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arXiv · 2006.08873

GO Hessian for Expectation-Based Objectives

Yulai Cong·Miaoyun Zhao·Jianqiao Li·Junya Chen·Lawrence Carin
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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.

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Yulai Cong, Miaoyun Zhao, Jianqiao Li, Junya Chen, Lawrence Carin. 2020-06-16. GO Hessian for Expectation-Based Objectives. https://arxiv.org/abs/2006.08873

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