GO Hessian for Expectation-Based Objectives

TL;DR

This paper introduces GO Hessian, an unbiased low-variance estimator for second-order derivatives of expectation-based objectives, facilitating efficient curvature-based optimization in stochastic computation graphs.

Contribution

The paper extends the GO gradient to a Hessian estimator, enabling practical second-order optimization for expectation objectives involving complex stochastic nodes.

Findings

GO Hessian is easy to implement with auto-differentiation.

It provides efficient curvature information for non-reparameterizable distributions.

Experimental results show improved optimization performance.

Abstract

An unbiased low-variance gradient estimator, termed GO gradient, was proposed recently for expectation-based objectives $\mathbb{E}_{q_{\boldsymbol{\gamma}}(\boldsymbol{y})} [f(\boldsymbol{y})]$, where the random variable (RV) $\boldsymbol{y}$ may be drawn from a stochastic computation graph with continuous (non-reparameterizable) internal nodes and continuous/discrete leaves. Upgrading the GO gradient, we present for $\mathbb{E}_{q_{\boldsymbol{\boldsymbol{\gamma}}}(\boldsymbol{y})} [f(\boldsymbol{y})]$ an unbiased low-variance Hessian estimator, named GO Hessian. Considering practical implementation, we reveal that GO Hessian is easy-to-use with auto-differentiation and Hessian-vector products, enabling efficient cheap exploitation of curvature information over stochastic computation graphs. As representative examples, we present the GO Hessian for non-reparameterizable gamma and negative binomial RVs/nodes. Based on the GO Hessian, we design a new second-order method for $\mathbb{E}_{q_{\boldsymbol{\boldsymbol{\gamma}}}(\boldsymbol{y})} [f(\boldsymbol{y})]$, with rigorous experiments conducted to verify its effectiveness and efficiency.