vae

Variational Autoencoders 3: Training, Inference and comparison with other models

Variational Autoencoders 1: Overview
Variational Autoencoders 2: Maths
Variational Autoencoders 3: Training, Inference and comparison with other models

Recalling that the backbone of VAEs is the following equation:

\log P\left(X\right) - \mathcal{D}\left[Q\left(z\vert X\right)\vert\vert P\left(z\vert X\right)\right] = E_{z\sim Q}\left[\log P\left(X\vert z\right)\right] - \mathcal{D}\left[Q\left(z\vert X\right) \vert\vert P\left(z\right)\right]

In order to use gradient descent for the right hand side, we need a tractable way to compute it:

  • The first part E_{z\sim Q}\left[\log P\left(X\vert z\right)\right] is tricky, because that requires passing multiple samples of z through f in order to have a good approximation for the expectation (and this is expensive). However, we can just take one sample of z, then pass it through f and use it as an estimation for E_{z\sim Q}\left[\log P\left(X\vert z\right)\right] . Eventually we are doing stochastic gradient descent over different sample X in the training set anyway.
  • The second part \mathcal{D}\left[Q\left(z\vert X\right) \vert\vert P\left(z\right)\right] is even more tricky. By design, we fix P\left(z\right) to be the standard normal distribution \mathcal{N}\left(0,I\right) (read part 1 to know why). Therefore, we need a way to parameterize Q\left(z\vert X\right) so that the KL divergence is tractable.

Here comes perhaps the most important approximation of VAEs. Since P\left(z\right) is standard Gaussian, it is convenient to have Q\left(z\vert X\right) also Gaussian. One popular way to parameterize Q is to make it also Gaussian with mean \mu\left(X\right) and diagonal covariance \sigma\left(X\right)I, i.e. Q\left(z\vert X\right) = \mathcal{N}\left(z;\mu\left(X\right), \sigma\left(X\right)I\right), where \mu\left(X\right) and \sigma\left(X\right) are two vectors computed by a neural network. This is the original formulation of VAEs in section 3 of this paper.

This parameterization is preferred because the KL divergence now becomes closed-form:

\displaystyle \mathcal{D}\left[\mathcal{N}\left(\mu\left(X\right), \sigma\left(X\right)I\right)\vert\vert P\left(z\right)\right] = \frac{1}{2}\left[\left(\sigma\left(X\right)\right)^T\left(\sigma\left(X\right)\right) +\left(\mu\left(X\right)\right)^T\left(\mu\left(X\right)\right) - k - \log \det \left(\sigma\left(X\right)I\right) \right]

Although this looks like magic, but it is quite natural if you apply the definition of KL divergence on two normal distributions. Doing so will teach you a bit of calculus.

So we have all the ingredients. We use a feedforward net to predict \mu\left(X\right) and \sigma\left(X\right) given an input sample X draw from the training set. With those vectors, we can compute the KL divergence and \log P\left(X\vert z\right), which, in term of optimization, will translate into something similar to \Vert X - f\left(z\right)\Vert^2.

It is worth to pause here for a moment and see what we just did. Basically we used a constrained Gaussian (with diagonal covariance matrix) to parameterize Q. Moreover, by using \Vert X - f\left(z\right)\Vert^2 for one of the training criteria, we implicitly assume P\left(X\vert z\right) to be also Gaussian. So although the maths that lead to VAEs are generic and beautiful, at the end of the day, to make things tractable, we ended up using those severe approximations. Whether those approximations are good enough totally depend on practical applications.

There is an important detail though. Once we have \mu\left(X\right) and \sigma\left(X\right) from the encoder, we will need to sample z from a Gaussian distribution parameterized by those vectors. z is needed for the decoder to reconstruct \hat{X}, which will then be optimized to be as close to X as possible via gradient descent. Unfortunately, the “sample” step is not differentiable, therefore we will need a trick call reparameterization, where we don’t sample z directly from \mathcal{N}\left(\mu\left(X\right), \sigma\left(X\right)\right), but first sample z' from \mathcal{N}\left(0, I\right), and then compute z = \mu\left(X\right) + \mu\left(X\right)Iz'. This will make the whole computation differentiable and we can apply gradient descent as usual.

