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We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) forsampling a target density on $\mathbb{R}^d$. We assume that the target densitysatisfies $\psi_\mu$-isoperimetry and that the operator norm and trace of itsHessian are bounded by $L$ and $\Upsilon$ respectively. Our main resultestablishes that, from a warm start, to achieve $\epsilon$-total variationdistance to the target density, MALA mixes in$O\left(\frac{(L\Upsilon)^{\frac12}}{\psi_\mu^2}\log\left(\frac{1}{\epsilon}\right)\right)$ iterations. Notably, this resultholds beyond the log-concave sampling setting and the mixing time depends ononly $\Upsilon$ rather than its upper bound $L d$. In the $m$-stronglylogconcave and $L$-log-smooth sampling setting, our bound recovers the previousminimax mixing bound of MALA~\cite{wu2021minimax}.

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