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 $\psi_\mu$-isoperimetry and that the operator norm and trace of its Hessian are bounded by $L$ and $\Upsilon$ respectively. Our main result establishes that, from a warm start, to achieve $\epsilon$-total variation distance 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 result holds beyond the log-concave sampling setting and the mixing time depends on only $\Upsilon$ 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}.

翻译：我们研究了用于在$\mathbb{R}^d$上采样目标密度的Metropolis-Adjusted Langevin算法（MALA）的混合时间。假设目标密度满足$\psi_\mu$-等周性，且其Hessian矩阵的算子范数和迹分别以$L$和$\Upsilon$为上界。我们主要的结果表明，从热启动出发，要达到与目标密度的$\epsilon$-总变差距离，MALA需进行$O\left(\frac{(L\Upsilon)^{\frac12}}{\psi_\mu^2} \log\left(\frac{1}{\epsilon}\right)\right)$次迭代混合。值得注意的是，这一结果不仅适用于对数凹采样设置，而且混合时间仅依赖于$\Upsilon$而非其上界$L d$。在$m$-强对数凹和$L$-对数光滑采样设置下，我们的界恢复了MALA~\cite{wu2021minimax}的先前极小化最优混合界。