The cool thing is during inference, you won’t need the encoder to compute \mu\left(X\right) and \sigma\left(X\right) at all! Remember that during training, we try to pull Q to be close to P\left(z\right) (which is standard normal), so during inference, we can just inject \epsilon \sim \mathcal{N}\left(0, I\right) directly into the decoder and get a sample of X. This is how we can leverage the power of “generation” from VAEs.

There are various extensions to VAEs like Conditional VAEs and so on, but once you understand the basic, everything else is just nuts and bolts.

To sum up the series, this is the conceptual graph of VAEs during training, compared to some other models. Of course there are many details in those graphs that are left out, but you should get a rough idea about how they work.

vae

In the case of VAEs, I added the additional cost term in blue to highlight it. The cost term for other models, except GANs, are the usual L2 norm \Vert X - \hat{X}\Vert^2.

GSN is an extension to Denoising Autoencoder with explicit hidden variables, however that requires to form a fairly complicated Markov Chain. We may have another post  for it.

With this diagram, hopefully you will see how lame GAN is. It is even simpler than the humble RBM. However, the simplicity of GANs makes it so powerful, while the complexity of VAE makes it quite an effort just to understand. Moreover, VAEs make quite a few severe approximation, which might explain why samples generated from VAEs are far less realistic than those from GANs.

That’s quite enough for now. Next time we will switch to another topic I’ve been looking into recently.

Variational Autoencoders 2: Maths

Variational Autoencoders 1: Overview
Variational Autoencoders 2: Maths
Variational Autoencoders 3: Training, Inference and comparison with other models

Last time we saw the probability distribution of X with a latent variable z as follows:

\displaystyle P(X) = \int P\left(X\vert z; \theta\right)P(z)dz  (1)

and we said the key idea behind VAEs is to not sample z from the whole distribution P\left(z\right), but actually from a simpler distribution Q\left(z\vert X\right). The reason is because most of z will likely to give P\left(X\vert z\right) close to zero, and therefore making little contribution to the estimation of P\left(X\right). Now if we sample z \sim Q\left(z\vert X\right), those values of z will more likely to generate X in the training set. Moreover, we hope that Q will has less modes than P\left(z\right), and therefore easier to sample from. The intuition of this is the locations of the modes of Q\left(z\vert X\right) depends on X, and this flexibility will compensate the limitation of the fact that Q\left(z\vert X\right) is simpler than P\left(z\right).

But how Q\left(z\vert X\right) can help with modelling P\left(X\right)? If z is sampled from Q, then using f we will get E_{z \sim Q}P\left(X\vert z\right). We will then need to show the relationship of this quantity with P\left(X\right), which is the actual quantity we want to estimate. The relationship between E_{z \sim Q}P\left(X\vert z\right) and P\left(X\right) is the backbone of VAEs.

We start with the KL divergence of Q\left(z\vert X\right) and P\left(z\vert X\right):

\mathcal{D}\left[Q\left(z\vert X\right) \vert\vert P\left(z\vert X\right)\right] = E_{z\sim Q}\left[\log Q\left(z\vert X\right) - log P\left(z\vert X\right)\right]

The unknown quantity in this equation is P\left(z\vert X\right), but at least we can use Bayes rule for it:

\mathcal{D}\left[Q\left(z\vert X\right) \vert\vert P\left(z\vert X\right)\right] = E_{z\sim Q}\left[\log Q\left(z\vert X\right) - log P\left(X\vert z\right) - \log P\left(z\right)\right] + \log P\left(X\right)

Rearrange things a bit, and apply the definition of KL divergence between Q\left(z\vert X\right) and P\left(z\right), we have:

