No Free Lunch for Stochastic Gradient Langevin Dynamics

TL;DR

This paper demonstrates that stochastic gradient Langevin dynamics often fails to improve efficiency with smaller data subsamples, as the accuracy loss offsets the speed gains, challenging its scalability benefits.

Contribution

It provides a theoretical analysis showing the limitations of SGLD with data subsampling, contrasting previous optimistic assumptions.

Findings

Smaller subsamples reduce per-step computation but decrease accuracy.

Accuracy loss offsets speed gains in typical datasets.

SGLD's scalability benefits are limited by this accuracy-speed trade-off.

Abstract

As sample sizes grow, scalability has become a central concern in the development of Markov chain Monte Carlo (MCMC) methods. One general approach to this problem, exemplified by the popular stochastic gradient Langevin dynamics (SGLD) algorithm, is to use a small random subsample of the data at every time step. This paper, building on recent work such as \cite{nagapetyan2017true,JohndrowJamesE2020NFLf}, shows that this approach often fails: while decreasing the sample size increases the speed of each MCMC step, for typical datasets this is balanced by a matching decrease in accuracy. This result complements recent work such as \cite{nagapetyan2017true} (which came to the same conclusion, but analyzed only specific upper bounds on errors rather than actual errors) and \cite{JohndrowJamesE2020NFLf} (which did not analyze nonreversible algorithms and allowed for logarithmic improvements).