stop_gradient(learning_signal) * tf.add_n(logQHard)\n                      + reinforce_model_grad)\n    optimizerLoss = tf.reduce_mean(optimizerLoss)\n\n    simple_muprop_gradient = (self.optimizer_class.\n                              compute_gradients(optimizerLoss))\n    debug = {\n        'ELBO': hardELBO,\n        'muELBO': muELBO,\n        'RMS': U.rms(learning_signal),\n    }\n\n    return simple_muprop_gradient, debug\n\n  def get_muprop_gradient(self):\n    \"\"\"\n    random sample function that actually returns mean\n    new forward pass that returns logQ as a list\n\n    can get x_i from samples\n    \"\"\"\n\n    # Hard loss\n    logQHard, hardSamples = self._recognition_network()\n    hardELBO, reinforce_model_grad = self._generator_network(hardSamples, logQHard)\n\n    # Soft loss\n    logQ, muSamples = self._recognition_network(sampler=self._mean_sample)\n    muELBO, _ = self._generator_network(muSamples, logQ)\n\n    # Compute gradients\n    muELBOGrads = tf.gradients(tf.reduce_sum(muELBO),\n                               [ muSamples[i]['activation'] for\n                                i in xrange(self.hparams.n_layer) ])\n\n    # Compute MuProp gradient estimates\n    learning_signal = hardELBO\n    optimizerLoss = 0.0\n    learning_signals = []\n    for i in xrange(self.hparams.n_layer):\n      dfDiff = tf.reduce_sum(\n          muELBOGrads[i] * (hardSamples[i]['activation'] -\n                            muSamples[i]['activation']),\n          axis=1)\n      dfMu = tf.reduce_sum(\n          tf.stop_gradient(muELBOGrads[i]) *\n          tf.nn.sigmoid(hardSamples[i]['log_param']),\n          axis=1)\n\n      scaling_baseline_0 = self._create_eta(collection='BASELINE')\n      scaling_baseline_1 = self._create_eta(collection='BASELINE')\n      learning_signals.append(learning_signal - scaling_baseline_0 * muELBO - scaling_baseline_1 * dfDiff - self._create_baseline())\n      self.baseline_loss.append(tf.square(learning_signals[i]))\n\n      optimizerLoss += (\n          logQHard[i] * tf.stop_gradient(learning_signals[i]) +\n          tf.stop_gradient(scaling_baseline_1) * dfMu)\n    optimizerLoss += reinforce_model_grad\n    optimizerLoss *= -1\n\n    optimizerLoss = tf.reduce_mean(optimizerLoss)\n\n    muprop_gradient = self.optimizer_class.compute_gradients(optimizerLoss)\n    debug = {\n        'ELBO': hardELBO,\n        'muELBO': muELBO,\n    }\n\n    debug.update(dict([\n        ('RMS learning signal layer %d' % i, U.rms(learning_signal))\n        for (i, learning_signal) in enumerate(learning_signals)]))\n\n    return muprop_gradient, debug\n\n  # REBAR gradient helper functions\n  def _create_gumbel_control_variate(self, logQHard, temperature=None):\n    '''Calculate gumbel control variate.\n    '''\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    logQ, softSamples = self._recognition_network(sampler=functools.partial(\n        self._random_sample_soft, temperature=temperature))\n    softELBO, _ = self._generator_network(softSamples, logQ)\n    logQ = tf.add_n(logQ)\n\n    # Generate the softELBO_v (should be the same value but different grads)\n    logQ_v, softSamples_v = self._recognition_network(sampler=functools.partial(\n        self._random_sample_soft_v, temperature=temperature))\n    softELBO_v, _ = self._generator_network(softSamples_v, logQ_v)\n    logQ_v = tf.add_n(logQ_v)\n\n    # Compute losses\n    learning_signal = tf.stop_gradient(softELBO_v)\n\n    # Control variate\n    h = (tf.stop_gradient(learning_signal) * tf.add_n(logQHard)\n          - softELBO + softELBO_v)\n\n    extra = (softELBO_v, -softELBO + softELBO_v)\n\n    return h, extra\n\n  def _create_relaxed_gumbel_control_variate(self, logQHard, temperature=None):\n    '''Calculate gumbel control variate.\n    '''\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    logQ, softSamples = self._recognition_network(sampler=functools.partial(\n        self._random_sample_soft, temperature=temperature))\n    softELBO, _ = self._generator_network(softSamples, logQ)\n    Q_func,_ = self._q_func(softSamples)\n    f_soft = softELBO + Q_func\n    logQ = tf.add_n(logQ)\n\n    # Generate the softELBO_v (should be the same value but different grads)\n    logQ_v, softSamples_v = self._recognition_network(sampler=functools.partial(\n        self._random_sample_soft_v, temperature=temperature))\n    softELBO_v, _ = self._generator_network(softSamples_v, logQ_v)\n    Q_func_v,_ = self._q_func(softSamples_v)\n    f_soft_v = softELBO_v + Q_func_v\n    logQ_v = tf.add_n(logQ_v)\n\n    # Compute losses\n    learning_signal = tf.stop_gradient(f_soft_v)\n\n    # Control variate\n    h = (tf.stop_gradient(learning_signal) * tf.add_n(logQHard)\n          - f_soft + f_soft_v)\n\n    extra = (f_soft_v, -f_soft + f_soft_v)\n\n    return h, extra\n\n  def _create_gumbel_control_variate_quadratic(self, logQHard, temperature=None):\n    '''Calculate gumbel control variate.\n    '''\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    h = 0\n    extra = []\n    for layer in xrange(self.hparams.n_layer):\n      logQ, softSamples = self._recognition_network(sampler=functools.partial(\n          self._random_sample_switch, switch_layer=layer, temperature=temperature))\n      softELBO, _ = self._generator_network(softSamples, logQ)\n\n      # Generate the softELBO_v (should be the same value but different grads)\n      logQ_v, softSamples_v = self._recognition_network(sampler=functools.partial(\n          self._random_sample_switch_v, switch_layer=layer, temperature=temperature))\n      softELBO_v, _ = self._generator_network(softSamples_v, logQ_v)\n\n      # Compute losses\n      learning_signal = tf.stop_gradient(softELBO_v)\n\n      # Control variate\n      h += (tf.stop_gradient(learning_signal) * logQHard[layer]\n            - softELBO + softELBO_v)\n\n      extra.append((softELBO_v, -softELBO + softELBO_v))\n\n    return h, extra\n  \n  def _create_relaxed_gumbel_control_variate_quadratic(self, logQHard, temperature=None):\n    '''Calculate gumbel control variate.\n    '''\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    h = 0\n    extra = []\n    for layer in xrange(self.hparams.n_layer):\n      logQ, softSamples = self._recognition_network(sampler=functools.partial(\n          self._random_sample_switch, switch_layer=layer, temperature=temperature))\n      softELBO, _ = self._generator_network(softSamples, logQ)\n      Q_func,_ = self._q_func(softSamples)\n      f_soft = softELBO + Q_func\n\n      # Generate the softELBO_v (should be the same value but different grads)\n      logQ_v, softSamples_v = self._recognition_network(sampler=functools.partial(\n          self._random_sample_switch_v, switch_layer=layer, temperature=temperature))\n      softELBO_v, _ = self._generator_network(softSamples_v, logQ_v)\n      Q_func_v,_ = self._q_func(softSamples_v)\n      f_soft_v = softELBO_v + Q_func_v\n\n      # Compute losses\n      learning_signal = tf.stop_gradient(f_soft_v)\n\n      # Control variate\n      h += (tf.stop_gradient(learning_signal) * logQHard[layer]\n            - f_soft + f_soft_v)\n\n      extra.append((f_soft_v, -f_soft + f_soft_v))\n\n    return h, extra\n\n  def _create_hard_elbo(self):\n    logQHard, hardSamples = self._recognition_network()\n    hardELBO, reinforce_model_grad = self._generator_network(hardSamples, logQHard)\n    reinforce_learning_signal = tf.stop_gradient(hardELBO)\n\n    # Center learning signal\n    baseline = self._create_baseline(collection='CV')\n    reinforce_learning_signal = tf.stop_gradient(reinforce_learning_signal) - baseline\n\n    nvil_gradient = (tf.stop_gradient(hardELBO) - baseline) * tf.add_n(logQHard) + reinforce_model_grad\n\n    return hardELBO, nvil_gradient, logQHard\n\n  def multiply_by_eta(self, h_grads, eta):\n    # Modifies eta\n    res = []\n    eta_statistics = []\n    for (g, v) in h_grads:\n      if g is None:\n        res.append((g, v))\n      else:\n        if 'network' not in eta:\n          eta['network'] = self._create_eta()\n        res.append((g*eta['network'], v))\n    eta_statistics.append(eta['network'])\n\n    return res, eta_statistics\n\n  def multiply_by_eta_per_layer(self, h_grads, eta):\n    # Modifies eta\n    res = []\n    eta_statistics = []\n    for (g, v) in h_grads:\n      if g is None:\n        res.append((g, v))\n      else:\n        if v not in eta:\n          eta[v] = self._create_eta()\n        res.append((g*eta[v], v))\n        eta_statistics.append(eta[v])\n\n    return res, eta_statistics\n\n  def multiply_by_eta_per_unit(self, h_grads, eta):\n    # Modifies eta\n    res = []\n    eta_statistics = []\n    for (g, v) in h_grads:\n      if g is None:\n        res.append((g, v))\n      else:\n        if v not in eta:\n          g_shape = g.shape_as_list()\n          assert len(g_shape) <= 2, 'Gradient has too many dimensions'\n          if len(g_shape) == 1:\n            eta[v] = self._create_eta(g_shape)\n          else:\n            eta[v] = self._create_eta([1, g_shape[1]])\n        h_grads.append((g*eta[v], v))\n        eta_statistics.extend(tf.nn.moments(tf.squeeze(eta[v]), axes=[0]))\n    return res, eta_statistics\n\n  def get_relaxed_dynamic_rebar_gradient(self):\n    \"\"\"Get the relaxed (Q=f+g) dynamic rebar gradient (t, eta optimized).\"\"\"\n    tiled_pre_temperature = tf.tile([self.pre_temperature_variable],\n                                [self.batch_size])\n    temperature = tf.exp(tiled_pre_temperature)\n\n    hardELBO, nvil_gradient, logQHard = self._create_hard_elbo()\n    if self.hparams.quadratic:\n      gumbel_cv, extra  = self._create_relaxed_gumbel_control_variate_quadratic(logQHard, temperature=temperature)\n    else:\n      gumbel_cv, extra  = self._create_relaxed_gumbel_control_variate(logQHard, temperature=temperature)\n\n    f_grads = self.optimizer_class.compute_gradients(tf.reduce_mean(-nvil_gradient))\n    f_grads = [(grad,var) for grad,var in f_grads if 'q_func' not in var.name]\n    print(len(f_grads))\n\n    eta = {}\n    h_grads, eta_statistics = self.multiply_by_eta_per_layer(\n        self.optimizer_class.compute_gradients(tf.reduce_mean(gumbel_cv)),\n        eta)\n    h_grads = [(grad,var) for grad,var in h_grads if 'q_func' not in var.name]\n    \n\n    model_grads = U.add_grads_and_vars(f_grads, h_grads)\n    total_grads = model_grads\n\n    # Construct the variance objective\n    g = U.vectorize(model_grads, set_none_to_zero=True)\n    self.maintain_ema_ops.append(self.ema.apply([g]))\n    gbar = 0  #tf.stop_gradient(self.ema.average(g))\n    variance_objective = tf.reduce_mean(tf.square(g - gbar))\n\n    reinf_g_t = 0\n    if self.hparams.quadratic:\n      for layer in xrange(self.hparams.n_layer):\n        gumbel_learning_signal, _ = extra[layer]\n        df_dt = tf.gradients(gumbel_learning_signal, tiled_pre_temperature)[0]\n        reinf_g_t_i, _ = self.multiply_by_eta_per_layer(\n            self.optimizer_class.compute_gradients(tf.reduce_mean(tf.stop_gradient(df_dt) * logQHard[layer])),\n            eta)\n        reinf_g_t_i = [(grad,var) for grad,var in reinf_g_t_i if 'q_func' not in var.name]\n        print(reinf_g_t_i)\n        reinf_g_t += U.vectorize(reinf_g_t_i, set_none_to_zero=True)\n\n      reparam = tf.add_n([reparam_i for _, reparam_i in extra])\n    else:\n      gumbel_learning_signal, reparam = extra\n      df_dt = tf.gradients(gumbel_learning_signal, tiled_pre_temperature)[0]\n      reinf_g_t, _ = self.multiply_by_eta_per_layer(\n          self.optimizer_class.compute_gradients(tf.reduce_mean(tf.stop_gradient(df_dt) * tf.add_n(logQHard))),\n          eta)\n      reinf_g_t = [(grad,var) for grad,var in reinf_g_t if 'q_func' not in var.name]\n      reinf_g_t = U.vectorize(reinf_g_t, set_none_to_zero=True)\n\n    reparam_g, _ = self.multiply_by_eta_per_layer(\n        self.optimizer_class.compute_gradients(tf.reduce_mean(reparam)),\n        eta)\n    reparam_g = [(grad,var) for grad,var in reparam_g if 'q_func' not in var.name]\n\n    reparam_g = U.vectorize(reparam_g, set_none_to_zero=True)\n    reparam_g_t = tf.gradients(tf.reduce_mean(2*tf.stop_gradient(g - gbar)*reparam_g), self.pre_temperature_variable)[0]\n    \n\n    variance_objective_grad = tf.reduce_mean(2*(g - gbar)*reinf_g_t) + reparam_g_t\n\n    debug = { 'ELBO': hardELBO,\n             'etas': eta_statistics,\n             'variance_objective': variance_objective,\n             }\n    return total_grads, debug, variance_objective, variance_objective_grad\n\n  def get_dynamic_rebar_gradient(self):\n    \"\"\"Get the dynamic rebar gradient (t, eta optimized).\"\"\"\n    tiled_pre_temperature = tf.tile([self.pre_temperature_variable],\n                                [self.batch_size])\n    temperature = tf.exp(tiled_pre_temperature)\n\n    hardELBO, nvil_gradient, logQHard = self._create_hard_elbo()\n    if self.hparams.quadratic:\n      gumbel_cv, extra  = self._create_gumbel_control_variate_quadratic(logQHard, temperature=temperature)\n    else:\n      gumbel_cv, extra  = self._create_gumbel_control_variate(logQHard, temperature=temperature)\n\n    f_grads = self.optimizer_class.compute_gradients(tf.reduce_mean(-nvil_gradient))\n\n    eta = {}\n    h_grads, eta_statistics = self.multiply_by_eta_per_layer(\n        self.optimizer_class.compute_gradients(tf.reduce_mean(gumbel_cv)),\n        eta)\n\n    model_grads = U.add_grads_and_vars(f_grads, h_grads)\n    total_grads = model_grads\n\n    # Construct the variance objective\n    g = U.vectorize(model_grads, set_none_to_zero=True)\n    self.maintain_ema_ops.append(self.ema.apply([g]))\n    gbar = 0  #tf.stop_gradient(self.ema.average(g))\n    variance_objective = tf.reduce_mean(tf.square(g - gbar))\n\n    reinf_g_t = 0\n    if self.hparams.quadratic:\n      for layer in xrange(self.hparams.n_layer):\n        gumbel_learning_signal, _ = extra[layer]\n        df_dt = tf.gradients(gumbel_learning_signal, tiled_pre_temperature)[0]\n        reinf_g_t_i, _ = self.multiply_by_eta_per_layer(\n            self.optimizer_class.compute_gradients(tf.reduce_mean(tf.stop_gradient(df_dt) * logQHard[layer])),\n            eta)\n        reinf_g_t += U.vectorize(reinf_g_t_i, set_none_to_zero=True)\n\n      reparam = tf.add_n([reparam_i for _, reparam_i in extra])\n    else:\n      gumbel_learning_signal, reparam = extra\n      df_dt = tf.gradients(gumbel_learning_signal, tiled_pre_temperature)[0]\n      reinf_g_t, _ = self.multiply_by_eta_per_layer(\n          self.optimizer_class.compute_gradients(tf.reduce_mean(tf.stop_gradient(df_dt) * tf.add_n(logQHard))),\n          eta)\n      reinf_g_t = U.vectorize(reinf_g_t, set_none_to_zero=True)\n\n    reparam_g, _ = self.multiply_by_eta_per_layer(\n        self.optimizer_class.compute_gradients(tf.reduce_mean(reparam)),\n        eta)\n    reparam_g = U.vectorize(reparam_g, set_none_to_zero=True)\n    reparam_g_t = tf.gradients(tf.reduce_mean(2*tf.stop_gradient(g - gbar)*reparam_g), self.pre_temperature_variable)[0]\n\n    variance_objective_grad = tf.reduce_mean(2*(g - gbar)*reinf_g_t) + reparam_g_t\n\n    debug = { 'ELBO': hardELBO,\n             'etas': eta_statistics,\n             'variance_objective': variance_objective,\n             }\n    return total_grads, debug, variance_objective, variance_objective_grad\n\n  def get_rebar_gradient(self):\n    \"\"\"Get the rebar gradient.\"\"\"\n    hardELBO, nvil_gradient, logQHard = self._create_hard_elbo()\n    if self.hparams.quadratic:\n      gumbel_cv, _ = self._create_gumbel_control_variate_quadratic(logQHard)\n    else:\n      gumbel_cv, _ = self._create_gumbel_control_variate(logQHard)\n\n    f_grads = self.optimizer_class.compute_gradients(tf.reduce_mean(-nvil_gradient))\n\n    eta = {}\n    h_grads, eta_statistics = self.multiply_by_eta_per_layer(\n        self.optimizer_class.compute_gradients(tf.reduce_mean(gumbel_cv)),\n        eta)\n\n    model_grads = U.add_grads_and_vars(f_grads, h_grads)\n    total_grads = model_grads\n\n    # Construct the variance objective\n    variance_objective = tf.reduce_mean(tf.square(U.vectorize(model_grads, set_none_to_zero=True)))\n\n    debug = { 'ELBO': hardELBO,\n             'etas': eta_statistics,\n             'variance_objective': variance_objective,\n             }\n    return total_grads, debug, variance_objective\n\n###\n# Create varaints\n###\nclass SBNSimpleMuProp(SBN):\n  def _create_loss(self):\n    simple_muprop_gradient, debug = self.get_simple_muprop_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n        debug['muELBO'],\n    ])\n\n    return debug['ELBO'], simple_muprop_gradient\n\n  def _create_network(self):\n    logF, loss_grads = self._create_loss()\n    self._create_train_op(loss_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\nclass SBNMuProp(SBN):\n  def _create_loss(self):\n    muprop_gradient, debug = self.get_muprop_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n        debug['muELBO'],\n    ])\n\n    return debug['ELBO'], muprop_gradient\n\n  def _create_network(self):\n    logF, loss_grads = self._create_loss()\n    self._create_train_op(loss_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\n\nclass SBNNVIL(SBN):\n  def _create_loss(self):\n    nvil_gradient, debug = self.get_nvil_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n    ])\n\n    return debug['ELBO'], nvil_gradient\n\n  def _create_network(self):\n    logF, loss_grads = self._create_loss()\n    self._create_train_op(loss_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\n\nclass SBNRebar(SBN):\n  def _create_loss(self):\n    rebar_gradient, debug, variance_objective = self.get_rebar_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n    ])\n    self.lHat.extend(map(tf.reduce_mean, debug['etas']))\n\n    return debug['ELBO'], rebar_gradient, variance_objective\n\n  def _create_network(self):\n    logF, loss_grads, variance_objective = self._create_loss()\n\n    # Create additional updates for control variates and temperature\n    eta_grads = (self.optimizer_class.compute_gradients(variance_objective,\n                                                        var_list=tf.get_collection('CV')))\n\n    self._create_train_op(loss_grads, eta_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\nclass SBNDynamicRebar(SBN):\n  def _create_loss(self):\n    rebar_gradient, debug, variance_objective, variance_objective_grad = self.get_dynamic_rebar_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n        self.temperature_variable,\n    ])\n    self.lHat.extend(debug['etas'])\n\n    return debug['ELBO'], rebar_gradient, variance_objective, variance_objective_grad\n\n  def _create_network(self):\n    logF, loss_grads, variance_objective, variance_objective_grad = self._create_loss()\n\n    # Create additional updates for control variates and temperature\n    eta_grads = (self.optimizer_class.compute_gradients(variance_objective,\n                                                        var_list=tf.get_collection('CV'))\n                 + [(variance_objective_grad, self.pre_temperature_variable)])\n\n    self._create_train_op(loss_grads, eta_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n    \nclass SBNRelaxedDynamicRebar(SBN):\n  def _create_loss(self):\n    rebar_gradient, debug, variance_objective, variance_objective_grad = self.get_relaxed_dynamic_rebar_gradient()\n\n    self.lHat = map(tf.reduce_mean, [\n        debug['ELBO'],\n        self.temperature_variable,\n    ])\n    self.lHat.extend(debug['etas'])\n\n    return debug['ELBO'], rebar_gradient, variance_objective, variance_objective_grad\n\n  def _create_network(self):\n    logF, loss_grads, variance_objective, variance_objective_grad = self._create_loss()\n\n    # Create additional updates for control variates and temperature\n    eta_grads = (self.optimizer_class.compute_gradients(variance_objective,\n                                                        var_list=tf.get_collection('CV'))\n                 + [(variance_objective_grad, self.pre_temperature_variable)]\n                 + self.optimizer_class.compute_gradients(variance_objective,\n                                                           var_list=tf.get_collection('Q_FUNC')))\n#    eta_grads = (self.optimizer_class.compute_gradients(variance_objective,\n#                                                        var_list=tf.get_collection('CV'))\n#                 + self.optimizer_class.compute_gradients(variance_objective,\n#                                                           var_list=tf.get_collection('Q_FUNC')))\n\n    self._create_train_op(loss_grads, eta_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\n\nclass SBNTrackGradVariances(SBN):\n  \"\"\"Follow NVIL, compute gradient variances for NVIL, MuProp and REBAR.\"\"\"\n  def compute_gradient_moments(self, grads_and_vars):\n    first_moment = U.vectorize(grads_and_vars, set_none_to_zero=True)\n    second_moment = tf.square(first_moment)\n    self.maintain_ema_ops.append(self.ema.apply([first_moment, second_moment]))\n\n    return self.ema.average(first_moment), self.ema.average(second_moment)\n\n  def _create_loss(self):\n    self.losses = [\n        ('NVIL', self.get_nvil_gradient),\n        ('SimpleMuProp', self.get_simple_muprop_gradient),\n        ('MuProp', self.get_muprop_gradient),\n    ]\n\n    moments = []\n    for k, v in self.losses:\n      print(k)\n      gradient, debug = v()\n      if k == 'SimpleMuProp':\n        ELBO = debug['ELBO']\n        gradient_to_follow = gradient\n\n      moments.append(self.compute_gradient_moments(\n          gradient))\n\n    self.losses.append(('DynamicREBAR', self.get_dynamic_rebar_gradient))\n    dynamic_rebar_gradient, _, variance_objective, variance_objective_grad = self.get_dynamic_rebar_gradient()\n    moments.append(self.compute_gradient_moments(dynamic_rebar_gradient))\n\n    self.losses.append(('REBAR', self.get_rebar_gradient))\n    rebar_gradient, _, variance_objective2 = self.get_rebar_gradient()\n    moments.append(self.compute_gradient_moments(rebar_gradient))\n    \n    self.losses.append(('RelaxedDynamicREBAR', self.get_relaxed_dynamic_rebar_gradient))\n    relaxed_dynamic_rebar_gradient, _, variance_objective3, variance_objective_grad2 = self.get_relaxed_dynamic_rebar_gradient()\n    moments.append(self.compute_gradient_moments(relaxed_dynamic_rebar_gradient))\n\n    mu = tf.reduce_mean(tf.stack([f for f, _ in moments]), axis=0)\n    self.grad_variances = []\n    deviations = []\n    for f, s in moments:\n      self.grad_variances.append(tf.reduce_mean(s - tf.square(mu)))\n      deviations.append(tf.reduce_mean(tf.square(f - mu)))\n\n    self.lHat = map(tf.reduce_mean, [\n        ELBO,\n        self.temperature_variable,\n        variance_objective_grad,\n        variance_objective_grad*variance_objective_grad,\n    ])\n    self.lHat.extend(deviations)\n    self.lHat.append(tf.log(tf.reduce_mean(mu*mu)))\n    #    self.lHat.extend(map(tf.log, grad_variances))\n\n    return ELBO, gradient_to_follow, variance_objective + variance_objective2 + variance_objective3, variance_objective_grad + variance_objective_grad2\n\n  def _create_network(self):\n    logF, loss_grads, variance_objective, variance_objective_grad = self._create_loss()\n    eta_grads = (self.optimizer_class.compute_gradients(variance_objective,\n                                                        var_list=tf.get_collection('CV'))\n                 + [(variance_objective_grad, self.pre_temperature_variable)]\n                 + self.optimizer_class.compute_gradients(variance_objective,\n                                                           var_list=tf.get_collection('Q_FUNC')))\n    self._create_train_op(loss_grads, eta_grads)\n\n    # Create IWAE lower bound for evaluation\n    self.logF = self._reshape(logF)\n    self.iwae = tf.reduce_mean(U.logSumExp(self.logF, axis=1) -\n                               tf.log(tf.to_float(self.n_samples)))\n\n\nclass SBNGumbel(SBN):\n  def _random_sample_soft(self, log_alpha, u, layer, temperature=None):\n    \"\"\"Returns sampled random variables parameterized by log_alpha.\"\"\"\n    if temperature is None:\n      temperature = self.hparams.temperature\n\n    # Sample random variable underlying softmax/argmax\n    x = log_alpha + U.safe_log_prob(u) - U.safe_log_prob(1 - u)\n    x /= temperature\n\n    if self.hparams.muprop_relaxation:\n      x += temperature/(temperature + 1)*log_alpha\n\n    y = tf.nn.sigmoid(x)\n\n    return {\n        'preactivation': x,\n        'activation': y,\n        'log_param': log_alpha\n    }\n\n  def _create_loss(self):\n    # Hard loss\n    logQHard, hardSamples = self._recognition_network()\n    hardELBO, _ = self._generator_network(hardSamples, logQHard)\n\n    logQ, softSamples = self._recognition_network(sampler=self._random_sample_soft)\n    softELBO, _ = self._generator_network(softSamples, logQ)\n\n    self.optimizerLoss = -softELBO\n    self.lHat = map(tf.reduce_mean, [\n        hardELBO,\n        softELBO,\n    ])\n\n    return hardELBO\n\ndefault_hparams = tf.contrib.training.HParams(model='SBNGumbel',\n                             n_hidden=200,\n                             n_input=784,\n                             n_layer=1,\n                             nonlinear=False,\n                             learning_rate=0.001,\n                             temperature=0.5,\n                             n_samples=1,\n                             batch_size=24,\n                             trial=1,\n                             muprop_relaxation=True,\n                             dynamic_b=False, # dynamic binarization\n                             quadratic=True,\n                             beta2=0.99999,\n                             task='sbn',\n                             )\n"
  },
  {
    "path": "rebar_baseline/rebar_train.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport json\nimport random\nimport sys\nimport os\n\nimport numpy as np\nimport tensorflow as tf\n\nimport rebar\nimport datasets\nimport logger as L\n\ntry:\n  xrange          # Python 2\nexcept NameError:\n  xrange = range  # Python 3\n\ngfile = tf.gfile\n\ntf.app.flags.DEFINE_string(\"working_dir\", \"/tmp/rebar\",\n                           \"\"\"Directory where to save data, write logs, etc.\"\"\")\ntf.app.flags.DEFINE_string('hparams', '',\n                           '''Comma separated list of name=value pairs.''')\ntf.app.flags.DEFINE_integer('eval_freq', 20,\n                           '''How often to run the evaluation step.''')\nFLAGS = tf.flags.FLAGS\n\ndef manual_scalar_summary(name, value):\n  value = tf.Summary.Value(tag=name, simple_value=value)\n  summary_str = tf.Summary(value=[value])\n  return summary_str\n\ndef eval(sbn, eval_xs, n_samples=100, batch_size=5):\n  n = eval_xs.shape[0]\n  i = 0\n  res = []\n  while i < n:\n    batch_xs = eval_xs[i:min(i+batch_size, n)]\n    res.append(sbn.partial_eval(batch_xs, n_samples))\n    i += batch_size\n  res = np.mean(res, axis=0)\n  return res\n\ndef train(sbn, train_xs, valid_xs, test_xs, training_steps, debug=False):\n  hparams = sorted(sbn.hparams.values().items())\n  hparams = (map(str, x) for x in hparams)\n  hparams = ('_'.join(x) for x in hparams)\n  hparams_str = '.'.join(hparams)\n\n  logger = L.Logger()\n\n  # Create the experiment name from the hparams\n  experiment_name = ([str(sbn.hparams.n_hidden) for i in xrange(sbn.hparams.n_layer)] +\n                     [str(sbn.hparams.n_input)])\n  if sbn.hparams.nonlinear:\n    experiment_name = '~'.join(experiment_name)\n  else:\n    experiment_name = '-'.join(experiment_name)\n  experiment_name = 'SBN_%s' % experiment_name\n  rowkey = {'experiment': experiment_name,\n            'model': hparams_str}\n\n  # Create summary writer\n  summ_dir = os.path.join(FLAGS.working_dir, hparams_str)\n  summary_writer = tf.summary.FileWriter(\n      summ_dir, flush_secs=15, max_queue=100)\n\n  sv = tf.train.Supervisor(logdir=os.path.join(\n      FLAGS.working_dir, hparams_str),\n                     save_summaries_secs=0,\n                     save_model_secs=1200,\n                     summary_op=None,\n                     recovery_wait_secs=30,\n                     global_step=sbn.global_step)\n  with sv.managed_session() as sess:\n    # Dump hparams to file\n    with gfile.Open(os.path.join(FLAGS.working_dir,\n                                 hparams_str,\n                                 'hparams.json'),\n                    'w') as out:\n      json.dump(sbn.hparams.values(), out)\n\n    sbn.initialize(sess)\n    batch_size = sbn.hparams.batch_size\n    scores = []\n    n = train_xs.shape[0]\n    index = range(n)\n\n    while not sv.should_stop():\n      lHats = []\n      grad_variances = []\n      temperatures = []\n      random.shuffle(index)\n      i = 0\n      while i < n:\n        batch_index = index[i:min(i+batch_size, n)]\n        batch_xs = train_xs[batch_index, :]\n\n        if sbn.hparams.dynamic_b:\n          # Dynamically binarize the batch data\n          batch_xs = (np.random.rand(*batch_xs.shape) < batch_xs).astype(float)\n\n        lHat, grad_variance, step, temperature = sbn.partial_fit(batch_xs,\n                                                    sbn.hparams.n_samples)\n        if debug:\n          print(i, lHat)\n          if i > 100:\n            return\n        lHats.append(lHat)\n        grad_variances.append(grad_variance)\n        temperatures.append(temperature)\n        i += batch_size\n\n      grad_variances = np.log(np.mean(grad_variances, axis=0)).tolist()\n      summary_strings = []\n      if isinstance(grad_variances, list):\n        grad_variances = dict(zip([k for (k, v) in sbn.losses], map(float, grad_variances)))\n        rowkey['step'] = step\n        logger.log(rowkey, {'step': step,\n                             'train': np.mean(lHats, axis=0)[0],\n                             'grad_variances': grad_variances,\n                             'temperature': np.mean(temperatures), })\n        grad_variances = '\\n'.join(map(str, sorted(grad_variances.iteritems())))\n      else:\n        rowkey['step'] = step\n        logger.log(rowkey, {'step': step,\n                             'train': np.mean(lHats, axis=0)[0],\n                             'grad_variance': grad_variances,\n                             'temperature': np.mean(temperatures), })\n        summary_strings.append(manual_scalar_summary(\"log grad variance\", grad_variances))\n\n      print('Step %d: %s\\n%s' % (step, str(np.mean(lHats, axis=0)), str(grad_variances)))\n\n      # Every few epochs compute test and validation scores\n      epoch = int(step / (train_xs.shape[0] / sbn.hparams.batch_size))\n      if epoch % FLAGS.eval_freq == 0:\n        valid_res = eval(sbn, valid_xs)\n        test_res= eval(sbn, test_xs)\n\n        print('\\nValid %d: %s' % (step, str(valid_res)))\n        print('Test %d: %s\\n' % (step, str(test_res)))\n        logger.log(rowkey, {'step': step,\n                             'valid': valid_res[0],\n                             'test': test_res[0]})\n        logger.flush()  # Flush infrequently\n\n      # Create summaries\n      summary_strings.extend([\n        manual_scalar_summary(\"Train ELBO\", np.mean(lHats, axis=0)[0]),\n        manual_scalar_summary(\"Temperature\", np.mean(temperatures)),\n      ])\n      for summ_str in summary_strings:\n        summary_writer.add_summary(summ_str, global_step=step)\n      summary_writer.flush()\n\n      sys.stdout.flush()\n      scores.append(np.mean(lHats, axis=0))\n\n      if step > training_steps:\n        break\n\n    return scores\n\n\ndef main():\n  # Parse hyperparams\n  hparams = rebar.default_hparams\n  hparams.parse(FLAGS.hparams)\n  print(hparams.values())\n\n  train_xs, valid_xs, test_xs = datasets.load_data(hparams)\n  mean_xs = np.mean(train_xs, axis=0)  # Compute mean centering on training\n\n  training_steps = 2000000\n  model = getattr(rebar, hparams.model)\n  sbn = model(hparams, mean_xs=mean_xs)\n\n  scores = train(sbn, train_xs, valid_xs, test_xs,\n                 training_steps=training_steps, debug=False)\n\nif __name__ == '__main__':\n  main()\n"
  },
  {
    "path": "rebar_baseline/utils.py",
    "content": "# Copyright 2017 Google Inc. All Rights Reserved.\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\n# ==============================================================================\n\n\"\"\"Basic data management and plotting utilities.\"\"\"\n\nfrom __future__ import absolute_import\nfrom __future__ import division\nfrom __future__ import print_function\n\nimport os\nimport cPickle as pickle\nimport getpass\nimport numpy as np\nimport gc\nimport tensorflow as tf\n\n#\n# Python utlities\n#\ndef exp_moving_average(x, alpha=0.9):\n  res = []\n  mu = 0\n  alpha_factor = 1\n  for x_i in x:\n    mu += (1 - alpha)*(x_i - mu)\n    alpha_factor *= alpha\n    res.append(mu/(1 - alpha_factor))\n\n  return np.array(res)\n\ndef sanitize(s):\n  return s.replace('.', '_')\n\n#\n# Tensorflow utilities\n#\ndef softplus(x):\n  '''\n  Let m = max(0, x), then,\n\n  sofplus(x) = log(1 + e(x)) = log(e(0) + e(x)) = log(e(m)(e(-m) + e(x-m)))\n             = m + log(e(-m) + e(x - m))\n\n  The term inside of the log is guaranteed to be between 1 and 2.\n  '''\n  m = tf.maximum(tf.zeros_like(x), x)\n  return m + tf.log(tf.exp(-m) + tf.exp(x - m))\n\ndef safe_log_prob(x, eps=1e-8):\n  return tf.log(tf.clip_by_value(x, eps, 1.0))\n\ndef rms(x):\n  return tf.sqrt(tf.reduce_mean(tf.square(x)))\n\ndef center(x):\n  mu = (tf.reduce_sum(x) - x)/tf.to_float(tf.shape(x)[0] - 1)\n  return x - mu\n\ndef vectorize(grads_and_vars, set_none_to_zero=False, skip_none=False):\n  if set_none_to_zero:\n    return tf.concat([tf.reshape(g, [-1]) if g is not None else\n                         tf.reshape(tf.zeros_like(v), [-1]) for g, v in grads_and_vars], 0)\n  elif skip_none:\n    return tf.concat([tf.reshape(g, [-1]) for g, v in grads_and_vars if g is not None], 0)\n  else:\n    return tf.concat([tf.reshape(g, [-1]) for g, v in grads_and_vars], 0)\n\ndef add_grads_and_vars(a, b):\n  '''Add grads_and_vars from two calls to tf.compute_gradients.'''\n  res = []\n  for (g_a, v_a), (g_b, v_b) in zip(a, b):\n    assert v_a == v_b\n    if g_a is None:\n      res.append((g_b, v_b))\n    elif g_b is None:\n      res.append((g_a, v_a))\n    else:\n      res.append((g_a + g_b, v_a))\n  return res\n\ndef binary_log_likelihood(y, log_y_hat):\n  \"\"\"Computes binary log likelihood.\n\n  Args:\n    y: observed data\n    log_y_hat: parameters of the binary variables\n\n  Returns:\n    log_likelihood\n  \"\"\"\n  return tf.reduce_sum(y*(-softplus(-log_y_hat)) +\n                       (1 - y)*(-log_y_hat-softplus(-log_y_hat)),\n                       1)\n\ndef cov(a, b):\n  \"\"\"Compute the sample covariance between two vectors.\"\"\"\n  mu_a = tf.reduce_mean(a)\n  mu_b = tf.reduce_mean(b)\n  n = tf.to_float(tf.shape(a)[0])\n\n  return tf.reduce_sum((a - mu_a)*(b - mu_b))/(n - 1.0)\n\ndef corr(a, b):\n  return cov(a, b)*tf.rsqrt(cov(a, a))*tf.rsqrt(cov(b, b))\n\ndef logSumExp(t, axis=0, keep_dims = False):\n  '''Computes the log(sum(exp(t))) numerically stabily.\n\n  Args:\n    t: input tensor\n    axis: which axis to sum over\n    keep_dims: whether to keep the dim or not\n\n  Returns:\n    tensor with result\n\n  '''\n  m = tf.reduce_max(t, [axis])\n  res = m + tf.log(tf.reduce_sum(tf.exp(t - tf.expand_dims(m, axis)), [axis]))\n\n  if keep_dims:\n    return tf.expand_dims(res, axis)\n  else:\n    return res\n"
  },
  {
    "path": "rebar_tf.py",
    "content": "import numpy as np\nimport tensorflow as tf\n\nimport matplotlib.pyplot as plt\nfrom matplotlib.pyplot import ion\nion()\nimport matplotlib\n\nmatplotlib.rc(\"savefig\") #, dpi=300)\n\n\ndef safe_log_prob(x, eps=1e-8):\n    return tf.log(tf.clip_by_value(x, eps, 1.0))\n\ndef safe_clip(x, eps=1e-8):\n    return tf.clip_by_value(x, eps, 1.0)\n\n\ndef gs(x):\n    return x.get_shape().as_list()\n\n\ndef softplus(x):\n    '''\n    Let m = max(0, x), then,\n\n    sofplus(x) = log(1 + e(x)) = log(e(0) + e(x)) = log(e(m)(e(-m) + e(x-m)))\n                         = m + log(e(-m) + e(x - m))\n\n    The term inside of the log is guaranteed to be between 1 and 2.\n    '''\n    m = tf.maximum(tf.zeros_like(x), x)\n    return m + tf.log(tf.exp(-m) + tf.exp(x - m))\n\n\ndef bernoulli_loglikelihood(b, log_alpha):\n    return b * (-softplus(-log_alpha)) + (1 - b) * (-log_alpha - softplus(-log_alpha))\n\n\ndef bernoulli_loglikelihood_derivitive(b, log_alpha):\n    assert gs(b) == gs(log_alpha)\n    sna = tf.sigmoid(-log_alpha)\n    return b * sna - (1-b) * (1 - sna)\n\n\nclass REBAROptimizer(object):\n    def __init__(self, sess, loss, log_alpha=None, dim=None, name=\"REBAR\", learning_rate=.01, n_samples=1):\n        self.name = name\n        self.sess = sess\n        self.loss = loss\n        self.dim = dim\n        self.log_alpha = log_alpha\n        self.learning_rate = learning_rate\n        self.n_samples = n_samples\n        self.variance_optimizer = tf.train.AdamOptimizer(learning_rate)\n\n        \"\"\" model parameters \"\"\"\n        self._create_model_parameters()\n        \"\"\" reparameterization noise \"\"\"\n        self._create_reparam_variables()\n        \"\"\" relaxed loss evaluations \"\"\"\n        self._create_loss_evaluations()\n        \"\"\" gradvars for optimizers \"\"\"\n        self._create_gradvars()\n        \"\"\" variance reduction optimization operation \"\"\"\n        self.variance_reduction_op = self.variance_optimizer.apply_gradients(self.variance_gradvars)\n\n    def _create_model_parameters(self):\n        # alpha = theta / (1 - theta)\n        if self.log_alpha is None:\n            \"no log alpha given, creating here\"\n            self.log_alpha = tf.Variable(\n                [0.0 for i in range(self.dim)],  # initial value\n                name='log_alpha', dtype=tf.float32\n            )\n            self.batch_log_alpha = self.log_alpha\n            self.batch_size = 1\n        else:\n            sh = gs(self.log_alpha)\n            if len(sh) > 1:\n                self.batch_log_alpha = self.log_alpha\n                self.log_alpha = tf.reshape(self.batch_log_alpha, [-1])\n            self.dim = gs(self.log_alpha)[0]\n            self.batch_size = sh[0]\n        a = tf.exp(self.log_alpha)\n        theta = a / (1 + a)\n        tf.summary.histogram(\"theta\", theta)\n        # expanded version for internal purposes\n        self._log_alpha = tf.expand_dims(self.log_alpha, 0)\n        n_vars = self.dim / self.batch_size\n        self.n_vars = n_vars\n        self.batch_log_temperature = tf.Variable(\n            [np.log(.5) for i in range(n_vars)],\n            trainable=False,\n            name='log_temperature',\n            dtype=tf.float32\n        )\n        self.log_temperature = tf.reshape(tf.tile(tf.expand_dims(self.batch_log_temperature, 0), [self.batch_size, 1]), [-1])\n        self.tiled_log_temperature = tf.tile([self.log_temperature], [self.n_samples, 1])\n        self.temperature = tf.exp(self.tiled_log_temperature)\n        tf.summary.histogram(\"temp\", self.temperature)\n        self.batch_eta = tf.Variable(\n            [1.0 for i in range(n_vars)],\n            trainable=False,\n            name='eta',\n            dtype=tf.float32\n        )\n        self.eta = tf.reshape(tf.tile(tf.expand_dims(self.batch_eta, 0), [self.batch_size, 1]), [-1])\n\n    def _create_reparam_variables(self, eps=1e-8):\n        # noise for generating z\n        u = tf.random_uniform([self.n_samples, self.dim], dtype=tf.float32)\n        log_alpha = self._log_alpha\n        # logistic reparameterization z = g(u, log_alpha)\n        z = log_alpha + safe_log_prob(u) - safe_log_prob(1 - u)\n        # b = H(z)\n        b = tf.to_float(tf.stop_gradient(z > 0))\n        # g(u', log_alpha) = 0\n        u_prime = tf.nn.sigmoid(-log_alpha)\n        v_1 = (u - u_prime) / tf.clip_by_value(1 - u_prime, eps, 1)\n        v_1 = tf.clip_by_value(v_1, 0, 1)\n        v_1 = tf.stop_gradient(v_1)\n        v_1 = v_1 * (1 - u_prime) + u_prime\n        v_0 = u / tf.clip_by_value(u_prime, eps, 1)\n        v_0 = tf.clip_by_value(v_0, 0, 1)\n        v_0 = tf.stop_gradient(v_0)\n        v_0 = v_0 * u_prime\n\n        v = tf.where(u > u_prime, v_1, v_0)\n        v = tf.check_numerics(v, 'v sampling is not numerically stable.')\n        v = v + tf.stop_gradient(-v + u)  # v and u are the same up to numerical errors\n        tf.summary.histogram(\"u-v\", u-v)\n\n        z_tilde = log_alpha + safe_log_prob(v) - safe_log_prob(1 - v)\n        self.b = b\n        self.z = z\n        self.z_tilde = z_tilde\n\n    def _create_loss_evaluations(self):\n        \"\"\"\n        produces f(b), f(sig(z)), f(sig(z_tilde))\n        \"\"\"\n        # relaxed inputs\n        log_alpha = self._log_alpha\n        sig_z = tf.nn.sigmoid(self.z / self.temperature + log_alpha)\n        sig_z_tilde = tf.nn.sigmoid(self.z_tilde / self.temperature + log_alpha)\n        # evaluate loss\n        f_b = tf.reshape(self.loss(tf.reshape(self.b, [self.batch_size, -1])), [-1])\n        f_z = tf.reshape(self.loss(tf.reshape(sig_z, [self.batch_size, -1])), [-1])\n        f_z_tilde = tf.reshape(self.loss(tf.reshape(sig_z_tilde, [self.batch_size, -1])), [-1])\n        self.f_b = f_b\n        self.f_z = f_z\n        self.f_z_tilde = f_z_tilde\n\n    def _create_gradvars(self):\n        \"\"\"\n        produces d[log p(b)]/d[log_alpha], d[f(sigma_theta(z))]/d[log_alpha], d[f(sigma_theta(z_tilde))]/d[log_alpha]\n        \"\"\"\n        log_alpha = self._log_alpha\n        eta = tf.expand_dims(self.eta, 0)\n        f_b = tf.expand_dims(self.f_b, 1)\n        f_z_tilde = tf.expand_dims(self.f_z_tilde, 1)\n        d_log_p_d_log_alpha = bernoulli_loglikelihood_derivitive(self.b, log_alpha)\n        term1 = ((f_b - eta * f_z_tilde) * d_log_p_d_log_alpha)[0]\n        # d[f(sigma_theta(z))]/d[log_alpha] - eta * d[f(sigma_theta(z_tilde))]/d[log_alpha]\n        term2 = tf.gradients(\n            tf.reduce_mean(self.f_z - self.f_z_tilde),\n            self.log_alpha\n        )[0]\n        # rebar gradient estimator\n        rebar = term1 + self.eta * term2\n        reinforce = (f_b * d_log_p_d_log_alpha)[0]\n        # now compute gradients of the variance of this wrt other parameters\n        # eta\n        d_var_d_eta = tf.gradients(\n            tf.reduce_sum(tf.square(rebar)) / self.batch_size,\n            self.batch_eta\n        )[0]\n        # temperature\n        d_var_d_temperature = tf.gradients(\n            tf.reduce_sum(tf.square(rebar)) / self.batch_size,\n            self.batch_log_temperature\n        )[0]\n        self._rebar = rebar\n        self.rebar = tf.reshape(rebar, [self.batch_size, -1])\n        self.reinforce = tf.reshape(reinforce, [self.batch_size, -1])\n        tf.summary.histogram(\"rebar_gradient\", rebar)\n        tf.summary.histogram(\"reinforce_gradient\", reinforce)\n        self.rebar_gradvars = [(rebar, self.log_alpha)]\n        self.variance_gradvars = [(d_var_d_eta, self.batch_eta), (d_var_d_temperature, self.batch_log_temperature)]\n\n    def train(self, n_steps=10000):\n        self.sess.run(tf.global_variables_initializer())\n        ave_loss = tf.reduce_mean(self.f_b)\n        summ_op = tf.summary.merge_all()\n        summary_writer = tf.summary.FileWriter(\"/tmp/rebar\")\n        for iter in xrange(n_steps):\n            if iter % 100 == 0:\n                _, sum_str, loss_val, g_val, g_val_r, la, t, e = sess.run(\n                    [self.train_op, summ_op, ave_loss, self.rebar, self.reinforce, self.log_alpha, self.log_temperature, self.eta])\n                summary_writer.add_summary(sum_str, iter)\n            else:\n                _, loss_val, g_val, g_val_r, la, t, e = sess.run(\n                [self.train_op, ave_loss, self.rebar, self.reinforce, self.log_alpha, self.log_temperature, self.eta])\n\n\nclass RelaxedREBAROptimizer(REBAROptimizer):\n    def __init__(self, sess, loss, q_func, log_alpha=None, dim=None, name=\"REBAR\", learning_rate=.01, n_samples=1):\n        self.q_func = q_func\n        super(RelaxedREBAROptimizer, self).__init__(sess, loss, log_alpha, dim, name, learning_rate, n_samples)\n        self.Q_optimizer = tf.train.AdamOptimizer(learning_rate)\n        self.Q_vars = [v for v in tf.trainable_variables() if \"Q_func\" in v.name]\n        self._Q_gradvars()\n        self.Q_opt_op = self.Q_optimizer.apply_gradients(self.Q_gradvars)\n        old_var_op = self.variance_reduction_op\n        with tf.control_dependencies([self.Q_opt_op, old_var_op]):\n            self.variance_reduction_op = tf.no_op()\n\n\n    def _create_loss_evaluations(self):\n        \"\"\"\n        produces f(b), f(sig(z)), f(sig(z_tilde))\n        \"\"\"\n        # relaxed inputs\n        log_alpha = self._log_alpha\n        sig_z = tf.nn.sigmoid(self.z / self.temperature + log_alpha)\n        sig_z_tilde = tf.nn.sigmoid(self.z_tilde / self.temperature + log_alpha)\n        # evaluate loss\n        f_b = tf.reshape(self.loss(tf.reshape(self.b, [self.batch_size, -1])), [-1])\n        z_inp = tf.reshape(sig_z, [self.batch_size, -1])\n        z_tilde_inp = tf.reshape(sig_z_tilde, [self.batch_size, -1])\n        l_z = self.loss(z_inp)\n        l_z_tilde = self.loss(z_tilde_inp)\n        with tf.variable_scope(\"Q_func\"):\n            f_z = tf.reshape(self.q_func(z_inp) + l_z, [-1])\n        with tf.variable_scope(\"Q_func\", reuse=True):\n            f_z_tilde = tf.reshape(self.q_func(z_tilde_inp) + l_z_tilde, [-1])\n\n        self.f_b = f_b\n        self.f_z = f_z\n        self.f_z_tilde = f_z_tilde\n        tf.summary.scalar(\"f_b\", tf.reduce_mean(self.f_b))\n        tf.summary.scalar(\"f_z_tilde\", tf.reduce_mean(self.f_z_tilde))\n        tf.summary.scalar(\"f_z\", tf.reduce_mean(self.f_z))\n\n    def _Q_gradvars(self):\n        \"\"\"\n        produces d[log p(b)]/d[log_alpha], d[f(sigma_theta(z))]/d[log_alpha], d[f(sigma_theta(z_tilde))]/d[log_alpha]\n        \"\"\"\n        print(gs(self._rebar), gs(self.rebar))\n        self.Q_gradvars = []\n        for var in self.Q_vars:\n            d_var_d_v = tf.gradients(\n                tf.reduce_sum(tf.square(self._rebar)) / self.batch_size,\n                var\n            )[0]\n            # d_var_d_v = tf.gradients(\n            #     tf.reduce_mean(tf.square(self.f_b - self.f_z_tilde) + tf.square(self.f_b - self.f_z)),\n            #     var\n            # )[0]\n            print(var.name, gs(d_var_d_v))\n            self.Q_gradvars.append((d_var_d_v, var))\n\n\n\nif __name__ == \"__main__\":\n    def loss(b):\n        bs, dim = gs(b)\n        t = np.expand_dims(np.array(range(dim+2)[1:-1], dtype=np.float32) / (dim+2), 0)\n        return tf.reduce_sum(tf.square(b - t), axis=1)\n    sess = tf.Session()\n    r_opt = REBAROptimizer(sess, loss, dim=10, learning_rate=.1, n_samples=1)\n\n    \"\"\"\n    Bias and Variance test\n    \"\"\"\n    r_opt.sess.run(tf.global_variables_initializer())\n    summ_op = tf.summary.merge_all()\n    summary_writer = tf.summary.FileWriter(\"/tmp/rebar\")\n    percent_dims = []\n    rebar_vars = []\n    rebar_means = []\n    reinforce_vars = []\n    reinforce_means = []\n    for iter in xrange(100):\n        rebars = []\n        reinforces = []\n        for i in range(10000):\n            [reb, ref] = sess.run([r_opt.rebar, r_opt.reinforce])\n            rebars.append(reb)\n            reinforces.append(ref)\n        rebar_vars.append(np.var(rebars, axis=0))\n        reinforce_vars.append(np.var(reinforces, axis=0))\n        rebar_means.append(np.mean(rebars, axis=0))\n        reinforce_means.append(np.mean(reinforces, axis=0))\n        print(\"vars\", np.mean(rebar_vars[-1]), np.mean(reinforce_vars[-1]))\n        print(\"means\", rebar_means[-1][3], reinforce_means[-1][3])\n        print()\n        sess.run(r_opt.train_op)\n\n\n\n\n#\n#     def train(self, n_steps=100000):\n#         sess.run(tf.global_variables_initializer())\n#         loss_vals = []\n#         ave_loss = tf.reduce_mean(self.f_b)\n#         summ_op = tf.summary.merge_all()\n#         summary_writer = tf.summary.FileWriter(\"/tmp/rebar\")\n#         for iter in xrange(n_steps):\n#             if iter % 100 == 0:\n#                 _, sum_str, loss_val, g_val, la, t, s, ls, e = sess.run(\n#                     [self.train_op, summ_op, ave_loss, self.rebar, self.log_alpha, self.log_temperature, self.log_scale,\n#                      self.len_scale, self.eta])\n#                 summary_writer.add_summary(sum_str, iter)\n#             else:\n#                 _, loss_val, g_val, la, t, s, ls, e = sess.run(\n#                 [self.train_op, ave_loss, self.rebar, self.log_alpha, self.log_temperature, self.log_scale, self.len_scale, self.eta])\n#             loss_vals.append(loss_val)\n#             a = np.exp(la)\n#             theta = a/(1+a)\n#             if iter % 100 == 0:\n#                 print(\n#                     \"iter {}, loss = {}\\n grad = {}\\n theta = {}\\n temp = {}\\n scale = {}\\n len_scale = {}\\n eta = {}\\n\".format(\n#                         iter, loss_val, g_val, theta, np.exp(t), np.exp(s), np.exp(ls), e\n#                     )\n#                 )\n#"
  },
  {
    "path": "rebar_toy.py",
    "content": "from __future__ import print_function\nfrom tensorflow.examples.tutorials.mnist import input_data\nfrom tqdm import tqdm\nimport tensorflow as tf\nimport numpy as np\nimport os\nimport matplotlib\nmatplotlib.use('Agg')\nimport matplotlib.pyplot as plt\nimport cPickle as pickle\nimport pandas\nimport seaborn as sns\nsns.set()\nsns.set_style(\"white\", {\"axes.edgecolor\": \".7\"})\nsns.set_style(\"ticks\")\n# sns.set_style(\"whitegrid\")\n\n# Tableau 20 Colors\ntableau20 = [(31, 119, 180), (174, 199, 232), (255, 127, 14), (255, 187, 120),\n             (44, 160, 44), (152, 223, 138), (214, 39, 40), (255, 152, 150),\n             (148, 103, 189), (197, 176, 213), (140, 86, 75), (196, 156, 148),\n             (227, 119, 194), (247, 182, 210), (127, 127, 127), (199, 199, 199),\n             (188, 189, 34), (219, 219, 141), (23, 190, 207), (158, 218, 229)]\nfor i in range(len(tableau20)):\n    r, g, b = tableau20[i]\n    tableau20[i] = (r / 255., g / 255., b / 255.)\n\n\nITERS = 5000\nRESOLUTION = 10\n\n\"\"\" Helper Functions \"\"\"\ndef safe_log_prob(x, eps=1e-8):\n    return tf.log(tf.clip_by_value(x, eps, 1.0))\n  \n\ndef safe_clip(x, eps=1e-8):\n    return tf.clip_by_value(x, eps, 1.0)\n\n\ndef gs(x):\n    return x.get_shape().as_list()\n\n\ndef softplus(x):\n    '''\n    Let m = max(0, x), then,\n\n    sofplus(x) = log(1 + e(x)) = log(e(0) + e(x)) = log(e(m)(e(-m) + e(x-m)))\n                         = m + log(e(-m) + e(x - m))\n\n    The term inside of the log is guaranteed to be between 1 and 2.\n    '''\n    m = tf.maximum(tf.zeros_like(x), x)\n    return m + tf.log(tf.exp(-m) + tf.exp(x - m))\n\n\ndef logistic_loglikelihood(z, loc, scale=1):\n    return tf.log(tf.exp(-(z-loc)/scale)/scale*tf.square((1+tf.exp(-(z-loc)/scale))))\n\n\ndef bernoulli_loglikelihood(b, log_alpha):\n    return b * (-softplus(-log_alpha)) + (1 - b) * (-log_alpha - softplus(-log_alpha))\n\n\ndef bernoulli_loglikelihood_derivitive(b, log_alpha):\n    assert gs(b) == gs(log_alpha)\n    sna = tf.sigmoid(-log_alpha)\n    return b * sna - (1-b) * (1 - sna)\n\n\ndef v_from_u(u, log_alpha, force_same=True, b=None, v_prime=None):\n    u_prime = tf.nn.sigmoid(-log_alpha)\n    if not force_same:\n        v = b*(u_prime+v_prime*(1-u_prime)) + (1-b)*v_prime*u_prime\n    else:\n        v_1 = (u - u_prime) / safe_clip(1 - u_prime)\n        v_1 = tf.clip_by_value(v_1, 0, 1)\n        v_1 = tf.stop_gradient(v_1)\n        v_1 = v_1 * (1 - u_prime) + u_prime\n        v_0 = u / safe_clip(u_prime)\n        v_0 = tf.clip_by_value(v_0, 0, 1)\n        v_0 = tf.stop_gradient(v_0)\n        v_0 = v_0 * u_prime\n    \n        v = tf.where(u > u_prime, v_1, v_0)\n        v = tf.check_numerics(v, 'v sampling is not numerically stable.')\n        if force_same:\n            v = v + tf.stop_gradient(-v + u)  # v and u are the same up to numerical errors\n    return v\n\n\ndef reparameterize(log_alpha, noise):\n    return log_alpha + safe_log_prob(noise) - safe_log_prob(1 - noise)\n\n\ndef concrete_relaxation(z, temp):\n    return tf.sigmoid(z / temp)\n\n\ndef assert_same_shapes(*args):\n    shapes = [gs(arg) for arg in args]\n    s0, sr = shapes[0], shapes[1:]\n    assert all([s == s0 for s in sr])\n\n\ndef neg_elbo(x, b, log_alpha, pred_x_log_alpha):\n    log_q_b_given_x = tf.reduce_sum(bernoulli_loglikelihood(b, log_alpha), axis=1)\n    log_p_b = tf.reduce_sum(bernoulli_loglikelihood(b, tf.zeros_like(log_alpha)), axis=1)\n    log_p_x_given_b = tf.reduce_sum(bernoulli_loglikelihood(x, pred_x_log_alpha), axis=1)\n    return -1. * (log_p_x_given_b + log_p_b - log_q_b_given_x)\n\n\n\"\"\" Networks \"\"\"\ndef Q_func(z):\n    h1 = tf.layers.dense(2. * z - 1., 10, tf.nn.tanh, name=\"q_1\", use_bias=True)\n    #h2 = tf.layers.dense(h1, 10, tf.nn.relu, name=\"q_2\", use_bias=True)\n    #h3 = tf.layers.dense(h2, 10, tf.nn.relu, name=\"q_3\", use_bias=True)\n    #h4 = tf.layers.dense(h3, 10, tf.nn.relu, name=\"q_4\", use_bias=True)\n    out = tf.layers.dense(h1, 1, name=\"q_out\", use_bias=True)\n    scale = tf.get_variable(\n        \"q_scale\", shape=[1], dtype=tf.float32,\n        initializer=tf.constant_initializer(0), trainable=True\n    )\n    return scale[0] * out\n#    return out\n\ndef loss_func(b, t):\n    return tf.reduce_mean(tf.square(b - t), axis=1)\n\n\ndef main(t=0.499, rand_seed=42, use_reinforce=False, relaxed=False, visualize=False,\n         log_var=False, tf_log=False, force_same=False, test_bias=False,\n         train_to_completion=False, use_exact_gradient=False, BAR=False, LAX=False, train_theta=True, square_loss=False):\n    with tf.Session() as sess:\n        TRAIN_DIR = \"./toy_problem\"\n        if os.path.exists(TRAIN_DIR):\n            print(\"Deleting existing train dir\")\n            import shutil\n\n            shutil.rmtree(TRAIN_DIR)\n        os.makedirs(TRAIN_DIR)\n        iters = ITERS # todo: change back\n        batch_size = 1\n        num_latents = 1\n        target = np.array([[t for i in range(num_latents)]], dtype=np.float32)\n        print(\"Target is {}\".format(target))\n        lr = .01\n\n        # encode data\n        log_alpha = tf.Variable(\n            [[0.0 for i in range(num_latents)]],\n            trainable=True,\n            name='log_alpha',\n            dtype=tf.float32\n        )\n        a = tf.exp(log_alpha)\n        theta = a / (1 + a)\n\n        tf.set_random_seed(rand_seed)  # fix for repeatable experiments\n\n        # reparameterization variables\n        u = tf.random_uniform([batch_size, num_latents], dtype=tf.float32)\n        v_p = tf.random_uniform([batch_size, num_latents], dtype=tf.float32)\n        z = reparameterize(log_alpha, u) # z(u)\n        b = tf.to_float(tf.stop_gradient(z > 0))\n        v = v_from_u(u, log_alpha, force_same, b, v_p)\n        z_tilde = reparameterize(log_alpha, v)\n\n        # rebar variables\n        eta = tf.Variable(\n            [1.0 for i in range(num_latents)],\n            trainable=True,\n            name='eta',\n            dtype=tf.float32\n        )\n        log_temperature = tf.Variable(\n            [np.log(.5) for i in range(num_latents)],\n            trainable=True,\n            name='log_temperature',\n            dtype=tf.float32\n        )\n        temperature = tf.exp(log_temperature)\n\n        # loss function evaluations\n        f_b = loss_func(b, target)\n\n        # if we are relaxing the relaxation\n        if relaxed == \"relaxation\":\n            with tf.variable_scope(\"Q_func\"):\n                sig_z = Q_func(z)\n            with tf.variable_scope(\"Q_func\", reuse=True):\n                sig_z_tilde = Q_func(z_tilde)\n            f_z = loss_func(sig_z, target)\n            f_z_tilde = loss_func(sig_z_tilde, target)\n\n        else:\n            # relaxation variables\n            batch_temp = tf.expand_dims(temperature, 0)\n            sig_z = concrete_relaxation(z, batch_temp)\n            sig_z_tilde = concrete_relaxation(z_tilde, batch_temp)\n\n            f_z = loss_func(sig_z, target)\n            f_z_tilde = loss_func(sig_z_tilde, target)\n\n            if relaxed != False:\n                with tf.variable_scope(\"Q_func\"):\n                    q_z = Q_func(sig_z)[:, 0]\n                with tf.variable_scope(\"Q_func\", reuse=True):\n                    q_z_tilde = Q_func(sig_z_tilde)[:, 0]\n                if relaxed == True:\n                    f_z = f_z + q_z\n                    f_z_tilde = f_z_tilde + q_z_tilde\n                elif relaxed == \"super\":\n                    f_z = q_z\n                    f_z_tilde = q_z_tilde\n\n\n        tf.summary.scalar(\"fb\", tf.reduce_mean(f_b))\n        tf.summary.scalar(\"fz\", tf.reduce_mean(f_z))\n        tf.summary.scalar(\"fzt\", tf.reduce_mean(f_z_tilde))\n        # loss function for generative model\n        loss = tf.reduce_mean(f_b)\n        tf.summary.scalar(\"loss\", loss)\n\n        # rebar construction\n        d_f_z_d_log_alpha = tf.gradients(f_z, log_alpha)[0]\n        d_f_z_tilde_d_log_alpha = tf.gradients(f_z_tilde, log_alpha)[0]\n#        d_log_pb_d_log_alpha = bernoulli_loglikelihood_derivitive(b, log_alpha)\n        d_log_pb_d_log_alpha = tf.gradients(bernoulli_loglikelihood(b, log_alpha), log_alpha)[0]\n        d_log_pz_d_log_alpha = tf.gradients(logistic_loglikelihood(z, log_alpha), log_alpha)[0]\n        # check shapes are alright\n        assert_same_shapes(d_f_z_d_log_alpha, d_f_z_tilde_d_log_alpha, d_log_pb_d_log_alpha, d_log_pz_d_log_alpha)\n        assert_same_shapes(f_b, f_z_tilde)\n        batch_eta = tf.expand_dims(eta, 0)\n        batch_f_b = tf.expand_dims(f_b, 1)\n        batch_f_z_tilde = tf.expand_dims(f_z_tilde, 1)\n        # do one of LAX, BAR, relaxed-REBAR, or REBAR\n        if LAX or BAR:\n            batch_f_z = tf.expand_dims(f_z, 1)\n            rebar = batch_f_b*d_log_pb_d_log_alpha - batch_eta*batch_f_z*d_log_pz_d_log_alpha + batch_eta*d_f_z_d_log_alpha\n#            rebar = (batch_f_b - batch_f_z) * d_log_pb_d_log_alpha + (d_f_z_d_log_alpha)\n        elif relaxed == \"super\":\n            rebar = (batch_f_b - batch_f_z_tilde) * d_log_pb_d_log_alpha + (d_f_z_d_log_alpha - d_f_z_tilde_d_log_alpha)\n        else:\n            rebar = (batch_f_b - batch_eta * batch_f_z_tilde) * d_log_pb_d_log_alpha + batch_eta * (d_f_z_d_log_alpha - d_f_z_tilde_d_log_alpha)\n        reinforce = batch_f_b * d_log_pb_d_log_alpha\n        exact_gradient = tf.stop_gradient(tf.square(1 - target) - tf.square(-target)) * tf.nn.sigmoid(log_alpha)\n        tf.summary.histogram(\"rebar\", rebar)\n        tf.summary.histogram(\"reinforce\", reinforce)\n\n        # variance reduction objective\n        variance_loss = tf.reduce_mean(tf.square(rebar))\n\n        # optimizers\n        inf_opt = tf.train.AdamOptimizer(lr)\n\n        # need to scale by batch size cuz tf.gradients sums\n        if use_reinforce:\n            log_alpha_grads = reinforce/batch_size\n        elif use_exact_gradient:\n            log_alpha_grads = exact_gradient/batch_size\n        else:\n            log_alpha_grads = rebar/batch_size\n          \n        inf_train_op = inf_opt.apply_gradients([(log_alpha_grads, log_alpha)])\n\n        var_opt = tf.train.AdamOptimizer(lr)\n        var_vars = [eta, log_temperature]\n        if relaxed:\n            print(\"Relaxed model\")\n            q_vars = [v for v in tf.trainable_variables() if \"Q_func\" in v.name]\n            var_vars = var_vars + q_vars\n        var_gradvars = var_opt.compute_gradients(variance_loss, var_list=var_vars)\n        var_train_op = var_opt.apply_gradients(var_gradvars)\n\n        print(\"Variance\")\n        for g, v in var_gradvars:\n            print(\"    {}\".format(v.name))\n            if g is not None:\n                tf.summary.histogram(v.name, v)\n                tf.summary.histogram(v.name + \"_grad\", g)\n\n        if use_reinforce or use_exact_gradient:\n            with tf.control_dependencies([inf_train_op]):\n                train_op = tf.no_op()\n        else:\n            with tf.control_dependencies([inf_train_op, var_train_op]):\n                train_op = tf.no_op()\n\n        test_loss = tf.Variable(600, trainable=False, name=\"test_loss\", dtype=tf.float32)\n        rebar_var = tf.Variable(np.zeros([batch_size, num_latents]), trainable=False, name=\"rebar_variance\", dtype=tf.float32)\n        reinforce_var = tf.Variable(np.zeros([batch_size, num_latents]), trainable=False, name=\"reinforce_variance\", dtype=tf.float32)\n        est_diffs = tf.Variable(np.zeros([batch_size, num_latents]), trainable=False, name=\"estimator_differences\", dtype=tf.float32)\n        tf.summary.scalar(\"test_loss\", test_loss)\n        tf.summary.histogram(\"rebar_variance\", rebar_var)\n        tf.summary.histogram(\"reinforace_variance\", reinforce_var)\n        tf.summary.histogram(\"estimator_diffs\", est_diffs)\n        summ_op = tf.summary.merge_all()\n        summary_writer = tf.summary.FileWriter(TRAIN_DIR)\n        sess.run(tf.global_variables_initializer())\n        \n        variances = []\n        losses = []\n        thetas = []\n        FBs = []\n        FZs = []\n        print(\"Collecting {} samples\".format(ITERS//RESOLUTION))\n        for i in tqdm(range(iters)):\n            if (i+1) % RESOLUTION == 0:\n                if train_to_completion:\n                    for _ in tqdm(range(1000)):\n                        sess.run(var_train_op)\n                        \n                if tf_log:\n                    if train_theta:\n                        loss_value, _, sum_str, theta_value = sess.run([loss, train_op, summ_op, theta])\n                    else:\n                        loss_value, _, sum_str, theta_value = sess.run([loss, var_train_op, summ_op, theta]) # just train eta and temp\n                    summary_writer.add_summary(sum_str, i)\n                else:\n                    if train_theta:\n                        loss_value, _, theta_value, temp = sess.run([loss, train_op, theta, temperature])\n                    else:\n                        loss_value, _, theta_value, temp = sess.run([loss, var_train_op, theta, temperature]) # just train eta and temp\n                    \n                tv = theta_value[0][0]\n                thetas.append(tv)\n                losses.append(tv*(1-target[0][0])**2+(1-tv)*target[0][0]**2)\n                print(i, loss_value, [t for t in theta_value[0]], [tmp for tmp in temp])\n\n\n                if log_var:\n                    grads = [sess.run([rebar, reinforce]) for i in tqdm(range(100))]\n                    rebars, reinforces = zip(*grads)\n                    re_m, re_v = np.mean(rebars), np.std(rebars)\n                    rf_m, rf_v = np.mean(reinforces), np.std(reinforces)\n                    if use_reinforce:\n                        variances.append(re_v)\n                    print(\"Reinforce mean = {}, Reinforce std = {}\".format(rf_m, rf_v))\n                    print(\"Rebar mean     = {}, Rebar std     = {}\".format(re_m, re_v))\n\n                if test_bias:\n                    n_variance_samples = 1000\n                    rebars = []\n                    reinforces = []\n                    for _ in tqdm(range(n_variance_samples)):\n                        rb, re = sess.run([rebar, reinforce])\n                        rebars.append(rb)\n                        reinforces.append(re)\n                    rebars = np.array(rebars)\n                    reinforces = np.array(reinforces)\n                    re_var = np.log(reinforces.var(axis=0))\n                    rb_var = np.log(rebars.var(axis=0))\n                    if use_reinforce:\n                      variances.append(np.mean(re_var))\n                    else:\n                      variances.append(np.mean(rb_var))\n                    diffs = np.abs(rebars.mean(axis=0) - reinforces.mean(axis=0))\n                    sess.run([rebar_var.assign(rb_var), reinforce_var.assign(re_var), est_diffs.assign(diffs)])\n                    print(\"rebar variance = {}\".format(rb_var.mean()))\n                    print(\"reinforce variance = {}\".format(re_var.mean()))\n                    print(\"rebar     = {}\".format(rebars.mean(axis=0)[0]))\n                    print(\"reinforce = {}\\n\".format(reinforces.mean(axis=0)[0]))\n\n                if visualize == \"f\":\n                    # run variance reduction operation\n                    for i in range(1000):\n                        sess.run(var_train_op)\n                    X = [float(i) / 100 for i in range(100)]\n                    FZ = []\n                    for x in X:\n                        fz = sess.run(f_z, feed_dict={sig_z: [[x]]})\n                        FZ.append(fz)\n                    plt.plot(X, FZ)\n                    plt.show()\n                elif visualize == \"sig\":\n                    us = np.linspace(0.0, 1.0, 1000, dtype=np.float32)\n                    FB = []\n                    FZ = []\n                    for _u in us:\n                        fb = sess.run(f_b, feed_dict={u: [[_u]], log_alpha:[[0.0]]})\n                        fz = sess.run(f_z, feed_dict={u: [[_u]], log_alpha:[[0.0]]})\n                        FB.append(fb)\n                        FZ.append(fz)\n                    FBs.append(FB)\n                    FZs.append(FZ)\n#                    plt.plot(us, FB, 'red', label='f(b=H(z))')\n#                    if not relaxed:\n#                      plt.plot(us, FZ, 'blue', label='f(sigmoid(z/temp))')\n#                    elif relaxed in [True, \"super\"]:\n#                      plt.plot(us, FZ, 'blue', label='Q(z)')\n#                    plt.xlabel('u')\n#                    plt.legend(bbox_to_anchor=(1.0,0.5))\n#                    plt.show()\n                    #plt.savefig('/home/damichoi/ml/relaxed-rebar/test.png')\n                        \n\n            else:\n                if train_to_completion:\n                    for _ in tqdm(range(100)):\n                        sess.run(var_train_op)\n                if train_theta:\n                    _, = sess.run([train_op])\n                else:\n                    _, = sess.run([var_train_op])\n                \n        tv = None\n        print(tv)\n#        return tv, thetas, losses, variances, FBs, FZs\n        return tv, thetas, losses, variances, FBs, FZs\n\n\nif __name__ == \"__main__\":\n    t = 0.499\n    rand_seed = 42\n\n    print(\"INFO: Collecting data for plots...\")\n\n## FIGURE 3\n    output_file_name = \"relaxations_plot_new_{}_{}.pkl\".format(ITERS, ITERS / RESOLUTION)  # file to save results\n    try:\n        with open(output_file_name, 'r') as f:\n            FB, FZ, QZ, us = pickle.load(f)\n            print(\"\\nINFO: Loaded precomputed data.\")\n    except IOError:\n        print(\"INFO: No precomputed data found, running toy example experiments.\")\n        _,relax_thetas,relax_losses,relax_variances,_,QZ = main(t=t, relaxed=\"super\", visualize=\"sig\", force_same=True, test_bias=False, train_to_completion=False, train_theta=False)\n        _,rebar_thetas,rebar_losses,rebar_variances,FB,FZ = main(t=t, relaxed=False, visualize=\"sig\", force_same=True, test_bias=False, train_to_completion=False, train_theta=False)\n        us = np.linspace(0,1,len(FB[0]))\n        with open(output_file_name, 'w') as f:\n            pickle.dump([FB, FZ, QZ, us], f)\n\n    # Outputs ###.png for composition into gif\n    print(\"INFO: Generating and saving plots...\")\n    try:\n        os.stat('./gif')\n    except:\n        os.mkdir('./gif')\n\n    for which in tqdm(range(len(FB))):\n        with sns.axes_style(\"whitegrid\"):\n            sns.set_style(\"ticks\")\n            sns.set_context(\"poster\")\n\n            f, (ax1, ax2) = plt.subplots(2, sharex=True)\n            # ax1.plot(us, FB[which], color=tableau20[0],  label=r'$f(b=H(z(u)))$',)\n            # ax1.legend(loc='best')\n            # plt.xlabel(r\"$u \\sim \\operatorname{Unif}[0,1]$\")\n\n            ax1.plot(us,  FZ[which], color=tableau20[4],label=r'$f(\\sigma_\\lambda(z(u)))$')\n            # ax1.legend(loc='best')\n            ax2.plot(us, QZ[which], color=tableau20[6], label=r'$c_\\phi(z(u))$')\n            # ax2.legend(loc='best')\n        # Fine-tune figure; make subplots close to each other and hide x ticks for\n        # all but bottom plot.