# Wasserstein Control of Mirror Langevin Monte Carlo

arXiv: https://arxiv.org/abs/2002.04363 · HTML: https://mlsanity.com/paper/2002.04363/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2020.

## Built on by (most cited first, 30 of 55)

- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Sampling with Mollified Interaction Energy Descent](https://mlsanity.com/paper/2210.13400/) (2022)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [Private Convex Optimization in General Norms](https://mlsanity.com/paper/2207.08347/) (2022)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [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)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Non-Reversible Langevin Algorithms for Constrained Sampling](https://mlsanity.com/paper/2501.11743/) (2025)
- [Constrained Langevin Algorithms with L-mixing External Random Variables](https://mlsanity.com/paper/2205.14192/) (2022)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [Sensing Cox Processes via Posterior Sampling and Positive Bases](https://mlsanity.com/paper/2110.11181/) (2021)
- [A Note on the Convergence of Mirrored Stein Variational Gradient Descent under (L0,L1)-Smoothness Condition](https://mlsanity.com/paper/2206.09709/) (2022)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems](https://mlsanity.com/paper/2203.10215/) (2022)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)

## Abstract (arXiv)

Discretized Langevin diffusions are efficient Monte Carlo methods for sampling from high dimensional target densities that are log-Lipschitz-smooth and (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlo sampling algorithm has received much attention lately, leading to a detailed understanding of its non-asymptotic convergence properties and of the role that smoothness and log-concavity play in the convergence rate. Distributions that do not possess these regularity properties can be addressed by considering a Riemannian Langevin diffusion with a metric capturing the local geometry of the log-density. However, the Monte Carlo algorithms derived from discretizations of such Riemannian Langevin diffusions are notoriously difficult to analyze. In this paper, we consider Langevin diffusions on a Hessian-type manifold and study a discretization that is closely related to the mirror-descent scheme. We establish for the first time a non-asymptotic upper-bound on the sampling error of the resulting Hessian Riemannian Langevin Monte Carlo algorithm. This bound is measured according to a Wasserstein distance induced by a Riemannian metric ground cost capturing the Hessian structure and closely related to a self-concordance-like condition. The upper-bound implies, for instance, that the iterates contract toward a Wasserstein ball around the target density whose radius is made explicit. Our theory recovers existing Euclidean results and can cope with a wide variety of Hessian metrics related to highly non-flat geometries.