Stochastic Variance-Reduced Hamilton Monte Carlo Methods

TL;DR

This paper introduces a stochastic Hamilton Monte Carlo method with variance reduction that efficiently samples from smooth, strongly log-concave distributions, outperforming existing methods in gradient complexity and demonstrating superior empirical results.

Contribution

The paper develops a novel stochastic HMC algorithm with variance reduction, achieving improved theoretical gradient complexity bounds and extending to general log-concave distributions.

Findings

Outperforms state-of-the-art HMC methods in gradient complexity.

Achieves $ ilde O(n+ ext{terms depending on } ppa, d, ext{and } \u03b5)$ complexity.

Demonstrates superior empirical performance on synthetic and real data.

Abstract

We propose a fast stochastic Hamilton Monte Carlo (HMC) method, for sampling from a smooth and strongly log-concave distribution. At the core of our proposed method is a variance reduction technique inspired by the recent advance in stochastic optimization. We show that, to achieve $\epsilon$ accuracy in 2-Wasserstein distance, our algorithm achieves $\tilde O(n+\kappa^{2}d^{1/2}/\epsilon+\kappa^{4/3}d^{1/3}n^{2/3}/\epsilon^{2/3})$ gradient complexity (i.e., number of component gradient evaluations), which outperforms the state-of-the-art HMC and stochastic gradient HMC methods in a wide regime. We also extend our algorithm for sampling from smooth and general log-concave distributions, and prove the corresponding gradient complexity as well. Experiments on both synthetic and real data demonstrate the superior performance of our algorithm.