The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables

TL;DR

This paper introduces the Concrete distribution, a continuous relaxation of discrete variables, enabling low-variance gradient estimation in stochastic computation graphs for improved density estimation and structured prediction.

Contribution

The paper presents the Concrete distribution with closed-form densities and a simple reparameterization, facilitating gradient-based optimization of discrete variables.

Findings

Effective in density estimation tasks

Improves structured prediction performance

Enables low-variance biased gradient estimates

Abstract

The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propagated by the chain rule through the graph are low variance unbiased estimators of the gradients of the expected loss. While many continuous random variables have such reparameterizations, discrete random variables lack useful reparameterizations due to the discontinuous nature of discrete states. In this work we introduce Concrete random variables---continuous relaxations of discrete random variables. The Concrete distribution is a new family of distributions with closed form densities and a simple reparameterization. Whenever a discrete stochastic node of a computation graph can be refactored into a one-hot bit representation that is treated continuously, Concrete stochastic nodes can be used with automatic differentiation to produce low-variance biased gradients of objectives (including objectives that depend on the log-probability of latent stochastic nodes) on the corresponding discrete graph. We demonstrate the effectiveness of Concrete relaxations on density estimation and structured prediction tasks using neural networks.