# Improved analysis for a proximal algorithm for sampling

arXiv: https://arxiv.org/abs/2202.06386 · HTML: https://mlsanity.com/paper/2202.06386/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 13 Feb 2022.

## Builds on (most cited first, 18 of 18)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [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)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (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)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020)
- [Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry](https://mlsanity.com/paper/1911.01469/) (2019)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [Efficient MCMC Sampling with Dimension-Free Convergence Rate using ADMM-type Splitting](https://mlsanity.com/paper/1905.11937/) (2019)
- [Auxiliary gradient-based sampling algorithms](https://mlsanity.com/paper/1610.09641/) (2016)
- [The query complexity of sampling from strongly log-concave distributions in one dimension](https://mlsanity.com/paper/2105.14163/) (2021)
- [Langevin Monte Carlo: random coordinate descent and variance reduction](https://mlsanity.com/paper/2007.14209/) (2020)

## Built on by (most cited first, 30 of 73)

- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences](https://mlsanity.com/paper/2503.11043/) (2025)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Diffusive Gibbs Sampling](https://mlsanity.com/paper/2402.03008/) (2024)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion](https://mlsanity.com/paper/2402.17886/) (2024)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Models](https://mlsanity.com/paper/2404.18444/) (2024)
- [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026)
- [Improved Sample Complexity Bounds for Diffusion Model Training](https://mlsanity.com/paper/2311.13745/) (2023)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)
- [Private (Stochastic) Non-Convex Optimization Revisited: Second-Order Stationary Points and Excess Risks](https://mlsanity.com/paper/2302.09699/) (2023)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions](https://mlsanity.com/paper/2508.16306/) (2025)
- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Faster logconcave sampling from a cold start in high dimension](https://mlsanity.com/paper/2505.01937/) (2025)

## Abstract (arXiv)

We study the proximal sampler of Lee, Shen, and Tian (2021) and obtain new convergence guarantees under weaker assumptions than strong log-concavity: namely, our results hold for (1) weakly log-concave targets, and (2) targets satisfying isoperimetric assumptions which allow for non-log-concavity. We demonstrate our results by obtaining new state-of-the-art sampling guarantees for several classes of target distributions. We also strengthen the connection between the proximal sampler and the proximal method in optimization by interpreting the proximal sampler as an entropically regularized Wasserstein proximal method, and the proximal point method as the limit of the proximal sampler with vanishing noise.