\n            f.subplots_adjust(hspace=0.5)\n\n            plt.setp([a.get_xticklabels() for a in f.axes[:-1]], visible=False)\n            plt.xlabel(r\"$u$\")\n            plt.ylim([0.24, 0.26])\n            sns.despine(offset=0.125, trim=True)\n            # plt.setp([a.get_yticklabels() for a in f.axes], visible=False)\n            plt.tight_layout()\n            plt.savefig(\"./gif/{}.png\".format(str(which).zfill(3), which), format='png')\n            plt.close(f)\n#\n#\n#\n    #\n    # f, axes = plt.subplots(3,3)\n    # for i, ax in enumerate(axes.flatten()):\n    #     with sns.axes_style(\"darkgrid\"):\n    #         plt.subplot(311)\n    #         ax.plot(us, (FB[i]-np.mean(FB[i]))/np.std(FB[i])-3, ls='-', color='red', alpha=0.8, label=r'$f(b=H(z(u)))$')\n    #         ax.plot(us, (FZ[i]-np.mean(FZ[i]))/np.std(FZ[i])-1, ls=':', color='blue', alpha=0.8, label=r'$f(\\sigma_\\lambda(z(u)))$')\n    #         ax.plot(us, (QZ[i]-np.mean(QZ[i]))/np.std(QZ[i])+1, ls='-.',  color='green', alpha=0.8, label=r'$Q(\\sigma(z(u)))$')\n    #         # ax.legend(loc='best')#bbox_to_anchor=(1.0, 0.65))\n    #     # ax.xlabel('u')\n    # plt.tight_layout()\n    # plt.savefig(\"relaxations_t_{}_which_{}.pdf\".format(t, which), format='pdf')\n#     #\n##    fig1_dict = {}\n##    fig1_dict[\"relax_losses_t0.1\"] = relax_losses\n##    fig1_dict[]\n#    us = np.linspace(0,1,len(FB[0]))\n#    for i in range(len(FB)):\n#        plt.figure(i)\n##        plt.subplot(2, 2, 1)\n#        plt.plot(us, FB[i], 'red', label=r'$f(b=H(z(u)))$')\n#        plt.plot(us, FZ[i], 'blue', label=r'$f(\\sigma_\\lambda(z(u)))$')\n#        plt.plot(us, QZ[i], 'green', label=r'$Q(\\sigma(z(u)))$')\n#        plt.legend(bbox_to_anchor=(1.0,0.65))\n#        plt.xlabel('u')\n#        plt.savefig('/home/damichoi/ml/relaxed-rebar/toy_problem/test'+str(i)+'.png', bbox_inches='tight')\n        \n#        plt.subplot(2, 2, 2)\n#        plt.xlim(0,10000)\n#        #plt.ylim(0.2489,0.2503)\n#        x = np.arange(0,100*(i+1),100)\n#        plt.plot(x, rebar_losses[:i+1], 'blue', label='REBAR')\n#        plt.plot(x, relax_losses[:i+1], 'green', label='RELAX')\n#        plt.legend(bbox_to_anchor=(1.7,0.65))\n#        plt.xlabel('steps')\n#        plt.ylabel('loss')\n#        plt.tight_layout()\n#        plt.subplots_adjust(wspace = 1.2)\n        \n#        plt.subplot(2, 2, 2)\n#        plt.xlim(0,10000)\n#        plt.ylim(-0.02,0.7)\n#        x = np.arange(0,100*(i+1),100)\n#        plt.plot(x, rebar_thetas[:i+1], 'blue', label='REBAR')\n#        plt.plot(x, relax_thetas[:i+1], 'green', label='RELAX')\n#        plt.legend(bbox_to_anchor=(1.65,0.65))\n#        plt.xlabel('steps')\n#        plt.ylabel('theta')\n#        plt.tight_layout()\n#        plt.subplots_adjust(wspace = 1.0)\n#        \n#        plt.subplot(2, 2, 2)\n#        plt.xlim(0,10000)\n##        plt.ylim(-19,-3)\n#        x = np.arange(0,100*(i+1),100)\n#        plt.plot(x, rebar_variances[:i+1], 'blue', label='REBAR')\n#        plt.plot(x, relax_variances[:i+1], 'green', label='RELAX')\n#        plt.legend(bbox_to_anchor=(1.65,0.65))\n#        plt.xlabel('steps')\n#        plt.ylabel('log variance')\n#        plt.tight_layout()\n#        plt.subplots_adjust(wspace = 1.0)\n#        plt.savefig('/home/damichoi/ml/relaxed-rebar/toy_problem/test'+str(i)+'.png', bbox_inches='tight')\n#        plt.show()\n# uncomment once to collect losses for Figs 1 and 2:\n#     file_name = \"toy_losses_{}_{}\".format(ITERS, t)\n#     try:\n#         with open(file_name+'.pkl', 'r') as f:\n#             unloaded = pickle.load(f)\n#     except IOError:\n#         _,ext_thetas, ext_losses, ext_variances,__,___ = main(t=t, use_reinforce=False, log_var=False, relaxed=False, visualize=None, force_same=True, test_bias=True, use_exact_gradient=True)\n#         tf.reset_default_graph()\n#         _,reinf_thetas, reinf_losses, reinf_variances,__,___ = main(t=t, use_reinforce=True,  log_var=False, relaxed=False, visualize=None, force_same=True, test_bias=True)\n#         tf.reset_default_graph()\n#         _,rebar_thetas, rebar_losses, rebar_variances,__,___ = main(t=t, relaxed=False, log_var=False, visualize=None, force_same=True, test_bias=True)\n#         tf.reset_default_graph()\n#     #    _,rebar_thetas_ttc, rebar_losses_ttc, rebar_variances_ttc,__,___ = main(relaxed=False, visualize=None, force_same=True, test_bias=False, train_to_completion=True)\n#     #    tf.reset_default_graph()\n#         _,relax_thetas, relax_losses, relax_variances,__,___ = main(t=t, relaxed=\"super\", log_var=False, visualize=None, force_same=True, test_bias=True)\n#         tf.reset_default_graph()\n#     #    _,relax_thetas_ttc, relax_losses_ttc, relax_variances_ttc,__,___ = main(relaxed=\"super\", visualize=None, force_same=True, test_bias=False, train_to_completion=True)\n#     #     tf.reset_default_graph()\n#     #     _,lax_thetas, lax_losses, lax_variances,__,___ = main(t=t, relaxed=\"super\", visualize=None, force_same=True, test_bias=False, train_to_completion=False, LAX=True)\n#         # tf.reset_default_graph()\n#     #    _,lax_thetas_ttc, lax_losses_ttc, lax_variances_ttc,__,___ = main(relaxed=\"super\", visualize=None, force_same=True, test_bias=False, train_to_completion=True, LAX=True)\n#     #     tf.reset_default_graph()\n#     #    _,bar_thetas, bar_losses, bar_variances,__,___ = main(relaxed=False, visualize=None, force_same=True, test_bias=False, train_to_completion=False, BAR=True)\n#     #     tf.reset_default_graph()\n#     #    _,bar_thetas_ttc, bar_losses_ttc, bar_variances_ttc,__,___ = main(relaxed=False, visualize=None, force_same=True, test_bias=False, train_to_completion=True, BAR=True)\n#\n#         with open(file_name+'.pkl', 'w') as f:\n#             pickle.dump([ext_losses, reinf_losses, rebar_losses, relax_losses, rebar_variances, reinf_variances, relax_variances, ext_variances], f)\n# #\n#     # UNCOMMENT HERE FOR FIGURE 1:\n#     sns.set_style(\"white\", {\"axes.edgecolor\": \".7\",\n#                         \"font_scale\": \"1.2\"})\n#     sns.set_context(\"talk\")\n#     x = np.arange(0, ITERS, RESOLUTION) #len(rebar_losses))\n#     ext_losses, reinf_losses, rebar_losses, relax_losses, _ = unloaded\n#     print(\"rebar_losses {}\".format(len(rebar_losses)))\n#     max_plot_iters = 10000\n#     plt.figure(1)\n#     plt.xlim(0,max_plot_iters)\n#     alpha=1.0\n#     sns.set_context(\"paper\")\n#     sns.set(font_scale=1.2)\n#\n#     plt.plot(x, reinf_losses, color=tableau20[0],label=\"REINFORCE\", alpha=alpha)\n#     plt.plot(x, rebar_losses,color=tableau20[4], label=\"REBAR\", alpha=alpha)\n# #    plt.plot(x, rebar_losses_ttc, 'orange', label=\"REBAR trained to completion\")\n#     plt.plot(x, relax_losses, color=tableau20[6],label=\"RELAX (ours)\", alpha=alpha)\n#\n#     plt.plot(x, ext_losses, color='black', ls='-.', label=\"Exact gradient\", alpha=0.5)\n# #    plt.plot(x, relax_losses_ttc, 'purple', label=\"RELAX trained to completion\")\n# #     plt.plot(x, lax_losses,color=tableau20[5], label=\"LAX\", alpha=alpha)\n#\n#     fill_alpha=0.25\n#     # plt.fill_between(x, reinf_losses, lax_losses, where=np.array(lax_losses) >= np.array(reinf_losses),\n#     #                  color=tableau20[5], alpha=fill_alpha, interpolate=True)\n#     # plt.fill_between(x, relax_losses, rebar_losses, facecolor=tableau20[3], alpha=fill_alpha)\n#     # # plt.fill_between(x, relax_losses, ext_losses, facecolor=tableau20[2], alpha=fill_alpha)\n#     # plt.fill_between(x, reinf_losses, rebar_losses,  where=np.array(reinf_losses) >= np.array(rebar_losses),\n#     #                  facecolor=tableau20[1], alpha=fill_alpha)\n#\n# #    plt.plot(x, lax_losses_ttc, 'black', label=\"LAX trained to completion\")\n# #    plt.plot(x, bar_losses, 'pink', label=\"BAR\")\n# #    plt.plot(x, bar_losses_ttc, 'yellow', label=\"BAR trained to completion\")\n#     plt.legend(loc=\"best\")#bbox_to_anchor=(1.0, 0.75))\n#     plt.xlabel(\"Steps\")\n#     plt.ylabel((\"Loss\"))\n#     # plt.rc('grid', linestyle=\"--\", color='black')\n#     # plt.grid(True)\n#     # plt.ylabel(\"Loss\")\n#     # plt.xlabel(\"Iteration\")\n#     ylims = plt.gca().get_ylim()\n#     sns.despine()\n#     plt.savefig(os.path.join('toy_problem', file_name +'no_envelope' + '.pdf'), format='pdf', bbox_inches='tight')\n\n# END FIGURE 1\n\n#    plt.figure(2)\n#    plt.xlim(0,10000)\n#    plt.plot(x, ext_thetas, 'green', label=\"exact_gradient\")\n#    plt.plot(x, reinf_thetas, 'magenta', label=\"REINFORCE\")\n#    plt.plot(x, rebar_thetas, 'red', label=\"REBAR\")\n##    plt.plot(x, rebar_thetas_ttc, 'orange', label=\"REBAR trained to completion\")\n#    plt.plot(x, relax_thetas, 'blue', label=\"RELAX\")\n#    plt.plot(x, relax_thetas_ttc, 'purple', label=\"RELAX trained to completion\")\n#    plt.plot(x, lax_thetas, 'cyan', label=\"LAX\")\n#    plt.plot(x, lax_thetas_ttc, 'black', label=\"LAX trained to completion\")\n##    plt.plot(x, bar_thetas, 'pink', label=\"BAR\")\n##    plt.plot(x, bar_thetas_ttc, 'yellow', label=\"BAR trained to completion\")\n#    plt.legend(bbox_to_anchor=(1.0,0.75))\n#    plt.rc('grid', linestyle=\"--\", color='black')\n#    plt.grid(True)\n#    plt.ylabel(\"theta\")\n#    plt.xlabel(\"Steps\")\n#    plt.savefig('/home/damichoi/ml/relaxed-rebar/theta.png', bbox_inches='tight')\n#    \n#    plt.figure(3)\n#    plt.xlim(0,10000)\n##    plt.plot(x, reinf_variances, 'magenta', label=\"REINFORCE\")\n#    plt.plot(x, rebar_variances, 'red', label=\"REBAR\")\n##    plt.plot(x, rebar_variances_ttc, 'orange', label=\"REBAR trained to completion\")\n#    plt.plot(x, relax_variances, 'blue', label=\"RELAX\")\n##    plt.plot(x, relax_variances_ttc, 'purple', label=\"RELAX trained to completion\")\n#    plt.plot(x, lax_variances, 'cyan', label=\"LAX\")\n##    plt.plot(x, lax_variances_ttc, 'black', label=\"LAX trained to completion\")\n##    plt.plot(x, bar_variances, 'pink', label=\"BAR\")\n##    plt.plot(x, bar_variances_ttc, 'yellow', label=\"BAR trained to completion\")\n#    plt.legend(bbox_to_anchor=(1.0,0.75))\n#    plt.rc('grid', linestyle=\"--\", color='black')\n#    plt.grid(True)\n#    plt.ylabel(\"log(Var(gradient estimator))\")\n#    plt.xlabel(\"Steps\")\n#    plt.savefig('/home/damichoi/ml/relaxed-rebar/variance.png', bbox_inches='tight')\n\n\n# UNCOMMENT HERE FOR LOG VARIANCE PLOT:\n#     _x = np.arange(0, ITERS, RESOLUTION)\n#     # _reinf_variances = [reinf_variances[i] for i in range(10000) if (i+1)%10 == 0]\n#     # _rebar_variances = [rebar_variances[i] for i in range(ITERS) if i%RESOLUTION == 0]\n#     # _relax_variances = [relax_variances[i] for i in range(ITERS) if i%RESOLUTION == 0]\n#     # _lax_variances =   [lax_variances[i] for i in range(10000) if (i+1)%10 == 0]\n#     # _bar_variances =   [bar_variances[i] for i in range(10000) if (i+1)%10 == 0]\n#\n#     plt.figure(4)\n#     # sns.set(font_scale=1.2)\n#     window = 50\n#     sns.set_style(\"white\", {\"axes.edgecolor\": \".7\",\n#                             \"font_scale\" : \"1.2\"})\n#     sns.set_context(\"talk\")\n#\n#     A = pandas.Series(reinf_variances, _x)\n#     B = pandas.Series(rebar_variances, _x)\n#     C = pandas.Series(relax_variances, _x)\n#     point_alpha=0.1\n#     line_alpha=0.8\n#     plt.plot(_x, pandas.rolling_mean(A, window),  color=tableau20[0], alpha=line_alpha, label=\"REINFORCE\")\n#     plt.plot(_x, pandas.rolling_mean(B, window), color=tableau20[4], alpha=line_alpha,  label=\"REBAR\")\n#     plt.plot(_x, pandas.rolling_mean(C, window), color=tableau20[6], alpha=line_alpha,  label=\"RELAX (ours)\")\n#     plt.legend(loc='best')# bbox_to_anchor=(1.0, 0.75))\n#     plt.ylabel(\"Log Variance of Gradient Estimates\")\n#     plt.xlabel(\"Steps\")\n#     plt.xlim([500, ITERS])\n#     sns.despine()\n#     plt.savefig(os.path.join('variance_100_{}.pdf'.format(t)), format='pdf', bbox_inches='tight')\n"
  },
  {
    "path": "relax-autograd/demo_concrete.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\nimport matplotlib.pyplot as plt\n\nimport autograd.numpy as np\nimport autograd.numpy.random as npr\nfrom autograd.scipy.special import expit\n\nfrom autograd import grad, value_and_grad\nfrom autograd.misc.optimizers import adam\n\nfrom relax import concrete\n\nif __name__ == '__main__':\n\n    D = 100\n    rs = npr.RandomState(0)\n    num_samples = 50\n    init_params = (np.zeros(D), 1.0)\n\n    def objective(b):\n        return np.sum((b - np.linspace(0, 1, D))**2, axis=-1, keepdims=True)\n\n    def mc_objective_and_var(combined_params, t):\n        params, est_params = combined_params\n        params_rep = np.tile(params, (num_samples, 1))\n        rs = npr.RandomState(t)\n        noise_u = rs.rand(num_samples, D)\n        objective_vals, grads = \\\n            value_and_grad(concrete)(params_rep, est_params, noise_u, objective)\n        return np.mean(objective_vals), np.var(grads, axis=0)\n\n    def combined_obj(combined_params, t):\n        # Combines objective value and variance of gradients.\n        obj_value, grad_variances = mc_objective_and_var(combined_params, t)\n        return obj_value\n\n    # Set up figure.\n    fig = plt.figure(figsize=(8, 8), facecolor='white')\n    ax1 = fig.add_subplot(411, frameon=False)\n    ax2 = fig.add_subplot(412, frameon=False)\n    ax3 = fig.add_subplot(413, frameon=False)\n    ax4 = fig.add_subplot(414, frameon=False)\n    plt.ion()\n    plt.show(block=False)\n\n    temperatures = []\n    def callback(combined_params, t, combined_grads):\n        params, temperature = combined_params\n        grad_params, grad_temperature = combined_grads\n        temperatures.append(temperature)\n        if t % 10 == 0:\n            objective_val, grad_vars = mc_objective_and_var(combined_params, t)\n            print(\"Iteration {} objective {}\".format(t, objective_val))\n            ax1.cla()\n            ax1.plot(expit(params), 'r')\n            ax1.set_ylabel('parameter values')\n            ax1.set_ylim([0, 1])\n            ax2.cla()\n            ax2.plot(grad_params, 'g')\n            ax2.set_ylabel('average gradient')\n            ax3.cla()\n            ax3.plot(grad_vars, 'b')\n            ax3.set_ylabel('gradient variance')\n            ax3.set_xlabel('parameter index')\n            ax4.cla()\n            ax4.plot(temperatures, 'b')\n            ax4.set_ylabel('temperature')\n\n            plt.draw()\n            plt.pause(1.0/30.0)\n\n    print(\"Optimizing...\")\n    adam(grad(combined_obj), init_params, step_size=0.1, num_iters=2000, callback=callback)\n    plt.pause(10.0)"
  },
  {
    "path": "relax-autograd/demo_rebar.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\nimport matplotlib.pyplot as plt\n\nimport autograd.numpy as np\nimport autograd.numpy.random as npr\nfrom autograd.scipy.special import expit\nfrom autograd.misc.optimizers import adam\n\nfrom relax import rebar_all\n\nif __name__ == '__main__':\n\n    D = 100\n    rs = npr.RandomState(0)\n    num_samples = 10\n    init_params = (np.zeros(D), (1.0, 1.0))\n\n    def objective(b):\n        return np.sum((b - np.linspace(0, 1, D))**2, axis=-1, keepdims=True)\n\n    def mc_objective_and_var(combined_params, t):\n        params, est_params = combined_params\n        params_rep = np.tile(params, (num_samples, 1))\n        rs = npr.RandomState(t)\n        noise_u = rs.rand(num_samples, D)\n        noise_v = rs.rand(num_samples, D)\n        return rebar_all(params_rep, est_params, noise_u, noise_v, objective)\n\n    def combined_grad(combined_params, t):\n        obj_value, grad_obj, grad_var = mc_objective_and_var(combined_params, t)\n        return (np.mean(grad_obj, axis=0), grad_var)\n\n    # Set up figure.\n    fig = plt.figure(figsize=(8, 8), facecolor='white')\n    ax1 = fig.add_subplot(411, frameon=False)\n    ax2 = fig.add_subplot(412, frameon=False)\n    ax3 = fig.add_subplot(413, frameon=False)\n    ax4 = fig.add_subplot(414, frameon=False)\n    plt.ion()\n    plt.show(block=False)\n\n    temperatures = []\n    etas = []\n    def callback(combined_params, t, combined_gradient):\n        params, est_params = combined_params\n        grad_params, grad_est = combined_gradient\n        log_temperature, log_eta = est_params\n        temperatures.append(np.exp(log_temperature))\n        etas.append(np.exp(log_eta))\n        if t % 10 == 0:\n            objective_val, grads, est_grads = mc_objective_and_var(combined_params, t)\n            print(\"Iteration {} objective {}\".format(t, np.mean(objective_val)))\n            ax1.cla()\n            ax1.plot(expit(params), 'r')\n            ax1.set_ylabel('parameter values')\n            ax1.set_ylim([0, 1])\n            ax2.cla()\n            ax2.plot(grad_params, 'g')\n            ax2.set_ylabel('average gradient')\n            ax3.cla()\n            ax3.plot(temperatures, 'b')\n            ax3.set_ylabel('temperature')\n            ax4.cla()\n            ax4.plot(etas, 'b')\n            ax4.set_ylabel('eta')\n            ax4.set_xlabel('iteration')\n\n            plt.draw()\n            plt.pause(1.0/30.0)\n\n    print(\"Optimizing...\")\n    adam(combined_grad, init_params, step_size=0.1, num_iters=2000, callback=callback)\n    plt.pause(10.0)"
  },
  {
    "path": "relax-autograd/demo_reinforce.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\nimport matplotlib.pyplot as plt\n\nimport autograd.numpy as np\nimport autograd.numpy.random as npr\nfrom autograd.scipy.special import expit\n\nfrom autograd import grad\nfrom autograd.misc.optimizers import adam\n\nfrom relax import bernoulli_sample, reinforce\n\nif __name__ == '__main__':\n\n    D = 100\n    rs = npr.RandomState(0)\n    num_samples = 50\n    init_params = np.zeros(D)\n\n    def objective(b):\n        return np.sum((b - np.linspace(0, 1, D))**2, axis=-1, keepdims=True)\n\n    def mc_objective_and_var(params, t):\n        params_rep = np.tile(params, (num_samples, 1))\n        rs = npr.RandomState(t)\n        noise_u = rs.rand(num_samples, D)\n        samples = bernoulli_sample(params_rep, noise_u)\n        objective_vals = objective(samples)\n        grads = reinforce(params_rep, noise_u, objective_vals)\n        return np.mean(objective_vals), np.mean(grads, axis=0), np.var(grads, axis=0)\n\n    def obj_grads(params, t):\n        obj_value, grads, grad_variances = mc_objective_and_var(params, t)\n        return grads\n\n    # Set up figure.\n    fig = plt.figure(figsize=(8, 8), facecolor='white')\n    ax1 = fig.add_subplot(311, frameon=False)\n    ax2 = fig.add_subplot(312, frameon=False)\n    ax3 = fig.add_subplot(313, frameon=False)\n    plt.ion()\n    plt.show(block=False)\n\n    temperatures = []\n    def callback(params, t, gradient):\n        grad_params = gradient[:D]\n        if t % 10 == 0:\n            objective_val, grads, grad_vars = mc_objective_and_var(params, t)\n            print(\"Iteration {} objective {}\".format(t, objective_val))\n            ax1.cla()\n            ax1.plot(expit(params), 'r')\n            ax1.set_ylabel('parameter values')\n            ax1.set_ylim([0, 1])\n            ax2.cla()\n            ax2.plot(grad_params, 'g')\n            ax2.set_ylabel('average gradient')\n            ax3.cla()\n            ax3.plot(grad_vars, 'b')\n            ax3.set_ylabel('gradient variance')\n            ax3.set_xlabel('parameter index')\n\n            plt.draw()\n            plt.pause(1.0/30.0)\n\n    print(\"Optimizing...\")\n    adam(obj_grads, init_params, step_size=0.1, num_iters=2000, callback=callback)\n    plt.pause(10.0)"
  },
  {
    "path": "relax-autograd/demo_relax.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\nimport matplotlib.pyplot as plt\n\nimport autograd.numpy as np\nimport autograd.numpy.random as npr\nfrom autograd.scipy.special import expit, logit\nfrom autograd.misc.optimizers import adam\n\nfrom relax import init_nn_params, nn_predict, relax_all\n\ndef make_one_d(f, d, full_d_input):\n    def oned(one_d_input):\n        c = full_d_input.copy()\n        c[d] = one_d_input\n        return f(c)\n    return oned\n\ndef map_and_stack(f):\n    def mapped(inputs):\n        return np.stack([f(a) for a in inputs])\n    return mapped\n\nif __name__ == '__main__':\n\n    D = 100\n    slice_dim = D // 2 - 1\n    num_hidden_units = 5\n    rs = npr.RandomState(0)\n    num_samples = 10\n    init_est_params = (0.0, init_nn_params(0.1, [D, num_hidden_units, 1]))\n    init_model_params = np.zeros(D)\n    init_combined_params = (init_model_params, init_est_params)\n\n    def objective(b):\n        return np.sum((b - np.linspace(0, 1, D))**2, axis=-1, keepdims=True)\n\n    def mc_objective_and_var(combined_params, t):\n        params, est_params = combined_params\n        params_rep = np.tile(params, (num_samples, 1))\n        rs = npr.RandomState(t)\n        noise_u = rs.rand(num_samples, D)\n        noise_v = rs.rand(num_samples, D)\n        return relax_all(params_rep, est_params, noise_u, noise_v, objective)\n\n    def combined_grad(combined_params, t):\n        obj_value, grad_obj, grad_var = mc_objective_and_var(combined_params, t)\n        return (np.mean(grad_obj, axis=0), grad_var)\n\n    # Set up figure.\n    fig = plt.figure(figsize=(8, 8), facecolor='white')\n    ax1 = fig.add_subplot(511, frameon=False)\n    ax2 = fig.add_subplot(512, frameon=False)\n    ax3 = fig.add_subplot(513, frameon=False)\n    ax4 = fig.add_subplot(514, frameon=False)\n    ax5 = fig.add_subplot(515, frameon=False)\n\n    plt.ion()\n    plt.show(block=False)\n\n    temperatures = []\n    nn_scales = []\n    def callback(combined_params, t, combined_gradient):\n        params, est_params = combined_params\n        grad_params, grad_est = combined_gradient\n        log_temperature, nn_params = est_params\n        temperatures.append(np.exp(log_temperature))\n        if t % 10 == 0:\n            objective_val, grads, est_grads = mc_objective_and_var(combined_params, t)\n            print(\"Iteration {} objective {}\".format(t, np.mean(objective_val)))\n            ax1.cla()\n            ax1.plot(expit(params), 'r')\n            ax1.set_ylabel('parameter values')\n            ax1.set_xlabel('parameter index')\n            ax1.set_ylim([0, 1])\n            ax2.cla()\n            ax2.plot(grad_params, 'g')\n            ax2.set_ylabel('average gradient')\n            ax2.set_xlabel('parameter index')\n            ax3.cla()\n            ax3.plot(np.var(grads), 'b')\n            ax3.set_ylabel('gradient variance')\n            ax3.set_xlabel('parameter index')\n            ax4.cla()\n            ax4.plot(temperatures, 'b')\n            ax4.set_ylabel('temperature')\n            ax4.set_xlabel('iteration')\n\n            ax5.cla()\n            xrange = np.linspace(0, 1, 200)\n            f_tilde = lambda x: nn_predict(nn_params, x)\n            f_tilde_map = map_and_stack(make_one_d(f_tilde, slice_dim, params))\n            ax5.plot(xrange, f_tilde_map(logit(xrange)), 'b')\n            ax5.set_ylabel('1d slide of surrogate')\n            ax5.set_xlabel('relaxed sample')\n            plt.draw()\n            plt.pause(1.0/30.0)\n\n    print(\"Optimizing...\")\n    adam(combined_grad, init_combined_params, step_size=0.1, num_iters=2000, callback=callback)\n    plt.pause(10.0)\n"
  },
  {
    "path": "relax-autograd/relax.py",
    "content": "import autograd.numpy as np\nimport autograd.numpy.random as npr\n\nfrom autograd.scipy.special import expit, logit\nfrom autograd import elementwise_grad, value_and_grad, make_vjp\n\n\ndef heaviside(z):\n    return z >= 0\n\ndef softmax(z, log_temperature):\n    temperature = np.exp(log_temperature)\n    return expit(z / temperature)\n\ndef logistic_sample(noise, mu=0, sigma=1):\n    return mu + logit(noise) * sigma\n\ndef logistic_logpdf(x, mu=0, scale=1):\n    y = (x - mu) / (2 * scale)\n    return -2 * np.logaddexp(y, -y) - np.log(scale)\n\ndef bernoulli_sample(logit_theta, noise):\n    return logit(noise) < logit_theta\n\ndef relaxed_bernoulli_sample(logit_theta, noise, log_temperature):\n    return softmax(logistic_sample(noise, expit(logit_theta)), log_temperature)\n\ndef conditional_noise(logit_theta, samples, noise):\n    # Computes p(u|b), where b = H(z), z = logit_theta + logit(noise), p(u) = U(0, 1)\n    uprime = expit(-logit_theta)  # u' = 1 - theta\n    return samples * (noise * (1 - uprime) + uprime) + (1 - samples) * noise * uprime\n\ndef bernoulli_logprob(logit_theta, targets):\n    # log Bernoulli(targets | theta), targets are 0 or 1.\n    return -np.logaddexp(0, -logit_theta * (targets * 2 - 1))\n\n\n############### REINFORCE ##################\n\ndef reinforce(params, noise, func_vals):\n    samples = bernoulli_sample(params, noise)\n    return func_vals * elementwise_grad(bernoulli_logprob)(params, samples)\n\n\n############### CONCRETE ###################\n\ndef concrete(params, log_temperature, noise, f):\n    relaxed_samples = relaxed_bernoulli_sample(params, noise, log_temperature)\n    return f(relaxed_samples)\n\n\n############### REBAR ######################\n\ndef rebar(params, est_params, noise_u, noise_v, f):\n    log_temperature, log_eta = est_params\n    eta = np.exp(log_eta)\n    samples = bernoulli_sample(params, noise_u)\n\n    def concrete_cond(params):\n        cond_noise = conditional_noise(params, samples, noise_v)\n        return concrete(params, log_temperature, cond_noise, f)\n\n    grad_concrete = elementwise_grad(concrete)(params, log_temperature, noise_u, f)\n    f_cond, grad_concrete_cond = value_and_grad(concrete_cond)(params)\n    return reinforce(params, noise_u, f(samples) - eta * f_cond) + \\\n           eta * (grad_concrete - grad_concrete_cond)\n\ndef rebar_all(params, est_params, noise_u, noise_v, f):\n    # Returns objective, gradients, and gradients of variance of gradients.\n    func_vals = f(bernoulli_sample(params, noise_u))\n    var_vjp, grads = make_vjp(rebar, argnum=1)(params, est_params, noise_u, noise_v, f)\n    d_var_d_est = var_vjp(2 * grads / grads.shape[0])\n    return func_vals, grads, d_var_d_est\n\n\n############### RELAX ######################\n# Uses a neural network for control variate instead of original objective\n\ndef init_nn_params(scale, layer_sizes, rs=npr.RandomState(0)):\n    \"\"\"Build a list of (weights, biases) tuples, one for each layer.\"\"\"\n    return [(rs.randn(insize, outsize) * scale,   # weight matrix\n             rs.randn(outsize) * scale)           # bias vector\n            for insize, outsize in zip(layer_sizes[:-1], layer_sizes[1:])]\n\nrelu = lambda x: np.maximum(0, x)\n\ndef nn_predict(params, inputs):\n    for W, b in params:\n        outputs = np.dot(inputs, W) + b\n        inputs = relu(outputs)\n    return outputs\n\ndef relax(params, est_params, noise_u, noise_v, func_vals):\n    samples = bernoulli_sample(params, noise_u)\n    log_temperature, nn_params = est_params\n\n    def surrogate(relaxed_samples):\n        return nn_predict(nn_params, relaxed_samples)\n\n    def surrogate_cond(params):\n        cond_noise = conditional_noise(params, samples, noise_v)  # z tilde\n        return concrete(params, log_temperature, cond_noise, surrogate)\n\n    grad_surrogate = elementwise_grad(concrete)(params, log_temperature, noise_u, surrogate)\n    surrogate_cond, grad_surrogate_cond = value_and_grad(surrogate_cond)(params)\n    return reinforce(params, noise_u, func_vals - surrogate_cond) + \\\n           grad_surrogate - grad_surrogate_cond\n\ndef relax_all(params, est_params, noise_u, noise_v, f):\n    # Returns objective, gradients, and gradients of variance of gradients.\n    func_vals = f(bernoulli_sample(params, noise_u))\n    var_vjp, grads = make_vjp(relax, argnum=1)(params, est_params, noise_u, noise_v, func_vals)\n    d_var_d_est = var_vjp(2 * grads / grads.shape[0])\n    return func_vals, grads, d_var_d_est\n"
  },
  {
    "path": "relax-autograd/tests.py",
