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Non-Reversible Langevin Algorithms for Constrained Sampling

Hengrong Du, Qi Feng, Changwei Tu, Xiaoyu Wang, Lingjiong Zhu

TL;DR

The paper addresses constrained sampling on a convex domain by introducing skew-reflected non-reversible Langevin dynamics (SRNLD) and its discretized variant SRNLMC. It proves that the Gibbs measure $\pi(x)\propto e^{-f(x)}$ is invariant under SRNLD and derives non-asymptotic convergence bounds in total variation and $1$-Wasserstein distances, with an acceleration relative to reversible dynamics. A detailed discretization analysis for SRNLMC yields $1$-Wasserstein error bounds and explicit iteration complexities, highlighting improved performance over projected Langevin Monte Carlo when the anti-symmetric component $J$ is active. Numerical experiments on a toy truncated Gaussian and constrained Bayesian linear/logistic regression with both synthetic and real data corroborate the theoretical gains and demonstrate practical efficiency of SRNLMC/SRNSGLD in constrained settings.

Abstract

We consider the constrained sampling problem where the goal is to sample from a target distribution on a constrained domain. We propose skew-reflected non-reversible Langevin dynamics (SRNLD), a continuous-time stochastic differential equation with skew-reflected boundary. We obtain non-asymptotic convergence rate of SRNLD to the target distribution in both total variation and 1-Wasserstein distances. By breaking reversibility, we show that the convergence is faster than the special case of the reversible dynamics. Based on the discretization of SRNLD, we propose skew-reflected non-reversible Langevin Monte Carlo (SRNLMC), and obtain non-asymptotic discretization error from SRNLD, and convergence guarantees to the target distribution in 1-Wasserstein distance. We show better performance guarantees than the projected Langevin Monte Carlo in the literature that is based on the reversible dynamics. Numerical experiments are provided for both synthetic and real datasets to show efficiency of the proposed algorithms.

Non-Reversible Langevin Algorithms for Constrained Sampling

TL;DR

The paper addresses constrained sampling on a convex domain by introducing skew-reflected non-reversible Langevin dynamics (SRNLD) and its discretized variant SRNLMC. It proves that the Gibbs measure is invariant under SRNLD and derives non-asymptotic convergence bounds in total variation and -Wasserstein distances, with an acceleration relative to reversible dynamics. A detailed discretization analysis for SRNLMC yields -Wasserstein error bounds and explicit iteration complexities, highlighting improved performance over projected Langevin Monte Carlo when the anti-symmetric component is active. Numerical experiments on a toy truncated Gaussian and constrained Bayesian linear/logistic regression with both synthetic and real data corroborate the theoretical gains and demonstrate practical efficiency of SRNLMC/SRNSGLD in constrained settings.

Abstract

We consider the constrained sampling problem where the goal is to sample from a target distribution on a constrained domain. We propose skew-reflected non-reversible Langevin dynamics (SRNLD), a continuous-time stochastic differential equation with skew-reflected boundary. We obtain non-asymptotic convergence rate of SRNLD to the target distribution in both total variation and 1-Wasserstein distances. By breaking reversibility, we show that the convergence is faster than the special case of the reversible dynamics. Based on the discretization of SRNLD, we propose skew-reflected non-reversible Langevin Monte Carlo (SRNLMC), and obtain non-asymptotic discretization error from SRNLD, and convergence guarantees to the target distribution in 1-Wasserstein distance. We show better performance guarantees than the projected Langevin Monte Carlo in the literature that is based on the reversible dynamics. Numerical experiments are provided for both synthetic and real datasets to show efficiency of the proposed algorithms.
Paper Structure (12 sections, 19 theorems, 118 equations, 9 figures)

This paper contains 12 sections, 19 theorems, 118 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Skorokhod Problem
  4. Continuous-time analysis
  5. Discretization analysis
  6. Numerical Experiments
  7. Toy Example: Truncated Standard Multivariate Normal Distribution
  8. Constrained Bayesian Linear Regression
  9. Constrained Bayesian Logistic Regression
  10. Synthetic data
  11. Real data
  12. Conclusion

Key Result

Lemma 2.3

For a non-reversible Langevin SDE takes the form of eqn:anti, there exists a skew-reflected non-reversible Langevin dynamics in the form of eqn:anti:reflected, and the soluiton is unique in the strong sense.

Figures (9)

  • Figure 1: Visualized density plots for the first 2 dimensions in ball constraint
  • Figure 2: $1$-Wasserstein distance in each dimension of PLMC and SRNLMC in ball constraint
  • Figure 3: Visualized density plots for the first 2 dimensions in cubic constraint
  • Figure 4: $1$-Wasserstein distance in each dimension of PLMC and SRNLMC in cubic constraint
  • Figure 5: Prior and posterior distributions plot with disk constraint
  • ...and 4 more figures

Theorems & Definitions (42)

  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • proof : Proof of Lemma \ref{['lem:generator']}
  • Theorem 2.6
  • proof
  • Lemma 2.7: Commutator Identity
  • proof
  • Remark 2.8
  • ...and 32 more