# Constrained Sampling with Primal-Dual Langevin Monte Carlo arXiv: https://arxiv.org/abs/2411.00568 ยท HTML: https://mlsanity.com/paper/2411.00568/ ## Key facts - Cited by 10 papers in the map (data through 14 Oct 2026). - Builds on 31 of its 76 references that are in the map. - First posted on arXiv 1 Nov 2024. ## Builds on (most cited first, 30 of 31) - [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016) - [Preventing Fairness Gerrymandering: Auditing and Learning for Subgroup Fairness](https://mlsanity.com/paper/1711.05144/) (2017) - [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019) - [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (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) - [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020) - [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017) - [Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals](https://mlsanity.com/paper/1809.04198/) (2018) - [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018) - [Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2002.00057/) (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) - [On the Convergence and Robustness of Training GANs with Regularized Optimal Transport](https://mlsanity.com/paper/1802.08249/) (2018) - [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020) - [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021) - [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019) - [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020) - [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020) - [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022) - [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) - [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019) - [Nest Your Adaptive Algorithm for Parameter-Agnostic Nonconvex Minimax Optimization](https://mlsanity.com/paper/2206.00743/) (2022) - [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023) - [Asymptotic convergence of constrained primal-dual dynamics](https://mlsanity.com/paper/1510.02150/) (2015) - [Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo](https://mlsanity.com/paper/2210.11925/) (2022) ## Built on by (most cited first, 10 of 10) - [Constrained Diffusers for Safe Planning and Control](https://mlsanity.com/paper/2506.12544/) (2025) - [Projected Coupled Diffusion for Test-Time Constrained Joint Generation](https://mlsanity.com/paper/2508.10531/) (2025) - [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025) - [Constrained Flow Optimization via Sequential Fine Tuning for Molecular Design](https://mlsanity.com/paper/2605.30610/) (2026) - [Unlearning in Diffusion Models: A Unified Framework with KL Divergence and Likelihood Constraints](https://mlsanity.com/paper/2605.30825/) (2026) - [DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising](https://mlsanity.com/paper/2606.15359/) (2026) - [Constrained Diffusion Models with Primal-Dual Inference](https://mlsanity.com/paper/2606.17192/) (2026) - [Sampling with Shielded Langevin Monte Carlo Using Navigation Potentials](https://mlsanity.com/paper/2512.22153/) (2025) - [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026) - [Generative Modeling by Value-Driven Transport](https://mlsanity.com/paper/2605.22507/) (2026) ## Abstract (arXiv) This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications.