GO Gradient for Expectation-Based Objectives

Yulai Cong, Miaoyun Zhao, Ke Bai, Lawrence Carin

Abstract: Within many machine learning algorithms, a fundamental problem concerns efficient calculation of an unbiased gradient wrt parameters $\boldsymbol{\gamma}$ for expectation-based objectives $\mathbb{E}_{q_{\boldsymbol{\gamma}} (\boldsymbol{y})} [f (\boldsymbol{y}) ]$. Most existing methods either ($i$) suffer from high variance, seeking help from (often) complicated variance-reduction techniques; or ($ii$) they only apply to reparameterizable continuous random variables and employ a reparameterization trick. To address these limitations, we propose a General and One-sample (GO) gradient that ($i$) applies to many distributions associated with non-reparameterizable continuous {\em or} discrete random variables, and ($ii$) has the same low-variance as the reparameterization trick. We find that the GO gradient often works well in practice based on only one Monte Carlo sample (although one can of course use more samples if desired). Alongside the GO gradient, we develop a means of propagating the chain rule through distributions, yielding statistical back-propagation, coupling neural networks to common random variables.

Keywords: generalized reparameterization gradient, variance reduction, non-reparameterizable, discrete random variable, GO gradient, general and one-sample gradient, expectation-based objective, variable nabla, statistical back-propagation, hierarchical, graphical model

TL;DR: a Rep-like gradient for non-reparameterizable continuous/discrete distributions; further generalized to deep probabilistic models, yielding statistical back-propagation

Code: [ YulaiCong/GOgradient](https://github.com/YulaiCong/GOgradient)

Data: [MNIST](https://paperswithcode.com/dataset/mnist)

Community Implementations: [ 4 code implementations](https://www.catalyzex.com/paper/arxiv:1901.06020/code)