\log P\left(X\right) - \mathcal{D}\left[Q\left(z\vert X\right)\vert\vert P\left(z\vert X\right)\right] = E_{z\sim Q}\left[\log P\left(X\vert z\right)\right] - \mathcal{D}\left[Q\left(z\vert X\right) \vert\vert P\left(z\right)\right]    (2)

If you forget everything, this formula is the thing you should remember. It is therefore important to understand what it means:

  • The left-hand-side is exactly what we want to optimize, plus an error term. The smaller this error term is, the better we are in mazimizing P\left(X\right). In other words, the left-hand-side is a lower-bound of what we want to optimize, hence the name variational (Bayesian).
  • If Q happens to be a differentiable function, the right-hand-side is something we can optimize with gradient descent (we will see how to do it later). Note that the right-hand-side happens to take the form of encoder and decoder, where Q encodes X into z, and then P decodes z to reconstruct X, hence the name “Autoencoder”. However, VAEs don’t really belong to the family of Denoising and Sparse Autoencoders, although there are indeed some connections.
  • Note that P\left(z\vert X\right) on the left hand side is something intractable. However, by maximizing the left hand side, we simultaneously minimize \mathcal{D}\left[Q\left(z\vert X\right)\vert\vert P\left(z\vert X\right)\right], and therefore pull Q\left(z\vert X\right) closer to P\left(z\vert X\right). If we use a flexible model for Q, then we can use Q as an approximation for P\left(z\vert X\right). This is a nice side effect of the whole framework.

Actually the above maths existed way before VAEs. However the trick was to use a feedforward network for Q, which gave rise to VAEs several years ago.

Next time, we will see how to do that, and hopefully conclude this series. Then we can move on with something more interesting.

Variational Autoencoders 1: Overview

In a previous post, we briefly mentioned some recent approaches for Generative Modeling. Among those, RBMs and DBMs are probably very tricky because the estimation of gradients in those models is based on a good mixing of MCMC, which tends to get worse during the course of training because the model distribution gets sharper. Autogressive models like PixelRNN, WaveNet, etc… are easier to train but have no latent variables, which makes them somewhat less powerful. Therefore, the current frontier in Generative Modelling is probably GANs and Variational Autoencoders (VAEs).

While GANs are too mainstream, I thought I can probably write a post or two about Variational Autoencoders, at least to clear up some confusions I am having about them.

Formally, generative modeling is the area in Machine Learning that deals with models of distributions P(X), defined over datapoints X in some high-dimensional space \mathcal{X}. The whole idea is to construct models of P(X) that assigns high probabilities to data points similar to those in the training set, and low probabilities every where else. For example, a generative models of images of cows should assign small probabilities to images of human.

However, computing the probability of a given example is not the most exciting thing about generative models. More often, we want to use the model to generate new samples that look like those in the training set. This “creativity” is something unique to generative models, and does not exist in, for instance, discriminative models. More formally, say we have a training set sampled from an unknown distribution P_\text{org}(X), and we want to train a model P which we can take sample from, such that P is as close as possible to P_\text{org}.

Needless to say, this is a difficult problem. To make it tractable, traditional approaches in Machine Learning often have to 1) make strong assumptions about the structure of the data, or 2) make severe approximation, leading to suboptimal models, or 3) rely on expensive sampling procedures like MCMC. Those are all limitations which make Generative modeling a long-standing problem in ML research.

Without further ado, let’s get to the point. When \mathcal{X} is a high-dimensional space, modeling is difficult mostly because it is tricky to handle the inter-dependencies between dimensions. For instance, if the left half of an image is a horse then probably the right half is likely another horse.

