Evidence Lower Bound Maximization

Introduction

Evidence lower bound (ELBO) maximization[JGJS1999] is a general family of algorithms that minimize the exclusive (or reverse) Kullback-Leibler (KL) divergence between the target distribution $\pi$ and a variational approximation $q_{\lambda}$. More generally, they aim to solve the following problem:

\[ \mathrm{minimize}_{q \in \mathcal{Q}}\quad \mathrm{KL}\left(q, \pi\right),\]

where $\mathcal{Q}$ is some family of distributions, often called the variational family. Since the target distribution $\pi$ is intractable in general, the KL divergence is also intractable. Instead, the ELBO maximization strategy maximizes a surrogate objective, the ELBO:

\[ \mathrm{ELBO}\left(q\right) \triangleq \mathbb{E}_{\theta \sim q} \log \pi\left(\theta\right) + \mathbb{H}\left(q\right),\]

which serves as a lower bound to the KL. The ELBO and its gradient can be readily estimated through various strategies. Overall, ELBO maximization algorithms aim to solve the problem:

\[ \mathrm{maximize}_{q \in \mathcal{Q}}\quad \mathrm{ELBO}\left(q\right).\]

Multiple ways to solve this problem exist, each leading to a different variational inference algorithm.

Algorithms

Currently, AdvancedVI only provides the approach known as black-box variational inference (also known as Monte Carlo VI, Stochastic Gradient VI). (Introduced independently by two groups [RGB2014][TL2014] in 2014.) In particular, AdvancedVI focuses on the reparameterization gradient estimator[TL2014][RMW2014][KW2014], which is generally superior compared to alternative strategies[XQKS2019], discussed in the following section:

  • JGJS1999Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1999). An introduction to variational methods for graphical models. Machine learning, 37, 183-233.
  • TL2014Titsias, M., & Lázaro-Gredilla, M. (2014). Doubly stochastic variational Bayes for non-conjugate inference. In International Conference on Machine Learning.
  • RMW2014Rezende, D. J., Mohamed, S., & Wierstra, D. (2014). Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning.
  • KW2014Kingma, D. P., & Welling, M. (2014). Auto-encoding variational bayes. In International Conference on Learning Representations.
  • XQKS2019Xu, M., Quiroz, M., Kohn, R., & Sisson, S. A. (2019). Variance reduction properties of the reparameterization trick. In *The International Conference on Artificial Intelligence and Statistics.
  • RGB2014Ranganath, R., Gerrish, S., & Blei, D. (2014). Black box variational inference. In Artificial Intelligence and Statistics.