On Convergence of the Alternating Directions SGHMC Algorithm

TL;DR

This paper analyzes the convergence rates of an extended Hamiltonian Monte Carlo algorithm that incorporates alternating directions and auxiliary distributions, providing detailed error analysis and dependence on key parameters.

Contribution

It introduces a novel alternating directions procedure for SGHMC, extending standard HMC with a comprehensive convergence analysis under mild stochastic gradient conditions.

Findings

Convergence rates depend explicitly on problem dimension and distribution properties.

The leapfrog integrator error is thoroughly characterized for general energy functions.

The method accommodates general auxiliary distributions, broadening HMC applicability.

Abstract

We study convergence rates of Hamiltonian Monte Carlo (HMC) algorithms with leapfrog integration under mild conditions on stochastic gradient oracle for the target distribution (SGHMC). Our method extends standard HMC by allowing the use of general auxiliary distributions, which is achieved by a novel procedure of Alternating Directions. The convergence analysis is based on the investigations of the Dirichlet forms associated with the underlying Markov chain driving the algorithms. For this purpose, we provide a detailed analysis on the error of the leapfrog integrator for Hamiltonian motions with both the kinetic and potential energy functions in general form. We characterize the explicit dependence of the convergence rates on key parameters such as the problem dimension, functional properties of both the target and auxiliary distributions, and the quality of the oracle.