# Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm arXiv: https://arxiv.org/abs/2312.08823 ยท HTML: https://mlsanity.com/paper/2312.08823/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Builds on 35 of its 64 references that are in the map. - First posted on arXiv 14 Dec 2023. ## Builds on (most cited first, 30 of 35) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019) - [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014) - [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015) - [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018) - [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018) - [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017) - [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016) - [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017) - [Gradient-Based Constrained Sampling from Language Models](https://mlsanity.com/paper/2205.12558/) (2022) - [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018) - [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020) - [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020) - [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020) - [Sampling from a log-concave distribution with Projected Langevin Monte Carlo](https://mlsanity.com/paper/1507.02564/) (2015) - [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017) - [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020) - [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021) - [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018) - [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019) - [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017) - [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022) - [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021) - [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022) - [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020) - [Fast MCMC sampling algorithms on polytopes](https://mlsanity.com/paper/1710.08165/) (2017) - [Alternating Mirror Descent for Constrained Min-Max Games](https://mlsanity.com/paper/2206.04160/) (2022) - [Strong Self-Concordance and Sampling](https://mlsanity.com/paper/1911.05656/) (2019) - [Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators](https://mlsanity.com/paper/2210.07219/) (2022) - [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023) ## Built on by (most cited first, 11 of 11) - [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024) - [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024) - [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024) - [R'enyi-infinity constrained sampling with d3 membership queries](https://mlsanity.com/paper/2407.12967/) (2024) - [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025) - [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025) - [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026) - [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024) - [Log-Concave Coupling for Sampling Neural Net Posteriors](https://mlsanity.com/paper/2407.18802/) (2024) - [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026) - [Generating Physically Consistent Molecules with Energy-Based Models](https://mlsanity.com/paper/2605.18381/) (2026) ## Abstract (arXiv) We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the Mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the Mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the Mirror Langevin dynamics including the Mirror Langevin algorithm have an asymptotic bias. For this algorithm, we also give upper bounds for the number of iterations taken to mix to a constrained distribution whose potential is relatively smooth, convex, and Lipschitz continuous with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the inclusion of the Metropolis-Hastings filter, we obtain an exponentially better dependence on the error tolerance for approximate constrained sampling. We also present numerical experiments that corroborate our theoretical findings.