On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators

TL;DR

This paper develops a theoretical framework for stochastic gradient MCMC algorithms using high-order integrators, demonstrating faster convergence and improved accuracy over traditional methods, supported by experiments on synthetic and real data.

Contribution

It introduces a new theory for high-order integrators in SG-MCMC, showing they achieve faster convergence and more accurate invariant measures than first-order methods.

Findings

Higher-order integrators lead to faster convergence rates.

The 2nd-order symmetric splitting integrator improves mean square error.

Experiments confirm theoretical advantages in large-scale applications.

Abstract

Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the {\em mean square error} (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of $L^{-4/5}$ at $L$ iterations, compared to $L^{-2/3}$ for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications.