Reducing Reparameterization Gradient Variance

TL;DR

This paper introduces a variance reduction technique for reparameterization gradients in Monte Carlo variational inference, significantly improving optimization efficiency by using control variates to reduce noise.

Contribution

It proposes an inexpensive control variate method that correlates with the noisy gradient to substantially reduce variance in gradient estimates.

Findings

Achieved 20-2000x reduction in gradient variance.

Improved convergence speed in hierarchical models.

Enhanced stability in Bayesian neural network training.

Abstract

Optimization with noisy gradients has become ubiquitous in statistics and machine learning. Reparameterization gradients, or gradient estimates computed via the "reparameterization trick," represent a class of noisy gradients often used in Monte Carlo variational inference (MCVI). However, when these gradient estimators are too noisy, the optimization procedure can be slow or fail to converge. One way to reduce noise is to use more samples for the gradient estimate, but this can be computationally expensive. Instead, we view the noisy gradient as a random variable, and form an inexpensive approximation of the generating procedure for the gradient sample. This approximation has high correlation with the noisy gradient by construction, making it a useful control variate for variance reduction. We demonstrate our approach on non-conjugate multi-level hierarchical models and a Bayesian neural net where we observed gradient variance reductions of multiple orders of magnitude (20-2,000x).