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Aug 22, 2016 · whitepaper

Under the Hood of the Variational Autoencoder (in Prose and Code)

The Variational Autoencoder (VAE) neatly synthesizes unsupervised deep learning and variational Bayesian methods into one sleek package. In Part I of this series, we introduced the theory and intuition behind the VAE, an exciting development in machine learning for combined generative modeling and inference—“machines that imagine and reason.”

from functional import compose, partial import numpy as np import tensorflow as tf


<p>One perk of these models is their modularity—VAEs are naturally amenable to swapping in whatever encoder/decoder architecture is most fitting for the task at hand: <a href="https://arxiv.org/abs/1502.04623">recurrent</a> <a href="https://arxiv.org/abs/1511.06349">neural</a> <a href="https://arxiv.org/abs/1412.6581">networks</a>, <a href="https://arxiv.org/abs/1411.5928">convolutional</a> and <a href="https://arxiv.org/abs/1503.03167">deconvolutional</a> networks, etc.</p>
<p>For our purposes, we will model the relatively simple <a href="http://yann.lecun.com/exdb/mnist/">MNIST</a> dataset using densely-connected layers, wired symmetrically around the hidden code.</p>

```python
class Dense():
    """Fully-connected layer"""
    def __init__(self, scope="dense_layer", size=None, dropout=1.,
                 nonlinearity=tf.identity):
        # (str, int, (float | tf.Tensor), tf.op)
        assert size, "Must specify layer size (num nodes)"
        self.scope = scope
        self.size = size
        self.dropout = dropout # keep_prob
        self.nonlinearity = nonlinearity

    def __call__(self, x):
        """Dense layer currying, to apply layer to any input tensor `x`"""
        # tf.Tensor -&gt; tf.Tensor
        with tf.name_scope(self.scope):
            while True:
                try: # reuse weights if already initialized
                    return self.nonlinearity(tf.matmul(x, self.w) + self.b)
                except(AttributeError):
                    self.w, self.b = self.wbVars(x.get_shape()[1].value, self.size)
                    self.w = tf.nn.dropout(self.w, self.dropout)
    ...
i.e. composed = composeAll([f, g, h])
     composed(x) # == f(g(h(x)))
"""
# adapted from https://docs.python.org/3.1/howto/functional.html
return partial(functools.reduce, compose)(*args)

<p>Now that we’ve defined our model primitives, we can tackle the VAE itself.</p>
<p>Keep in mind: the TensorFlow computational graph is cleanly divorced from the numerical computations themselves. In other words, a <code>tf.Graph</code> wireframes the underlying skeleton of the model, upon which we may hang values only within the context of a <code>tf.Session</code>.</p>
<p>Below, we initialize class <code>VAE</code> and activate a session for future convenience (so we can initialize and evaluate tensors within a single session, e.g. to persist weights and biases across rounds of training).</p>
<p>Here are some relevant snippets, cobbled together from the <a href="https://github.com/fastforwardlabs/vae-tf/blob/master/vae.py">full source code</a>:</p>

```python
class VAE():
    """Variational Autoencoder

    see: Kingma &amp; Welling - Auto-Encoding Variational Bayes
    (https://arxiv.org/abs/1312.6114)
    """
    DEFAULTS = {
        "batch_size": 128,
        "learning_rate": 1E-3,
        "dropout": 1., # keep_prob
        "lambda_l2_reg": 0.,
        "nonlinearity": tf.nn.elu,
        "squashing": tf.nn.sigmoid
    }
    RESTORE_KEY = "to_restore"

    def __init__(self, architecture, d_hyperparams={}, meta_graph=None,
                 save_graph_def=True, log_dir="./log"):
        """(Re)build a symmetric VAE model with given:

         * architecture (list of nodes per encoder layer); e.g.
           [1000, 500, 250, 10] specifies a VAE with 1000-D inputs, 10-D latents,
           &amp; end-to-end architecture [1000, 500, 250, 10, 250, 500, 1000]

         * hyperparameters (optional dictionary of updates to `DEFAULTS`)
        """
        self.architecture = architecture
        self.__dict__.update(VAE.DEFAULTS, **d_hyperparams)
        self.sesh = tf.Session()

        if not meta_graph: # new model
            handles = self._buildGraph()
            ...
            self.sesh.run(tf.initialize_all_variables())
    # encoding / "recognition": q(z|x)
    encoding = [Dense("encoding", hidden_size, dropout, self.nonlinearity)
                # hidden layers reversed for function composition: outer -&gt; inner
                for hidden_size in reversed(self.architecture[1:-1])]
    h_encoded = composeAll(encoding)(x_in)

    # latent distribution parameterized by hidden encoding
    # z ~ N(z_mean, np.exp(z_log_sigma)**2)
    z_mean = Dense("z_mean", self.architecture[-1], dropout)(h_encoded)
    z_log_sigma = Dense("z_log_sigma", self.architecture[-1], dropout)(h_encoded)

<p>Here, we build a pipe from <code>x_in</code> (an empty placeholder for input data <span class="math inline">\(x\)</span>), through the sequential hidden encoding, to the corresponding distribution over latent space—the variational approximate posterior, or hidden representation, <span class="math inline">\(z \sim q_\phi(z|x)\)</span>.</p>
<p>As observed in lines <code>14</code> - <code>15</code>, latent <span class="math inline">\(z\)</span> is distributed as a multivariate <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2465539/figure/fig1/">normal</a> with mean <span class="math inline">\(\mu\)</span> and diagonal covariance values <span class="math inline">\(\sigma^2\)</span> (the square of the “sigma” in <code>z_log_sigma</code>) directly parameterized by the encoder: <span class="math inline">\(\mathcal{N}(\mu, \sigma^2I)\)</span>. In other words, we set out to “explain” highly complex observations as the consequence of an unobserved collection of simplified latent variables, i.e. independent Gaussians. (This is dictated by our choice of a conjugate spherical Gaussian prior over <span class="math inline">\(z\)</span>—see <a href="http://blog.fastforwardlabs.com/2016/08/12/introducing-variational-autoencoders-in-prose-and.html">Part I</a>.)</p>
<p>Next, we sample from this latent distribution (in practice, <a href="https://arxiv.org/abs/1312.6114">one draw is enough</a> given sufficient minibatch size, i.e. &gt;100). This method involves a trick—can you figure out why?—that we will explore in more detail later.</p>
```python
        z = self.sampleGaussian(z_mean, z_log_sigma)
        # decoding / "generative": p(x|z)
        decoding = [Dense("decoding", hidden_size, dropout, self.nonlinearity)
                    for hidden_size in self.architecture[1:-1]] # assumes symmetry
        # final reconstruction: restore original dims, squash outputs [0, 1]
        decoding.insert(0, Dense( # prepend as outermost function
            "reconstruction", self.architecture[0], dropout, self.squashing))
        x_reconstructed = tf.identity(composeAll(decoding)(z), name="x_reconstructed")
        # ops to directly explore latent space
        # defaults to prior z ~ N(0, I)
        z_ = tf.placeholder_with_default(tf.random_normal([1, self.architecture[-1]]),
                                         shape=[None, self.architecture[-1]],
                                         name="latent_in")
        x_reconstructed_ = composeAll(decoding)(z_)
    def sampleGaussian(self, mu, log_sigma):
        """Draw sample from Gaussian with given shape, subject to random noise epsilon"""
        with tf.name_scope("sample_gaussian"):
            # reparameterization trick
            epsilon = tf.random_normal(tf.shape(log_sigma), name="epsilon")
            return mu + epsilon * tf.exp(log_sigma) # N(mu, sigma**2)
    @staticmethod
    def crossEntropy(obs, actual, offset=1e-7):
        """Binary cross-entropy, per training example"""
        # (tf.Tensor, tf.Tensor, float) -&gt; tf.Tensor
        with tf.name_scope("cross_entropy"):
            # bound by clipping to avoid nan
            obs_ = tf.clip_by_value(obs, offset, 1 - offset)
            return -tf.reduce_sum(actual * tf.log(obs_) +
                                  (1 - actual) * tf.log(1 - obs_), 1)
    @staticmethod
    def kullbackLeibler(mu, log_sigma):
        """(Gaussian) Kullback-Leibler divergence KL(q||p), per training example"""
        # (tf.Tensor, tf.Tensor) -&gt; tf.Tensor
        with tf.name_scope("KL_divergence"):
            # = -0.5 * (1 + log(sigma**2) - mu**2 - sigma**2)
            return -0.5 * tf.reduce_sum(1 + 2 * log_sigma - mu**2 -
                                        tf.exp(2 * log_sigma), 1)
        # reconstruction loss: mismatch b/w x &amp; x_reconstructed
        # binary cross-entropy -- assumes p(x) &amp; p(x|z) are iid Bernoullis
        rec_loss = VAE.crossEntropy(x_reconstructed, x_in)

        # Kullback-Leibler divergence: mismatch b/w approximate posterior &amp; imposed prior
        # KL[q(z|x) || p(z)]
        kl_loss = VAE.kullbackLeibler(z_mean, z_log_sigma)

        # average over minibatch
        cost = tf.reduce_mean(rec_loss + kl_loss, name="cost")
        # optimization
        global_step = tf.Variable(0, trainable=False)
        with tf.name_scope("Adam_optimizer"):
            optimizer = tf.train.AdamOptimizer(self.learning_rate)
            tvars = tf.trainable_variables()
            grads_and_vars = optimizer.compute_gradients(cost, tvars)
            clipped = [(tf.clip_by_value(grad, -5, 5), tvar) # gradient clipping
                    for grad, tvar in grads_and_vars]
            train_op = optimizer.apply_gradients(clipped, global_step=global_step,
                                                 name="minimize_cost") # back-prop
        return (x_in, dropout, z_mean, z_log_sigma, x_reconstructed,
                z_, x_reconstructed_, cost, global_step, train_op)
    def train(self, X, max_iter=np.inf, max_epochs=np.inf, cross_validate=True,
              verbose=True, save=False, outdir="./out", plots_outdir="./png"):
        try:
            err_train = 0
            now = datetime.now().isoformat()[11:]
            print("------- Training begin: {} -------\n".format(now))

            while True:
                x, _ = X.train.next_batch(self.batch_size)
                feed_dict = {self.x_in: x, self.dropout_: self.dropout}
                fetches = [self.x_reconstructed, self.cost, self.global_step, self.train_op]
                x_reconstructed, cost, i, _ = self.sesh.run(fetches, feed_dict)

                err_train += cost

                if i%1000 == 0 and verbose:
                    print("round {} --&gt; avg cost: ".format(i), err_train / i)

                if i &gt;= max_iter or X.train.epochs_completed &gt;= max_epochs:
                    print("final avg cost (@ step {} = epoch {}): {}".format(
                        i, X.train.epochs_completed, err_train / i))
                    now = datetime.now().isoformat()[11:]
                    print("------- Training end: {} -------\n".format(now))
                    break

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