# Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning

arXiv: https://arxiv.org/abs/2012.12137 · HTML: https://mlsanity.com/paper/2012.12137/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 30 of 33)

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- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Non-Reversible Langevin Algorithms for Constrained Sampling](https://mlsanity.com/paper/2501.11743/) (2025)
- [Constrained Langevin Algorithms with L-mixing External Random Variables](https://mlsanity.com/paper/2205.14192/) (2022)
- [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)
- [Constructing Semantics-Aware Adversarial Examples with a Probabilistic Perspective](https://mlsanity.com/paper/2306.00353/) (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)
- [Optimal Scaling for the Proximal Langevin Algorithm in High Dimensions](https://mlsanity.com/paper/2204.10793/) (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)
- [Concept-based Adversarial Attack: a Probabilistic Perspective](https://mlsanity.com/paper/2507.02965/) (2025)
- [Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds](https://mlsanity.com/paper/2602.17773/) (2026)
- [Decentralized Proximal Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2605.00723/) (2026)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Subsampling Error in Stochastic Gradient Langevin Diffusions](https://mlsanity.com/paper/2305.13882/) (2023)
- [How to beat a Bayesian adversary](https://mlsanity.com/paper/2407.08678/) (2024)
- [Measurement-Consistent Langevin Corrector for Stabilizing Latent Diffusion Inverse Problem Solvers](https://mlsanity.com/paper/2601.04791/) (2026)
- [Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control](https://mlsanity.com/paper/2303.16251/) (2023)
- [Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management](https://mlsanity.com/paper/2502.19210/) (2025)
- [Quantitative Convergence Analysis of Projected Stochastic Gradient Descent for Non-Convex Losses via the Goldstein Subdifferential](https://mlsanity.com/paper/2510.02735/) (2025)
- [Mean Estimation from Coarse Data: Characterizations and Efficient Algorithms](https://mlsanity.com/paper/2602.23341/) (2026)
- [Conditional Diffusion Under Linear Constraints: Langevin Mixing and Information-Theoretic Guarantees](https://mlsanity.com/paper/2605.05387/) (2026)
- [Unlearning with Asymmetric Sources: Improved Unlearning-Utility Trade-off with Public Data](https://mlsanity.com/paper/2605.11170/) (2026)

## Abstract (arXiv)

Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of O(T-1/4(log T)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves ε-suboptimal solutions, on average, provided that it is run for a time that is polynomial in ε-1 and slightly super-exponential in the problem dimension.