    "content": "from __future__ import absolute_import\nfrom __future__ import print_function\nimport itertools\n\nimport autograd.numpy as np\nimport autograd.numpy.random as npr\nfrom autograd.scipy.special import expit, logit\nfrom autograd import grad\n\nfrom relax import reinforce, concrete, bernoulli_sample,\\\n    relax_all, init_nn_params, rebar, rebar_all\n\n\nif __name__ == '__main__':\n    rs = npr.RandomState(0)\n    num_samples = 10000\n    D = 3\n    params = logit(rs.rand(D))\n\n    def objective(b):\n        return np.sum((b - np.linspace(0.2, 0.9, D))**2, axis=-1, keepdims=True)\n\n    def expected_objective(params):\n        lst = list(itertools.product([0.0, 1.0], repeat=D))\n        return sum([objective(np.array(b)) * np.prod([expit(params[i] * (b[i] * 2.0 - 1.0))\n                    for i in range(D)]) for b in lst])\n\n    def mc(params, estimator):  # Simple Monte Carlo\n        rs = npr.RandomState(0)\n        noise = rs.rand(num_samples, D)\n        params_rep = np.tile(params, (num_samples, 1))\n        objective_vals = estimator(params_rep, noise, objective)\n        return np.mean(objective_vals, axis=0)\n\n    print(\"Gradient estimators:\")\n    print(\"Exact              : {}\".format(grad(expected_objective)(params)))\n    print(\"Reinforce          : {}\".format(mc(params, lambda p, n, o: reinforce(p, n, objective(bernoulli_sample(p, n))))))\n    print(\"Concrete, temp = 1 : {}\".format(grad(mc)(params, lambda p, n, o: concrete(p, np.log(1), n, o))))\n    print(\"Rebar, temp = 1    : {}\".format(mc(params, lambda p, n, o: rebar(p, (np.log(1.0),  np.log(0.3)), n, rs.rand(num_samples, D), o))))\n    print(\"Rebar, temp = 10   : {}\".format(mc(params, lambda p, n, o: rebar(p, (np.log(10.0), np.log(0.3)), n, rs.rand(num_samples, D), o))))\n    print(\"Rebar, eta = 0     : {}\".format(mc(params, lambda p, n, o: rebar(p, (np.log(1.0),  np.log(0.0001)), n, rs.rand(num_samples, D), o))))\n    nn_params = init_nn_params(0.1, [D, 5, 1])\n    print(\"Relax              : {}\".format(mc(params, lambda p, n, o: relax_all(p, (0.0, nn_params), n, rs.rand(num_samples, D), o)[1])))\n\n    def var_naive(est_params, method):\n        rs = npr.RandomState(0)\n        noise_u = rs.rand(num_samples, D)\n        noise_v = rs.rand(num_samples, D)\n        params_rep = np.tile(params, (num_samples, 1))\n        obj, grads, vargrads = method(params_rep, est_params, noise_u, noise_v, objective)\n        return np.sum(np.var(grads, axis=0))\n\n    def var_grads(est_params, method):\n        rs = npr.RandomState(0)\n        noise_u = rs.rand(num_samples, D)\n        noise_v = rs.rand(num_samples, D)\n        params_rep = np.tile(params, (num_samples, 1))\n        obj, grads, vargrads = method(params_rep, est_params, noise_u, noise_v, objective)\n        return vargrads\n\n    print(\"\\n\\nGradient of variance of REBAR gradient:\")\n    print(\"Autodiff through variance : {}\".format(grad(var_naive)((1.0,  0.3), rebar_all)))\n    print(\"Single-sample unbiased    : {}\".format(var_grads((1.0,  0.3), rebar_all)))\n\n    print(\"\\n\\nGradient of variance of RELAX gradient:\")\n    print(\"Autodiff through variance : {}\".format(grad(var_naive)((0.0, nn_params), relax_all)))\n    print(\"Single-sample unbiased    : {}\".format(var_grads((0.0, nn_params), relax_all)))\n"
  },
  {
    "path": "toy.py",
    "content": "from __future__ import print_function\nfrom tensorflow.examples.tutorials.mnist import input_data\nfrom tqdm import tqdm\nimport tensorflow as tf\nimport numpy as np\nimport os\nimport matplotlib\n\nmatplotlib.use('Agg')\nimport matplotlib.pyplot as plt\n\n\nITERS = 5000\nRESOLUTION = 10\n\n\"\"\" Helper Functions \"\"\"\n\n\ndef safe_log_prob(x, eps=1e-8):\n    return tf.log(tf.clip_by_value(x, eps, 1.0))\n\n\ndef safe_clip(x, eps=1e-8):\n    return tf.clip_by_value(x, eps, 1.0)\n\n\ndef gs(x):\n    return x.get_shape().as_list()\n\n\ndef softplus(x):\n    '''\n    Let m = max(0, x), then,\n\n    sofplus(x) = log(1 + e(x)) = log(e(0) + e(x)) = log(e(m)(e(-m) + e(x-m)))\n                         = m + log(e(-m) + e(x - m))\n\n    The term inside of the log is guaranteed to be between 1 and 2.\n    '''\n    m = tf.maximum(tf.zeros_like(x), x)\n    return m + tf.log(tf.exp(-m) + tf.exp(x - m))\n\n\ndef logistic_loglikelihood(z, loc, scale=1):\n    return tf.log(tf.exp(-(z - loc) / scale) / scale * tf.square((1 + tf.exp(-(z - loc) / scale))))\n\n\ndef bernoulli_loglikelihood(b, log_alpha):\n    return b * (-softplus(-log_alpha)) + (1 - b) * (-log_alpha - softplus(-log_alpha))\n\n\ndef bernoulli_loglikelihood_derivitive(b, log_alpha):\n    assert gs(b) == gs(log_alpha)\n    sna = tf.sigmoid(-log_alpha)\n    return b * sna - (1 - b) * (1 - sna)\n\n\ndef v_from_u(u, log_alpha, force_same=True, b=None, v_prime=None):\n    u_prime = tf.nn.sigmoid(-log_alpha)\n    if not force_same:\n        v = b * (u_prime + v_prime * (1 - u_prime)) + (1 - b) * v_prime * u_prime\n    else:\n        v_1 = (u - u_prime) / safe_clip(1 - u_prime)\n        v_1 = tf.clip_by_value(v_1, 0, 1)\n        v_1 = tf.stop_gradient(v_1)\n        v_1 = v_1 * (1 - u_prime) + u_prime\n        v_0 = u / safe_clip(u_prime)\n        v_0 = tf.clip_by_value(v_0, 0, 1)\n        v_0 = tf.stop_gradient(v_0)\n        v_0 = v_0 * u_prime\n\n        v = tf.where(u > u_prime, v_1, v_0)\n        v = tf.check_numerics(v, 'v sampling is not numerically stable.')\n        if force_same:\n            v = v + tf.stop_gradient(-v + u)  # v and u are the same up to numerical errors\n    return v\n\n\ndef reparameterize(log_alpha, noise):\n    return log_alpha + safe_log_prob(noise) - safe_log_prob(1 - noise)\n\n\ndef concrete_relaxation(z, temp):\n    return tf.sigmoid(z / temp)\n\n\ndef assert_same_shapes(*args):\n    shapes = [gs(arg) for arg in args]\n    s0, sr = shapes[0], shapes[1:]\n    assert all([s == s0 for s in sr])\n\n\ndef neg_elbo(x, b, log_alpha, pred_x_log_alpha):\n    log_q_b_given_x = tf.reduce_sum(bernoulli_loglikelihood(b, log_alpha), axis=1)\n    log_p_b = tf.reduce_sum(bernoulli_loglikelihood(b, tf.zeros_like(log_alpha)), axis=1)\n    log_p_x_given_b = tf.reduce_sum(bernoulli_loglikelihood(x, pred_x_log_alpha), axis=1)\n    return -1. * (log_p_x_given_b + log_p_b - log_q_b_given_x)\n\n\n\"\"\" Networks \"\"\"\n\n\ndef Q_func(z):\n    h1 = tf.layers.dense(2. * z - 1., 10, tf.nn.tanh, name=\"q_1\", use_bias=True)\n    out = tf.layers.dense(h1, 1, name=\"q_out\", use_bias=True)\n    return out\n\n\ndef loss_func(b, t):\n    return tf.reduce_mean(tf.square(b - t), axis=1)\n\n\ndef main(t=0.499, rand_seed=42, use_reinforce=False, relaxed=False,\n         log_var=False, tf_log=False, force_same=False):\n    with tf.Session() as sess:\n        TRAIN_DIR = \"./toy_problem\"\n        if os.path.exists(TRAIN_DIR):\n            print(\"Deleting existing train dir\")\n            import shutil\n\n            shutil.rmtree(TRAIN_DIR)\n        os.makedirs(TRAIN_DIR)\n        iters = ITERS  # todo: change back\n        batch_size = 1\n        num_latents = 1\n        target = np.array([[t for i in range(num_latents)]], dtype=np.float32)\n        print(\"Target is {}\".format(target))\n        lr = .01\n\n        # encode data\n        log_alpha = tf.Variable(\n            [[0.0 for i in range(num_latents)]],\n            trainable=True,\n            name='log_alpha',\n            dtype=tf.float32\n        )\n        a = tf.exp(log_alpha)\n        theta = a / (1 + a)\n\n        tf.set_random_seed(rand_seed)  # fix for repeatable experiments\n\n        # reparameterization variables\n        u = tf.random_uniform([batch_size, num_latents], dtype=tf.float32)\n        v_p = tf.random_uniform([batch_size, num_latents], dtype=tf.float32)\n        z = reparameterize(log_alpha, u)  # z(u)\n        b = tf.to_float(tf.stop_gradient(z > 0))\n        v = v_from_u(u, log_alpha, force_same, b, v_p)\n        z_tilde = reparameterize(log_alpha, v)\n\n        # rebar variables\n        eta = tf.Variable(\n            [1.0 for i in range(num_latents)],\n            trainable=True,\n            name='eta',\n            dtype=tf.float32\n        )\n        log_temperature = tf.Variable(\n            [np.log(.5) for i in range(num_latents)],\n            trainable=True,\n            name='log_temperature',\n            dtype=tf.float32\n        )\n        temperature = tf.exp(log_temperature)\n\n        # loss function evaluations\n        f_b = loss_func(b, target)\n\n        # if we are relaxing the relaxation\n        if relaxed == \"THETA_U\":\n            z_inp = tf.concat([theta, u], 1)\n            z_tilde_inp = tf.concat([theta, v], 1)\n            with tf.variable_scope(\"Q_func\"):\n                f_z = Q_func(z_inp)[:, 0]\n            with tf.variable_scope(\"Q_func\", reuse=True):\n                f_z_tilde = Q_func(z_tilde_inp)[:, 0]\n\n        else:\n            # relaxation variables\n            batch_temp = tf.expand_dims(temperature, 0)\n            sig_z = concrete_relaxation(z, batch_temp)\n            sig_z_tilde = concrete_relaxation(z_tilde, batch_temp)\n\n            if relaxed:\n                with tf.variable_scope(\"Q_func\"):\n                    f_z = Q_func(sig_z)[:, 0]\n                with tf.variable_scope(\"Q_func\", reuse=True):\n                    f_z_tilde = Q_func(sig_z_tilde)[:, 0]\n            else:\n                f_z = loss_func(sig_z, target)\n                f_z_tilde = loss_func(sig_z_tilde, target)\n\n        tf.summary.scalar(\"fb\", tf.reduce_mean(f_b))\n        tf.summary.scalar(\"fz\", tf.reduce_mean(f_z))\n        tf.summary.scalar(\"fzt\", tf.reduce_mean(f_z_tilde))\n        # loss function for generative model\n        loss = tf.reduce_mean(f_b)\n        tf.summary.scalar(\"loss\", loss)\n\n        # rebar construction\n        d_f_z_d_log_alpha = tf.gradients(f_z, log_alpha)[0]\n        d_f_z_tilde_d_log_alpha = tf.gradients(f_z_tilde, log_alpha)[0]\n        #        d_log_pb_d_log_alpha = bernoulli_loglikelihood_derivitive(b, log_alpha)\n        d_log_pb_d_log_alpha = tf.gradients(bernoulli_loglikelihood(b, log_alpha), log_alpha)[0]\n        # check shapes are alright\n        assert_same_shapes(d_f_z_d_log_alpha, d_f_z_tilde_d_log_alpha, d_log_pb_d_log_alpha)\n        assert_same_shapes(f_b, f_z_tilde)\n        batch_eta = tf.expand_dims(eta, 0)\n        batch_f_b = tf.expand_dims(f_b, 1)\n        batch_f_z_tilde = tf.expand_dims(f_z_tilde, 1)\n        # do one of LAX, BAR, relaxed-REBAR, or REBAR\n        if relaxed:\n            rebar = (batch_f_b - batch_f_z_tilde) * d_log_pb_d_log_alpha + (d_f_z_d_log_alpha - d_f_z_tilde_d_log_alpha)\n        else:\n            rebar = (batch_f_b - batch_eta * batch_f_z_tilde) * d_log_pb_d_log_alpha + batch_eta * (\n            d_f_z_d_log_alpha - d_f_z_tilde_d_log_alpha)\n        reinforce = batch_f_b * d_log_pb_d_log_alpha\n        tf.summary.histogram(\"rebar\", rebar)\n        tf.summary.histogram(\"reinforce\", reinforce)\n\n        # variance reduction objective\n        variance_loss = tf.reduce_mean(tf.square(rebar))\n\n        # optimizers\n        inf_opt = tf.train.AdamOptimizer(lr)\n\n        # need to scale by batch size cuz tf.gradients sums\n        if use_reinforce:\n            log_alpha_grads = reinforce / batch_size\n        else:\n            log_alpha_grads = rebar / batch_size\n\n        inf_train_op = inf_opt.apply_gradients([(log_alpha_grads, log_alpha)])\n\n        var_opt = tf.train.AdamOptimizer(lr)\n        var_vars = [eta, log_temperature]\n        if relaxed:\n            print(\"Relaxed model\")\n            q_vars = [v for v in tf.trainable_variables() if \"Q_func\" in v.name]\n            var_vars = var_vars + q_vars\n        var_gradvars = var_opt.compute_gradients(variance_loss, var_list=var_vars)\n        var_train_op = var_opt.apply_gradients(var_gradvars)\n\n        print(\"Variance\")\n        for g, v in var_gradvars:\n            print(\"    {}\".format(v.name))\n            if g is not None:\n                tf.summary.histogram(v.name, v)\n                tf.summary.histogram(v.name + \"_grad\", g)\n\n        if use_reinforce:\n            with tf.control_dependencies([inf_train_op]):\n                train_op = tf.no_op()\n        else:\n            with tf.control_dependencies([inf_train_op, var_train_op]):\n                train_op = tf.no_op()\n\n        summ_op = tf.summary.merge_all()\n        summary_writer = tf.summary.FileWriter(TRAIN_DIR)\n        sess.run(tf.global_variables_initializer())\n\n        variances = []\n        losses = []\n        thetas = []\n\n        print(\"Collecting {} samples\".format(ITERS // RESOLUTION))\n        for i in tqdm(range(iters)):\n            if (i + 1) % RESOLUTION == 0:\n                if tf_log:\n                    loss_value, _, sum_str, theta_value = sess.run([loss, train_op, summ_op, theta])\n                    summary_writer.add_summary(sum_str, i)\n                else:\n                    loss_value, _, theta_value, temp = sess.run([loss, train_op, theta, temperature])\n\n                tv = theta_value[0][0]\n                thetas.append(tv)\n                losses.append(tv * (1 - target[0][0]) ** 2 + (1 - tv) * target[0][0] ** 2)\n                print(i, loss_value, [t for t in theta_value[0]], [tmp for tmp in temp])\n\n                if log_var:\n                    grads = [sess.run([rebar, reinforce]) for i in tqdm(range(100))]\n                    rebars, reinforces = zip(*grads)\n                    re_m, re_v = np.mean(rebars), np.std(rebars)\n                    rf_m, rf_v = np.mean(reinforces), np.std(reinforces)\n                    if use_reinforce:\n                        variances.append(re_v)\n                    print(\"Reinforce mean = {}, Reinforce std = {}\".format(rf_m, rf_v))\n                    print(\"Rebar mean     = {}, Rebar std     = {}\".format(re_m, re_v))\n\n\n            else:\n                _, = sess.run([train_op])\n\n        tv = None\n        print(tv)\n        return tv, thetas, losses, variances\n\n\nif __name__ == \"__main__\":\n    t = 0.499\n    rand_seed = np.random.randint(1, 1000)\n\n    main(t=t, relaxed=\"THETA_U\", rand_seed=rand_seed)\n"
  }
]