Stochastic Gradient Descent as Approximate Bayesian Inference

TL;DR

This paper explores how constant stochastic gradient descent can be used as an approximate Bayesian inference method, introduces new algorithms, and analyzes their theoretical properties and practical performance.

Contribution

It presents a novel perspective on using constant SGD for Bayesian inference, introduces a variational EM algorithm, and proposes a scalable approximate MCMC method.

Findings

Constant SGD can approximate Bayesian posteriors by tuning parameters.

A new variational EM algorithm for hyperparameter optimization.

A scalable approximate MCMC algorithm called Averaged Stochastic Gradient Sampler.

Abstract

Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results. (1) We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specifically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distributions. (2) We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models. (3) We also propose SGD with momentum for sampling and show how to adjust the damping coefficient accordingly. (4) We analyze MCMC algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally (5), we use the stochastic process perspective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler.