Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev
This work addresses the problem of analyzing Langevin Monte Carlo for non-convex targets in machine learning and statistics, representing a significant but incremental improvement over prior assumptions.
The paper tackled the challenge of providing convergence guarantees for the discrete-time Langevin Monte Carlo algorithm under weaker assumptions than strong log-concavity, by assuming the target distribution satisfies a Latała-Oleszkiewicz or modified log-Sobolev inequality, and achieved results that do not require convexity or dissipativity conditions.
Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $π$ under the sole assumption that $π$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $π$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.