The True Cost of Stochastic Gradient Langevin Dynamics

TL;DR

This paper analyzes the bias and computational cost of Stochastic Gradient Langevin Dynamics (SGLD) in Bayesian inference, showing how stepsize choices affect accuracy and proposing methods to reduce costs while maintaining credible interval coverage.

Contribution

It provides a theoretical analysis of SGLD bias and cost, demonstrating the impact of stepsize and batchsize, and introduces a control variate approach to improve efficiency.

Findings

Bias in SGLD depends on stepsize and batchsize.

Cost to achieve target accuracy is similar across batchsizes without control variates.

Control variates significantly reduce computational cost.

Abstract

The problem of posterior inference is central to Bayesian statistics and a wealth of Markov Chain Monte Carlo (MCMC) methods have been proposed to obtain asymptotically correct samples from the posterior. As datasets in applications grow larger and larger, scalability has emerged as a central problem for MCMC methods. Stochastic Gradient Langevin Dynamics (SGLD) and related stochastic gradient Markov Chain Monte Carlo methods offer scalability by using stochastic gradients in each step of the simulated dynamics. While these methods are asymptotically unbiased if the stepsizes are reduced in an appropriate fashion, in practice constant stepsizes are used. This introduces a bias that is often ignored. In this paper we study the mean squared error of Lipschitz functionals in strongly log- concave models with i.i.d. data of growing data set size and show that, given a batchsize, to control the bias of SGLD the stepsize has to be chosen so small that the computational cost of reaching a target accuracy is roughly the same for all batchsizes. Using a control variate approach, the cost can be reduced dramatically. The analysis is performed by considering the algorithms as noisy discretisations of the Langevin SDE which correspond to the Euler method if the full data set is used. An important observation is that the 1scale of the step size is determined by the stability criterion if the accuracy is required for consistent credible intervals. Experimental results confirm our theoretical findings.