Boosting Black Box Variational Inference

TL;DR

This paper introduces a novel boosting approach for Variational Inference that guarantees convergence with minimal assumptions, enabling black box implementation and improving approximation quality in Bayesian inference.

Contribution

It provides theoretical guarantees for boosting VI using a relaxed smoothness assumption and proposes a RELBO-based greedy step for black box applicability.

Findings

Convergence of boosting VI under relaxed assumptions

Effective RELBO-based greedy step for model-agnostic implementation

Empirical results demonstrating improved inference quality

Abstract

Approximating a probability density in a tractable manner is a central task in Bayesian statistics. Variational Inference (VI) is a popular technique that achieves tractability by choosing a relatively simple variational family. Borrowing ideas from the classic boosting framework, recent approaches attempt to \emph{boost} VI by replacing the selection of a single density with a greedily constructed mixture of densities. In order to guarantee convergence, previous works impose stringent assumptions that require significant effort for practitioners. Specifically, they require a custom implementation of the greedy step (called the LMO) for every probabilistic model with respect to an unnatural variational family of truncated distributions. Our work fixes these issues with novel theoretical and algorithmic insights. On the theoretical side, we show that boosting VI satisfies a relaxed smoothness assumption which is sufficient for the convergence of the functional Frank-Wolfe (FW) algorithm. Furthermore, we rephrase the LMO problem and propose to maximize the Residual ELBO (RELBO) which replaces the standard ELBO optimization in VI. These theoretical enhancements allow for black box implementation of the boosting subroutine. Finally, we present a stopping criterion drawn from the duality gap in the classic FW analyses and exhaustive experiments to illustrate the usefulness of our theoretical and algorithmic contributions.