Contraction Rate Estimates of Stochastic Gradient Kinetic Langevin Integrators

In previous work, we introduced a method for determining convergence rates for integration methods for the kinetic Langevin equation for M-▽ Lipschitz m -log-concave densities Leimkuhler et al., SIAM J. Numer. Anal . 62 (2024) 1226–1258. In this article, we exploit this method to treat several additional schemes including the method of Brunger, Brooks and Karplus (BBK) and stochastic position/velocity Verlet. We introduce a randomized midpoint scheme for kinetic Langevin dynamics, inspired by the recent scheme of Bou-Rabee and Marsden arXiv:2211.11003, 2022. We also extend our approach to stochastic gradient variants of these schemes under minimal extra assumptions. We provide convergence rates of O ( m/M ), with explicit stepsize restriction, which are of the same order as the stability thresholds for Gaussian targets and are valid for a large interval of the friction parameter. We compare the contraction rate estimates of many kinetic Langevin integrators from molecular dynamics and machine learning. Finally, we present numerical experiments for a Bayesian logistic regression example.