Variance reduction properties of the reparameterization trick

TL;DR

This paper analyzes why the reparameterization trick reduces variance in gradient estimates within variational inference, under idealized assumptions, and demonstrates its effectiveness through real-world examples.

Contribution

It provides a theoretical explanation for the variance reduction of the reparameterization trick under specific assumptions and validates the findings with practical examples.

Findings

Reparameterization gradient estimators have smaller variance than score function estimators.

Theoretical analysis under Gaussian and quadratic assumptions explains variance reduction.

Empirical results support the theoretical insights in real-world scenarios.

Abstract

The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.