# Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry

arXiv: https://arxiv.org/abs/1911.01469 · HTML: https://mlsanity.com/paper/1911.01469/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2019.

## Built on by (most cited first, 30 of 34)

- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods](https://mlsanity.com/paper/2305.15027/) (2023)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [The query complexity of sampling from strongly log-concave distributions in one dimension](https://mlsanity.com/paper/2105.14163/) (2021)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [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)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Parallel Simulation for Log-concave Sampling and Score-based Diffusion Models](https://mlsanity.com/paper/2412.07435/) (2024)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions](https://mlsanity.com/paper/2306.10506/) (2023)
- [On the query complexity of sampling from non-log-concave distributions](https://mlsanity.com/paper/2502.06200/) (2025)
- [Federated Learning with a Sampling Algorithm under Isoperimetry](https://mlsanity.com/paper/2206.00920/) (2022)
- [ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compression](https://mlsanity.com/paper/2303.04622/) (2023)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)
- [Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regression](https://mlsanity.com/paper/2412.07999/) (2024)
- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Beyond Scores: Proximal Diffusion Models](https://mlsanity.com/paper/2507.08956/) (2025)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [A Proximal Algorithm for Sampling from Non-convex Potentials](https://mlsanity.com/paper/2205.10188/) (2022)
- [A Dynamical System View of Langevin-Based Non-Convex Sampling](https://mlsanity.com/paper/2210.13867/) (2022)
- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)
- [An Improved Analysis of Langevin Algorithms with Prior Diffusion for Non-Log-Concave Sampling](https://mlsanity.com/paper/2403.06183/) (2024)
- [Mixing Times and Privacy Analysis for the Projected Langevin Algorithm under a Modulus of Continuity](https://mlsanity.com/paper/2501.04134/) (2025)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)

## Abstract (arXiv)

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution nu = e-f on ℝn under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when nu satisfies log-Sobolev inequality (LSI) and f has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in R'enyi divergence of order q > 1 when the biased limit satisfies either LSI or Poincar'e inequality.