Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo

For the task of sampling from a density pi proportional to exp(-V) on R-d, where V is possibly non-convex but L-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with epsilon-relative Fisher information after O(L(2)d(2)/epsilon(2)) iterations. This is the sampling analogue of complexity bounds for finding an epsilon-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-ofthe-art guarantee for sampling from distributions which satisfy a Poincar ' e inequality.