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

We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for sampling a target density on ℝd. We assume that the target density satisfies psiμ-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 ε-total variation distance to the target density, MALA mixes in Oleft(frac(LUpsilon)frac12psiμ2 logleft(frac1εright)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~citewu2021minimax.