For the task of sampling from a density \\(\u03c0\u221dexp(\u2212V)\\) on \\(R^d\\), where \\(V\\) is possibly non-convex but \\(L\\)-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with \\(\u03b5\\)-relative Fisher information after \\(O(L^2d^2\/\u03b5^2)\\) iterations. This is the sampling analogue of complexity bounds for finding an \\(\u03b5\\)-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 Poincar\u00e9 inequality.<\/p>\n","protected":false},"excerpt":{"rendered":"
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. 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