Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo - Balasubramanian, Krishnakumar; Chewi, Sinho; Erdogdu, Murat A.; Salim, Adil; Zhang, Matthew (COLT 2022, 2022)
For the task of sampling from a density on , where is possibly non-convex but -gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with -relative Fisher information after iterations. This is the sampling analogue of complexity bounds for finding an -approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincaré inequality.