GO Hessian for Expectation-Based Objectives
This provides a second-order optimization method for stochastic computation graphs, addressing a bottleneck in machine learning for probabilistic models, but it appears incremental as an upgrade to an existing gradient estimator.
The paper tackles the problem of estimating Hessians for expectation-based objectives with non-reparameterizable random variables, presenting an unbiased low-variance estimator called GO Hessian and demonstrating its effectiveness through experiments.
An unbiased low-variance gradient estimator, termed GO gradient, was proposed recently for expectation-based objectives $\mathbb{E}_{q_{\boldsymbolγ}(\boldsymbol{y})} [f(\boldsymbol{y})]$, 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 $\mathbb{E}_{q_{\boldsymbol{\boldsymbolγ}}(\boldsymbol{y})} [f(\boldsymbol{y})]$ 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 $\mathbb{E}_{q_{\boldsymbol{\boldsymbolγ}}(\boldsymbol{y})} [f(\boldsymbol{y})]$, with rigorous experiments conducted to verify its effectiveness and efficiency.