Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case
This work addresses sampling challenges in optimization for machine learning and statistics, offering theoretical guarantees under relaxed assumptions, but it is incremental as it builds on existing Hamiltonian and tamed Euler methods.
The paper tackles sampling from log-concave distributions in Hamiltonian settings without requiring globally Lipschitz gradients, proposing two algorithms based on monotone polygonal Euler schemes and providing non-asymptotic 2-Wasserstein distance bounds for sampling and excess risk optimization error bounds.
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.