Stochastic Gradient Hamiltonian Monte Carlo with Variance Reduction for Bayesian Inference

TL;DR

This paper introduces a variance reduction technique for Hamiltonian Monte Carlo, improving convergence and performance in Bayesian inference tasks, especially in large-scale settings.

Contribution

The paper extends variance reduction methods to Hamiltonian Monte Carlo, achieving better theoretical convergence and empirical performance than Langevin dynamics.

Findings

Variance-reduced Hamiltonian Monte Carlo outperforms Langevin dynamics in experiments.

Theoretical convergence results are improved with symmetric splitting schemes.

Experimental results confirm the effectiveness in real-world Bayesian tasks.

Abstract

Gradient-based Monte Carlo sampling algorithms, like Langevin dynamics and Hamiltonian Monte Carlo, are important methods for Bayesian inference. In large-scale settings, full-gradients are not affordable and thus stochastic gradients evaluated on mini-batches are used as a replacement. In order to reduce the high variance of noisy stochastic gradients, Dubey et al. [2016] applied the standard variance reduction technique on stochastic gradient Langevin dynamics and obtained both theoretical and experimental improvements. In this paper, we apply the variance reduction tricks on Hamiltonian Monte Carlo and achieve better theoretical convergence results compared with the variance-reduced Langevin dynamics. Moreover, we apply the symmetric splitting scheme in our variance-reduced Hamiltonian Monte Carlo algorithms to further improve the theoretical results. The experimental results are also consistent with the theoretical results. As our experiment shows, variance-reduced Hamiltonian Monte Carlo demonstrates better performance than variance-reduced Langevin dynamics in Bayesian regression and classification tasks on real-world datasets.