# Log-concave sampling: Metropolis-Hastings algorithms are fast

## TL;DR

This paper proves that Metropolis-Hastings algorithms, specifically MALA, significantly improve sampling efficiency for strongly log-concave densities by providing non-asymptotic mixing time bounds and demonstrating practical advantages over unadjusted Langevin algorithms.

## Contribution

The paper establishes non-asymptotic mixing time bounds for MALA, showing exponential improvements over ULA, and compares its performance with the Metropolized random walk.

## Key findings

- MALA requires O(κd log(1/δ)) steps for desired TV error δ.
- MALA outperforms ULA in strongly and weakly log-concave settings.
- Metropolized random walk mixes slower than MALA by a factor of O(κ).

## Abstract

We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. 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 $\delta$ for a density with condition number $\kappa$, we show that MALA requires $\mathcal{O} \big(\kappa d \log(1/\delta) \big)$ steps, as compared to the $\mathcal{O} \big(\kappa^2 d/\delta^2 \big)$ 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 the Metropolized random walk (MRW) and obtain $\mathcal{O}(\kappa)$ mixing time slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.