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

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arxiv 2304.04095 v2 pith:A4IVBCN3 submitted 2023-04-08 stat.ML cs.CCcs.LGstat.CO

Yuansi Chen , Khashayar Gatmiry This is my paper · ORCID
classification stat.MLcs.CCcs.LGstat.CO
keywords mixingbounddensitymalasamplingtargetupsilonalgorithm
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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 $\psi_\mu$-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}.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Beyond the $d^{2.5}$-mixing bound for Dikin walks on polytopes

    cs.DS 2026-07 conditional novelty 7.0 of 10

    The Dikin walk with a scaled Lee-Sidford metric provably mixes on a polytope in O~(d^2.25) iterations from a warm start, improving the decade-old d^2.5 bound and taking a step toward the conjectured d^2.

  2. Fast Score-Based Sampling via Log-Concave Reductions

    math.ST 2025-12 conditional novelty 7.0 of 10

    Score-based sampling reduces to a short sequence of strongly log-concave sampling problems, giving √d polylog(1/ε) complexity bounds and logarithmic dependence on the condition number for log-concave targets.

  3. Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

    cs.IT 2025-02 accept novelty 7.0 of 10

    The Proximal Sampler has exponential convergence in relative Fisher information for strongly log-concave targets, matching the rate of continuous-time Langevin dynamics.

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