Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectives

TL;DR

This paper analyzes importance weighted variational inference, revealing how properties of Monte Carlo estimates, such as negative correlation, influence the tightness of bounds and providing new theoretical insights into their behavior.

Contribution

It extends the monotonicity theorem for Monte Carlo objectives by considering non-uniform weights and links Monte Carlo estimate properties to variational bound tightness.

Findings

Negative correlation reduces the variational gap.

Increasing importance samples tightens the bound.

Non-uniform weights affect Monte Carlo estimate properties.

Abstract

We revisit the theory of importance weighted variational inference (IWVI), a promising strategy for learning latent variable models. IWVI uses new variational bounds, known as Monte Carlo objectives (MCOs), obtained by replacing intractable integrals by Monte Carlo estimates -- usually simply obtained via importance sampling. Burda, Grosse and Salakhutdinov (2016) showed that increasing the number of importance samples provably tightens the gap between the bound and the likelihood. Inspired by this simple monotonicity theorem, we present a series of nonasymptotic results that link properties of Monte Carlo estimates to tightness of MCOs. We challenge the rationale that smaller Monte Carlo variance leads to better bounds. We confirm theoretically the empirical findings of several recent papers by showing that, in a precise sense, negative correlation reduces the variational gap. We also generalise the original monotonicity theorem by considering non-uniform weights. We discuss several practical consequences of our theoretical results. Our work borrows many ideas and results from the theory of stochastic orders.