Sobolev.space

Previously we covered Variational Autoencoders (VAE) — a popular inference tool based on neural networks. In this post we'll consider a followup work from Toronto by Y. Burda, R. Grosse and R. Salakhutdinov, Importance Weighted Autoencoders (IWAE). The crucial contribution of this work is the introduction of a new lower-bound on the marginal log-likelihood $\log p(x)$ which generalizes ELBO, but also allows one to use less accurate approximate posteriors $q(z \mid x, \Lambda)$.

For dessert we'll discuss another paper, Variational inference for Monte Carlo objectives by A. Mnih and D. Rezende which aims to broaden the applicability of this approach to models where the reparametrization trick cannot be used (e.g. for discrete variables).

So far we had little of "neural" in our VI methods. Now it's time to fix it, as we're going to consider Variational Autoencoders (VAE), a paper by D. Kingma and M. Welling, which made a lot of buzz in the ML community. It has 2 main contributions: a new approach (AEVB) to large-scale inference in non-conjugate models with continuous latent variables, and a probabilistic model of autoencoders as an example of this approach. We then discuss connections to Helmholtz machines — a predecessor of VAEs.

In the previous post we covered Stochastic VI: an efficient and scalable variational inference method for exponential family models. However, there're many more distributions than those belonging to the exponential family. Inference in these cases requires a significant amount of model analysis. In this post we consider Black Box Variational Inference by Ranganath et al. This work just as the previous one comes from David Blei's lab — one of the leading researchers in VI. And, just for dessert, we'll touch upon another paper, which will finally introduce some neural networks in VI.

In the previous post I covered well-established classical theory developed in the early 2000s. Since then technology has made huge progress: now we have much more data, and a great need to process it and process it fast. In the big data era we have huge datasets, and cannot afford too many full passes over them, which might render classical VI methods impractical. Recently M. Hoffman et al. dissected classical Mean-Field VI to introduce stochasticity right into its heart, which resulted in Stochastic Variational Inference.

As a member of the Bayesian methods research group I'm heavily interested in the Bayesian approach to machine learning. One of the strengths of this approach is the ability to work with hidden (unobserved) variables which are interpretable. This power however comes at a cost of generally intractable exact inference, which limits the scope of solvable problems.

Another topic which gained lots of momentum in Machine Learning recently is Deep Learning, of course. With Deep Learning we can now build big and complex models that outperform most hand-engineered approaches given lots of data and computational power. The fact that Deep Learning needs a considerable amount of data also requires these methods to be scalable — a really nice property for any algorithm to have, especially in a Big Data epoch.

Given how appealing both topics are it's not a surprise there's been some work to marry these two recently. In this series of blogposts I'd like to summarize recent advances, particularly in variational inference. This is not meant to be an introductory discussion as prior familiarity with classical topics (Latent variable models, Variational Inference, Mean-field approximation) is required, though I'll introduce these ideas anyway just to remind you of them and set up the notation.