A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetry
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We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for sampling a target density on 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((LΥ)^1/2/ψ_μ^2log(1/ϵ)) 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 <cit.>.
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