To further reduce this complexity, we add a latent variable z in a high-dimensional space \mathcal{Z} that we can easily sample from, according to a distribution P(z) defined over \mathcal{Z}. Then say we have a family of deterministic function f(z;\theta) parameterized by a vector \theta in some space \Theta where f: \mathcal{Z} \times \Theta \rightarrow \mathcal{X}. Now f is deterministic, but since z is a random variable, f(z;\theta) is a random variable in \mathcal{X}.

During inference, we will sample z from P(z), and then train \theta such that f(z;\theta) is close to samples in the training set. Mathematically, we want to maximize the following probability for every sample X in the training set:

\displaystyle P(X) = \int P\left(X\vert z; \theta\right)P(z)dz   (1)

This is the good old maximum likelihood framework, but we replace f(z;\theta) by P\left(X\vert z;\theta\right) (called the output distribution) to explicitly indicate that X depends on z, so that we can use the integral to make it a proper probability distribution.

There are a few things to note here:

  • In VAEs, the choice of the output distribution is often Gaussian, i.e. P\left(X\vert z;\theta\right) = \mathcal{N}\left(X; f(z;\theta), \sigma^2 * I\right), meaning it is a Gaussian distribution with mean f(z;\theta) and diagonal covariance matrix where \sigma is a scalar hyper-parameter. This particular choice has some important motivations:
    • We need the output distribution to be continuous, so that we can use gradient descent on the whole model. It wouldn’t be possible if we use discontinuous function like the Dirac distribution, meaning to use exactly the output value of f(z;\theta) for X.
    • We don’t really need to train our model such that f(z;\theta) produces exactly some sample X in the training set. Instead, we want it to produce samples that are merely like X. In the beginning of training, there is no way for f to gives exact samples in the training set. Hence by using a Gaussian, we allow the model to gradually (and gently) learn to produce samples that are more and more like those in the training set.
    • It doesn’t have to be Gaussian though. For instance, if X is binary, we can make P\left(X\vert z;\theta\right) a Bernoulli parameterized by f(z;\theta). The important property is that P\left(X\vert z\right) can be computed, and is continuous in the domain of \theta.
  • The distribution of z is simply the normal distribution, i.e. P(z) = \mathcal{N}\left(0,I\right). Why? How is it possible? Is there any limitation with this? A related question is why don’t we have several levels of latent variables. which potentially might help modelling complicated processes?
    All those question can be answered by the key observation that any distribution in d dimensions can be generated by taking d variables from the normal distribution and mapping them through a sufficiently complicated function.
    Let that sink for a moment. Readers who are interested in the mathematical details can have a look at the conditional distribution method described in this paper. You can also convince yourself if you remember how we can sample from any Gaussian as described in an earlier post.
    Now, this observation means we don’t need to go to more than one level of latent variable, with a condition that we need a sufficiently complicated function for f(z;\theta). Since deep neural nets has been shown to be a powerful function approximator, it makes a lot of sense to use deep neural nets for modeling f.
  • Now the only business is to maximize (1). Using the law of large numbers, we can approximate the integral by the expected value over a large number of samples. So the plan will be to take a very large sample \left\{z_1, ..., z_n\right\} from P(z), then compute P(X) \approx \frac{1}{n}\sum_i P\left(X\vert z_i;\theta\right). Unfortunately the plan is infeasible because in high dimensional spaces, n needs to be very large in order to have a good enough approximation of P(X) (imagine how much samples you would need for 200 \times 200 \times 3 images, which is in 120K dimensional space?)
    Now the key to realize is that we don’t need to sample z from all over P(z). In fact, we only need to sample z such that f(z;\theta) is more likely to be similar to samples in the training set. Moreover, it is likely that for most of z, P(X\vert z) is nearly zero, and therefore contribute very little into the estimation of P(X). So the question is: is there any way to sample z such that it is likely to generate X, and only estimate P(X) from those?
    It is the key idea behind VAEs.

That’s quite enough for an overview. Next time we will do some maths and see how we go about maximizing (1). Hopefully I can then convince you that VAEs, GANs and GSNs are really not far away from each other, at least in their core ideas.