# Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm

arXiv: https://arxiv.org/abs/2006.09270 · HTML: https://mlsanity.com/paper/2006.09270/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

## Built on by (most cited first, 30 of 35)

- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
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- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Query lower bounds for log-concave sampling](https://mlsanity.com/paper/2304.02599/) (2023)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (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)
- [Towards a Better Theoretical Understanding of Independent Subnetwork Training](https://mlsanity.com/paper/2306.16484/) (2023)
- [Proximal Oracles for Optimization and Sampling](https://mlsanity.com/paper/2404.02239/) (2024)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers](https://mlsanity.com/paper/2512.07150/) (2025)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis](https://mlsanity.com/paper/2503.19066/) (2025)
- [Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems](https://mlsanity.com/paper/2203.10215/) (2022)
- [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)
- [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024)
- [Accelerating Constrained Sampling: A Large Deviations Approach](https://mlsanity.com/paper/2506.07816/) (2025)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Decentralized Proximal Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2605.00723/) (2026)
- [Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2608.13467/) (2026)
- [Covid19 Reproduction Number: Credibility Intervals by Blockwise Proximal Monte Carlo Samplers](https://mlsanity.com/paper/2203.09142/) (2022)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [A proximal gradient algorithm for composite log-concave sampling](https://mlsanity.com/paper/2605.12461/) (2026)
- [Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2609.12594/) (2026)

## Abstract (arXiv)

We consider the task of sampling with respect to a log concave probability distribution. The potential of the target distribution is assumed to be composite, textiti.e., written as the sum of a smooth convex term, and a nonsmooth convex term possibly taking infinite values. The target distribution can be seen as a minimizer of the Kullback-Leibler divergence defined on the Wasserstein space (textiti.e., the space of probability measures). In the first part of this paper, we establish a strong duality result for this minimization problem. In the second part of this paper, we use the duality gap arising from the first part to study the complexity of the Proximal Stochastic Gradient Langevin Algorithm (PSGLA), which can be seen as a generalization of the Projected Langevin Algorithm. Our approach relies on viewing PSGLA as a primal dual algorithm and covers many cases where the target distribution is not fully supported. In particular, we show that if the potential is strongly convex, the complexity of PSGLA is O(1/varepsilon2) in terms of the 2-Wasserstein distance. In contrast, the complexity of the Projected Langevin Algorithm is O(1/varepsilon12) in terms of total variation when the potential is convex.