A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetry

Abstract

We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for sampling a target density on $\mathbb{R}^d$. We assume that the target density satisfies $ψ_μ$-isoperimetry and that the operator norm and trace of its Hessian are bounded by $L$ and $Υ$ respectively. Our main result establishes that, from a warm start, to achieve $ε$-total variation distance to the target density, MALA mixes in $O\left(\frac{(LΥ)^{\frac12}}{ψ_μ^2} \log\left(\frac{1}ε\right)\right)$ iterations. Notably, this result holds beyond the log-concave sampling setting and the mixing time depends on only $Υ$ rather than its upper bound $L d$. In the $m$-strongly logconcave and $L$-log-smooth sampling setting, our bound recovers the previous minimax mixing bound of MALA~\cite{wu2021minimax}.