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Convergence of Reflected Langevin Diffusion for Constrained Sampling

Tarika Mane

TL;DR

The paper tackles constrained sampling by studying a reflected Langevin diffusion on a closed convex domain and approximating it with penalized SDEs whose invariant measures converge to the reflected diffusion's invariant law in Wasserstein-2 distance with polynomial rates. It proves contraction properties and existence/uniqueness of invariant measures for both the penalized and reflected processes, establishing a quantitative link between penalized and constrained dynamics via $f_n(x)= g(x) + \frac{n}{2}\mathrm{dist}^2(x,D)$. The discrete-time analysis introduces PCULA, an Euler–Maruyama scheme with a penalized potential, showing that its invariant measure converges to the penalized one at rate $O(h)$ and, in turn, to the constrained invariant measure with a term depending on $n^{-\delta}$. Numerical experiments on a truncated Gaussian within an ellipse illustrate rapid convergence to the constrained target as the penalty grows, and the work outlines extensions to non-convex domains and other penalty schemes.

Abstract

We examine the Langevin diffusion confined to a closed, convex domain $D\subset\mathbb{R}^d$, represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that their invariant measures converge, in Wasserstein-2 distance and with explicit polynomial rate, to the invariant measure of the reflected Langevin diffusion. We also analyze a time-discretization of the penalized process obtained via the Euler-Maruyama scheme and demonstrate the convergence to the original constrained measure. These results provide a rigorous approximation framework for reflected Langevin dynamics in both continuous and discrete time.

Convergence of Reflected Langevin Diffusion for Constrained Sampling

TL;DR

The paper tackles constrained sampling by studying a reflected Langevin diffusion on a closed convex domain and approximating it with penalized SDEs whose invariant measures converge to the reflected diffusion's invariant law in Wasserstein-2 distance with polynomial rates. It proves contraction properties and existence/uniqueness of invariant measures for both the penalized and reflected processes, establishing a quantitative link between penalized and constrained dynamics via . The discrete-time analysis introduces PCULA, an Euler–Maruyama scheme with a penalized potential, showing that its invariant measure converges to the penalized one at rate and, in turn, to the constrained invariant measure with a term depending on . Numerical experiments on a truncated Gaussian within an ellipse illustrate rapid convergence to the constrained target as the penalty grows, and the work outlines extensions to non-convex domains and other penalty schemes.

Abstract

We examine the Langevin diffusion confined to a closed, convex domain , represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that their invariant measures converge, in Wasserstein-2 distance and with explicit polynomial rate, to the invariant measure of the reflected Langevin diffusion. We also analyze a time-discretization of the penalized process obtained via the Euler-Maruyama scheme and demonstrate the convergence to the original constrained measure. These results provide a rigorous approximation framework for reflected Langevin dynamics in both continuous and discrete time.

Paper Structure

This paper contains 6 sections, 7 theorems, 57 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reflected Langevin Diffusion via Penalization
  3. Convergence to a Unique Invariant Measure in Wasserstein-2 Distance
  4. Discrete Approximation of the Penalized Langevin Diffusion
  5. Penalized Constrained Unadjusted Langevin Algorithm (PCULA)
  6. Truncated Gaussian Distribution via PCULA and Further Work

Key Result

Theorem 2.1

Let $D \subset \mathbb{R}^d$ be closed and convex, and $g \in C^1(\mathbb{R}^d)$ be our energy function. For $n \geq 1$, define Suppose $X^n$ satisfies the penalized SDE Xn for a Brownian motion $W^n$, with $K^n_t$ defined by K. Then, $X^n$ is a solution of the Langevin SDE Conversely, any solution of standard_penalized satisfies Xn (with $K^n$ given by K). We refer to $f_n$ as the penalized ene

Figures (2)

  • Figure 1: Truncated Gaussian supported on elliptical domain.
  • Figure 2: Empirical density of truncated Gaussian distribution via PCULA with $10^5$ iterations and step size $h = 10^{-4}.$

Theorems & Definitions (18)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1: Coupling
  • Definition 3.2: Wasserstein Distance of order 2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 8 more