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.