After quite some time spent on the pull request, I’m proud to announce that the VAE model is now integrated in Pylearn2. In this post, I’ll go over the main features of the VAE framework and how to extend it. I will assume the reader is familiar with the VAE model. If not, have a look at my VAE demo webpage as well as the (Kingma and Welling) and (Rezende et al.) papers.

# The model

A VAE comes with three moving parts:

• the prior distribution $$p_\theta(\mathbf{z})$$ on latent vector $$\mathbf{z}$$
• the conditional distribution $$p_\theta(\mathbf{x} \mid \mathbf{z})$$ on observed vector $$\mathbf{x}$$
• the approximate posterior distribution $$q_\phi(\mathbf{z} \mid \mathbf{x})$$ on latent vector $$\mathbf{z}$$

The parameters $$\phi$$ and $$\theta$$ are arbitrary functions of $$\mathbf{x}$$ and $$\mathbf{z}$$ respectively.

The model is trained to minimize the expected reconstruction loss of $$\mathbf{x}$$ under $$q_\phi(\mathbf{z} \mid \mathbf{x})$$ and the KL-divergence between the prior and posterior distributions on $$\mathbf{z}$$ at the same time.

In order to backpropagate the gradient on the reconstruction loss through the function mapping $$\mathbf{x}$$ to parameters $$\phi$$, the reparametrization trick is used, which allows sampling from $$\mathbf{z}$$ by considering it as a deterministic function of $$\mathbf{x}$$ and some noise $$\mathbf{\epsilon}$$.

# The VAE framework

## Overview

### pylearn2.models.vae.VAE

The VAE model is represented in Pylearn2 by the VAE class. It is responsible for high-level computation, such as computing the log-likelihood lower bound or an importance sampling estimate of the log-likelihood, and acts as the interface between the model and other parts of Pylearn2.

It delegates much of its functionality to three objects:

• pylearn2.models.vae.conditional.Conditional
• pylearn2.models.vae.prior.Prior
• pylearn2.models.vae.kl.KLIntegrator

### pylearn2.models.vae.conditional.Conditional

Conditional is used to represent conditional distributions in the VAE framework (namely the approximate posterior on $$\mathbf{z}$$ and the conditional on $$\mathbf{x}$$). It is responsible for mapping its input to parameters of the conditional distribution it represents, sampling from the conditional distribution with or without the reparametrization trick and computing the conditional log-likelihood of the distribution it represents given some samples.

Internally, the mapping from input to parameters of the conditional distribution is done via an MLP instance. This allows users familiar with the MLP framework to easily switch between different architectures for the encoding and decoding networks.

### pylearn2.models.vae.prior.Prior

Prior is used to represent the prior distribution on $$\mathbf{z}$$ in the VAE framework. It is responsible for sampling from the prior distribution and computing the log-likelihood of the distribution it represents given some samples.

### pylearn2.models.vae.kl.KLIntegrator

Some combinations of prior and posterior distributions (e.g. a gaussian prior with diagonal covariance matrix and a gaussian posterior with diagonal covariance matrix) allow the analytic integration of the KL term in the VAE criterion. KLIntegrator is responsible for representing this analytic expression and optionally representing it as a sum of elementwise KL terms, when such decomposition is allowed by the choice of prior and posterior distributions.

This allows the VAE framework to be more modular: otherwise, the analytical computation of the KL term would require that the prior and the posterior distributions are defined in the same class.

Subclasses of KLIntegrator define one subclass of Prior and one subclass of Conditional as class attributes and can carry out the analytic computation of the KL term for these two subclasses only. The pylearn2.models.vae.kl module also contains a method which can automatically infer which subclass of KLIntegrator is compatible with the current choice of prior and posterior, and VAE automatically falls back to a stochastic approximation of the KL term when the analytical computation is not possible.

### pylearn2.costs.vae.{VAE,ImportanceSampling}Criterion

Two Cost objects are compatible with the VAE framework: VAECriterion and ImportanceSamplingCriterion. VAECriterion represent the VAE criterion as defined in (Kingma and Welling), while ImportanceSamplingCriterion defines a cost based on the importance sampling approximation of the marginal log-likelihood which allows backpropagation through $$q_\phi(\mathbf{z} \mid \mathbf{x})$$ via the reparametrization trick.

