A Dynamical System View of Langevin-Based Non-Convex Sampling

Mohammad Reza Karimi Jaghargh, Ya-Ping Hsieh, Andreas Krause

Keywords: Non-Convex Sampling, Langevin Dynamics, Dynamical Systems

TL;DR: We develop a novel framework that guarantees last-iterate convergence in Wasserstein distance for many advanced sampling algorithms at once.

Abstract: Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain some important challenges: Existing guarantees suffer from the drawback of lacking guarantees for the last-iterates, and little is known beyond the elementary schemes of stochastic gradient Langevin dynamics. To address these issues, we develop a novel framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as mirror Langevin, proximal, randomized mid-point, and Runge-Kutta methods.

Submission Number: 6504