# Penalized Overdamped and Underdamped Langevin Monte Carlo Algorithms for Constrained Sampling

arXiv: https://arxiv.org/abs/2212.00570 · HTML: https://mlsanity.com/paper/2212.00570/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2022.

## Built on by (most cited first, 13 of 13)

- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Non-Reversible Langevin Algorithms for Constrained Sampling](https://mlsanity.com/paper/2501.11743/) (2025)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis](https://mlsanity.com/paper/2503.19066/) (2025)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [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)
- [Decentralized Proximal Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2605.00723/) (2026)
- [Alternating Diffusion for Proximal Sampling with Zeroth Order Queries](https://mlsanity.com/paper/2603.19633/) (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)
- [Exact Schur-Sylvester Dimensionality Reductions for Non-Smooth Stochastic Complexity and Manifold Sampling](https://mlsanity.com/paper/2606.23867/) (2026)

## Abstract (arXiv)

We consider the constrained sampling problem where the goal is to sample from a target distribution π(x)propto e-f(x) when x is constrained to lie on a convex body C. Motivated by penalty methods from continuous optimization, we propose penalized Langevin Dynamics (PLD) and penalized underdamped Langevin Monte Carlo (PULMC) methods that convert the constrained sampling problem into an unconstrained sampling problem by introducing a penalty function for constraint violations. When f is smooth and gradients are available, we get tildeO(d/varepsilon10) iteration complexity for PLD to sample the target up to an varepsilon-error where the error is measured in the TV distance and tildeO(·) hides logarithmic factors. For PULMC, we improve the result to tildeO(√d/varepsilon7) when the Hessian of f is Lipschitz and the boundary of C is sufficiently smooth. To our knowledge, these are the first convergence results for underdamped Langevin Monte Carlo methods in the constrained sampling that handle non-convex f and provide guarantees with the best dimension dependency among existing methods with deterministic gradient. If unbiased stochastic estimates of the gradient of f are available, we propose PSGLD and PSGULMC methods that can handle stochastic gradients and are scaleable to large datasets without requiring Metropolis-Hasting correction steps. For PSGLD and PSGULMC, when f is strongly convex and smooth, we obtain tildeO(d/varepsilon18) and tildeO(d√d/varepsilon39) iteration complexity in W2 distance. When f is smooth and can be non-convex, we provide finite-time performance bounds and iteration complexity results. Finally, we illustrate the performance on Bayesian LASSO regression and Bayesian constrained deep learning problems.