## Using the framework

### Training the example model

Let’s go over a small example on how to train a VAE on MNIST digits.

In this example I’ll be using Salakhutdinov and Murray’s binarized version of the MNIST dataset. Make sure the PYLEARN2_DATA_PATH environment variable is set properly, and download the data using

Here’s the YAML file we’ll be using for the example:

Give it a try:

This might take a while, but you can accelerate things using the appropriate Theano flags to train using a GPU.

You’ll see a couple things being monitored while the model learns:

• {train,valid,test}_objective tracks the value of the VAE criterion for the training, validation and test sets.
• {train,valid,test}_expectation_term tracks the expected reconstruction of the input under the posterior distribution averaged across the training, validation and test sets.
• {train,valid,test}_kl_divergence_term tracks the KL-divergence between the posterior and the prior distributions averaged across the training, validation and test sets.

### Evaluating the trained model

N.B.: At the moment of writing this post, there are no scripts in Pylearn2 to evaluate trained models by looking at samples or computing an approximate NLL. This is definitely something that will be included in the future, but for the moment here are some workarounds taken from my personal scripts.

When training is complete, you can look at samples from the model by running the following bit of Python code:

Look at samples by typing

You can also make use of VAE.log_likelihood_approximation to compute approximate NLL performance measures of the trained model:

All you have to do is type

### More details

Let’s concentrate on this part of the YAML file:

We define the dimensionality of $$\mathbf{x}$$ through nvis and the dimensionality of $$\mathbf{z}$$ through nhid.

At a high level, the form of the prior, posterior and conditional distributions is selected through the choice of which subclasses to instantiate. Here we chose a gaussian prior with diagonal covariance matrix, a gaussian posterior with diagonal covariance matrix and a product of bernoulli as the conditional for $$\mathbf{x}$$.

Note that we did not explicitly tell the model how to integrate the KL: it was able to find it on its own by calling pylearn2.models.vae.kl.find_integrator_for, which searched pylearn2.models.vae.kl for a match and returned an instance of DiagonalGaussianPriorPosteriorKL. If you were to explicitly tell the model how to integrate the KL term (for instance, if you have defined a new prior and a new KLIntegrator subclass to go with it), you would need to add

as a parameter to VAE’s constructor.

Conditional instances (passed as conditional and posterior parameters) need a name upon instantiation. This is to avoid key collisions in the monitoring channels.

They’re also given nested MLP instance. Why this is needed will become clear soon. Notice how the last layers’ dimensionality does not match either nhid or nvis. This is because they represent the last hidden representation from which the conditional parameters will be computed. You did not have to specify the layer mapping the last hidden representation to the conditional parameters because it was automatically inferred: after everything is instantiated, VAE calls initialize_parameters on prior, conditional and posterior and gives them relevant information about their input and output spaces. At that point, Conditional has enough information to infer how the last layer should look like. It calls its private _get_default_output_layer method, which returns a sane default output layer, and adds it to its MLP’s list of layers. This is why a nested MLP is required: this allows Conditional to delay the initialization of the MLP’s input space in order to add a layer to it in a clean fashion.

Naturally, you may want to decide on your own how parameters should be computed based on the last hidden representation. This can be done through Conditional’s output_layer_required constructor parameter. It is set to True by default, but you can switch it off and explicitly put the last layer in the MLP. For instance, you could decide that the gaussian posterior’s $$\log \sigma$$ should not be too big or too small and want to force it to be between -1 and 1 by using a tanh non-linearity. It can be done like so:

There are safeguards in place to make sure your code won’t crash without explanation in the case of a mistake: Conditional will verify that the custom output layer you put in MLP has the same output space as what it expects and raise an exception otherwise. Every Conditional subclass need to define how should the conditional parameters look like through a private _get_required_mlp_output_space method, and you should make sure that your custom output layer has the right output space by looking at the code. Moreover, you should have a look at the subclass’ _get_default_output_space implementation to see what is the nature and order of the conditional parameters being computed.

## Extending the VAE framework

This post will be updated soon with more information on how to write your own subclasses of Prior, Conditional and KLIntegrator.