Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectives
This work provides theoretical insights for researchers in variational inference, but it is incremental as it builds on existing monotonicity theorems.
The paper revisits importance weighted variational inference (IWVI) and its Monte Carlo objectives (MCOs), showing that negative correlation in Monte Carlo estimates reduces the variational gap, generalizing monotonicity results with non-uniform weights.
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.