# Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions

arXiv: https://arxiv.org/abs/2106.05480 · HTML: https://mlsanity.com/paper/2106.05480/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions](https://mlsanity.com/paper/1905.02313/) (2019)
- [Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2002.04121/) (2020)
- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds](https://mlsanity.com/paper/1911.03043/) (2019)
- [Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo](https://mlsanity.com/paper/1802.08898/) (2018)
- [Oracle Lower Bounds for Stochastic Gradient Sampling Algorithms](https://mlsanity.com/paper/2002.00291/) (2020)

## Built on by (most cited first, 19 of 19)

- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps](https://mlsanity.com/paper/2209.12771/) (2022)
- [Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants](https://mlsanity.com/paper/2210.06539/) (2022)
- [Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time](https://mlsanity.com/paper/2207.02189/) (2022)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://mlsanity.com/paper/2408.13115/) (2024)
- [On Accelerated Mixing of the No-U-turn Sampler](https://mlsanity.com/paper/2507.13259/) (2025)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)
- [A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets](https://mlsanity.com/paper/2603.18640/) (2026)
- [Learning variational autoencoders via MCMC speed measures](https://mlsanity.com/paper/2308.13731/) (2023)

## Abstract (arXiv)

We give lower bounds on the performance of two of the most popular sampling methods in practice, the Metropolis-adjusted Langevin algorithm (MALA) and multi-step Hamiltonian Monte Carlo (HMC) with a leapfrog integrator, when applied to well-conditioned distributions. Our main result is a nearly-tight lower bound of widetildeOmega(kappa d) on the mixing time of MALA from an exponentially warm start, matching a line of algorithmic results up to logarithmic factors and answering an open question of Chewi et. al. We also show that a polynomial dependence on dimension is necessary for the relaxation time of HMC under any number of leapfrog steps, and bound the gains achievable by changing the step count. Our HMC analysis draws upon a novel connection between leapfrog integration and Chebyshev polynomials, which may be of independent interest.