Functional Central Limit Theorem and Strong Law of Large Numbers for Stochastic Gradient Langevin Dynamics

TL;DR

This paper establishes a strong law of large numbers and a functional central limit theorem for stochastic gradient Langevin dynamics (SGLD) with fixed step size, even when data are dependent, advancing theoretical understanding of SGLD's long-term behavior.

Contribution

It provides the first rigorous analysis of SGLD's asymptotic properties under dependent data streams, modeling it as a Markov chain in a random environment.

Findings

Proves a strong law of large numbers for SGLD.

Establishes a functional central limit theorem for SGLD.

Handles dependent data streams in the analysis.

Abstract

We study the mixing properties of an important optimization algorithm of machine learning: the stochastic gradient Langevin dynamics (SGLD) with a fixed step size. The data stream is not assumed to be independent hence the SGLD is not a Markov chain, merely a \emph{Markov chain in a random environment}, which complicates the mathematical treatment considerably. We derive a strong law of large numbers and a functional central limit theorem for SGLD.