Fast Non-Log-Concave Sampling under Nonconvex Equality and Inequality Constraints with Landing

2025-10-24

AI Summary

Sampling from non-log-concave distributions subject to non-convex equality and inequality constraints remains challenging: existing methods rely on costly projections, fail to handle both constraint types uniformly, and lack rigorous convergence guarantees. This paper proposes the Overdamped Langevin with LAnding (OLLA) framework, which dynamically corrects trajectories along the normal direction of the constraint manifold to implicitly satisfy constraints—eliminating explicit projections entirely. OLLA is the first method to unify treatment of non-convex equality and inequality constraints without projection. We establish exponential convergence in Wasserstein-2 distance under mild regularity conditions, applicable even to non-log-concave target densities. Experiments demonstrate that OLLA significantly outperforms both projection-based and slack-variable-based approaches in computational efficiency and mixing performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers the cases where the constrained distribution is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods for nonconvex constraint set $Σsubset mathbb{R}^d$ defined by equality or inequality constraints commonly rely on costly projection steps. Moreover, they cannot handle equality and inequality constraints simultaneously as each method only specialized in one case. In addition, rigorous and quantitative convergence guarantee is often lacking. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and $Σ$, OLLA converges exponentially fast in $W_2$ distance to the constrained target density $ρ_Σ(x) propto exp(-f(x))dσ_Σ$. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to projection-based constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and reasonable empirical mixing.

Problem

• Sampling from nonconvex constrained distributions efficiently
• Handling both equality and inequality constraints simultaneously
• Avoiding costly projection steps in constrained sampling

Innovation

• OLLA framework handles both equality and inequality constraints
• Eliminates projection steps via deterministic trajectory correction
• Provides exponential convergence guarantees in W2 distance