On Convergence of the Alternating Directions SGHMC Algorithm
Soumyadip Ghosh, Yingdong Lu, Tomasz Nowicki
TL;DR
This paper establishes quantitative convergence rates for SGHMC algorithms that employ alternating direction steps and leapfrog integration, under general auxiliary distributions and stochastic gradient oracles. By leveraging Dirichlet forms and a meticulous leapfrog error analysis for general energy functions, it derives explicit geometric convergence bounds that depend on problem dimension, energy function properties, and gradient oracle quality. The results extend the convergence theory beyond Gaussian auxiliaries to general auxiliary forms and show that, with appropriate step size and number of leapfrog steps, the Markov chain exhibits exponential convergence in total variation. The framework also provides practical bounds on leapfrog numerical errors, KL divergences between pushforward measures, and MH acceptance probabilities, enabling robust deployment in high-dimensional, stochastic-gradient settings.
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.