HMC and underdamped Langevin united in the unadjusted convex smooth case

TL;DR

This paper analyzes unadjusted generalized HMC samplers, demonstrating improved convergence rates through parameter optimization and extending results to stochastic gradient versions with dimension-free convergence for log-concave targets.

Contribution

It provides a detailed analysis of parameter optimization for unadjusted HMC and Langevin samplers, and introduces a unified framework for stochastic gradient HMC with dimension-free convergence.

Findings

Improved convergence rate from 1/κ to 1/√κ using partial velocity refreshment.

Empirical validation of similar effects in Metropolis-adjusted gHMC and kinetic PDMPs.

Dimension-free convergence rates established for stochastic gradient HMC on log-concave targets.

Abstract

We consider a family of unadjusted generalized HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case, which shows an improvement from $1/\kappa$ to $1/\sqrt{\kappa}$ for the convergence rate in terms of the condition number $\kappa$ by using partial velocity refreshment, with respect to classical full refreshments. A similar effect is observed empirically for two related algorithms, namely Metropolis-adjusted gHMC and kinetic piecewise-deterministic Markov processes. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets over a large range of parameters, gathering in a unified framework previous results on position HMC and underdamped Langevin and extending them to HMC with inertia.