On Convergence of the Alternating Directions Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) Algorithms

TMLR Paper6094 Authors

Abstract:

We study convergence rates of practical Hamiltonian Monte Carlo (HMC) style algorithms where the Hamiltonian motion is approximated with leapfrog integration and where gradients of the log target density are accessed via a stochastic gradient (SG) oracle.
Importantly, our analysis extends to allowing the use of general auxiliary distributions via a novel HMC procedure of alternating directions (AD).

The convergence analysis is based on the investigation 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 when both the kinetic and potential energy functions are in general form. We characterize the explicit dependence of the convergence rates on key parameters such as the problem dimension, functional properties of the target and auxiliary distributions and the quality of the SG oracle. Our analysis also identifies a crucial derivative condition on the log density of the auxiliary distribution, and we show that Gaussians (auxiliaries for standard HMC) as well as common choices of general auxiliaries for ADHMC satisfy this condition.

Submission Type: Regular submission (no more than 12 pages of main content)

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Assigned Action Editor: ~Yingzhen_Li1

Submission Number: 6094