Differentiable Sampling of Categorical Distributions Using the CatLog-Derivative Trick

TL;DR

This paper introduces the CatLog-Derivative trick, a new method for differentiating categorical distributions, and proposes IndeCateR, an unbiased gradient estimator with lower variance, improving learning in models with categorical latent variables.

Contribution

The paper presents the CatLog-Derivative trick tailored for categorical distributions and introduces IndeCateR, a novel unbiased gradient estimator with lower variance than REINFORCE.

Findings

IndeCateR achieves lower variance than REINFORCE.

Gradient estimates with IndeCateR have reduced bias and variance.

Efficient implementation of IndeCateR demonstrates practical advantages.

Abstract

Categorical random variables can faithfully represent the discrete and uncertain aspects of data as part of a discrete latent variable model. Learning in such models necessitates taking gradients with respect to the parameters of the categorical probability distributions, which is often intractable due to their combinatorial nature. A popular technique to estimate these otherwise intractable gradients is the Log-Derivative trick. This trick forms the basis of the well-known REINFORCE gradient estimator and its many extensions. While the Log-Derivative trick allows us to differentiate through samples drawn from categorical distributions, it does not take into account the discrete nature of the distribution itself. Our first contribution addresses this shortcoming by introducing the CatLog-Derivative trick - a variation of the Log-Derivative trick tailored towards categorical distributions. Secondly, we use the CatLog-Derivative trick to introduce IndeCateR, a novel and unbiased gradient estimator for the important case of products of independent categorical distributions with provably lower variance than REINFORCE. Thirdly, we empirically show that IndeCateR can be efficiently implemented and that its gradient estimates have significantly lower bias and variance for the same number of samples compared to the state of the art.