Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case

TL;DR

This paper introduces two new algorithms for sampling from log-concave distributions using Hamiltonian dynamics without requiring the gradient to be globally Lipschitz, providing theoretical guarantees and applications to optimization.

Contribution

It presents novel tamed Euler schemes for Hamiltonian-based sampling without gradient Lipschitz assumptions, with non-asymptotic convergence bounds and optimization error analysis.

Findings

Non-asymptotic 2-Wasserstein bounds established

Algorithms work without global Lipschitz gradient assumption

Applications to excess risk optimization error

Abstract

In this article we consider sampling from log concave distributions in Hamiltonian setting, without assuming that the objective gradient is globally Lipschitz. We propose two algorithms based on monotone polygonal (tamed) Euler schemes, to sample from a target measure, and provide non-asymptotic 2-Wasserstein distance bounds between the law of the process of each algorithm and the target measure. Finally, we apply these results to bound the excess risk optimization error of the associated optimization problem.