Log-concave sampling: Metropolis-Hastings algorithms are fast!
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We consider the problem of sampling from a strongly log-concave density in R^d, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by running a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step to ensure the correct stationary distribution. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most δ for a density with condition number κ, we show that MALA requires O(κ d (1/δ) ) steps, as compared to the O(κ^2 d/δ^2 ) steps established in past work on ULA. We also demonstrate the gains of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for a zeroth-order method Metropolized random walk (MRW) and show that it mixes O(κ d) slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the potential gains of Metropolis-Hastings adjustment for Langevin-type algorithms.
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