Learning with Importance Weighted Variational Inference: Asymptotics for Gradient Estimators of the VR-IWAE Bound

TL;DR

This paper analyzes gradient estimators for importance weighted variational bounds, unifying and comparing ELBO, IWAE, and VR bounds, and provides insights into their advantages and limitations through theoretical and empirical studies.

Contribution

It introduces analyses of gradient estimators for the VR-IWAE bound, clarifying their properties and how different bounds relate in variational inference.

Findings

Reparameterized gradient estimators have specific advantages and limitations.

The VR-IWAE bound unifies ELBO, IWAE, and VR bounds methodologies.

Empirical results support the theoretical insights on gradient estimator performance.

Abstract

Several popular variational bounds involving importance weighting ideas have been proposed to generalize and improve on the Evidence Lower BOund (ELBO) in the context of maximum likelihood optimization, such as the Importance Weighted Auto-Encoder (IWAE) and the Variational R\'enyi (VR) bounds. The methodology to learn the parameters of interest using these bounds typically amounts to running gradient-based variational inference algorithms that incorporate the reparameterization trick. However, the way the choice of the variational bound impacts the outcome of variational inference algorithms can be unclear. Recently, the VR-IWAE bound was introduced as a variational bound that unifies the ELBO, IWAE and VR bounds methodologies. In this paper, we provide two analyses for the reparameterized and doubly-reparameterized gradient estimators of the VR-IWAE bound, which reveal the advantages and limitations of these gradient estimators while enabling us to compare of the ELBO, IWAE and VR bounds methodologies. Our work advances the understanding of importance weighted variational inference methods and we illustrate our theoretical findings empirically.