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Randomized Midpoint Method for Log-Concave Sampling under Constraints

Yifeng Yu, Lu Yu

TL;DR

This work addresses constrained log-concave sampling by pairing a proximal, projection-based framework with randomized midpoint discretizations of Langevin dynamics to sample from $\nu(\boldsymbol\theta) \propto e^{-U(\boldsymbol\theta)}$ on a convex set $\mathcal{K}$. A two-stage approach first constructs smooth surrogate densities $\nu^{\lambda}$ via projections (Euclidean, Bregman, or Gauge) and then analyzes the Wasserstein distance between $\nu^{\lambda}$ and the target $\nu$, establishing both upper and lower bounds. The authors introduce CRLMC and CRKLMC algorithms for constrained vanilla and kinetic Langevin diffusion, provide sharp non-asymptotic convergence guarantees in $W_1$ and $W_2$, and then derive improved error bounds for CLMC and CKLMC that leverage tighter surrogate-target gaps; empirical results in 2D corroborate the theoretical advantages and show midpoint methods outperform Euler-based schemes. Collectively, the paper advances constrained sampling with a flexible projection framework and faster, provably accurate midpoint-based diffusion samplers, enhancing practicality for high-dimensional, constrained log-concave targets.

Abstract

In this paper, we study the problem of sampling from log-concave distributions supported on convex, compact sets, with a particular focus on the randomized midpoint discretization of both vanilla and kinetic Langevin diffusions in this constrained setting. We propose a unified proximal framework for handling constraints via a broad class of projection operators, including Euclidean, Bregman, and Gauge projections. Within this framework, we establish non-asymptotic bounds in both $\mathcal{W}_1$ and $\mathcal{W}_2$ distances, providing precise complexity guarantees and performance comparisons. In addition, our analysis leads to sharper convergence guarantees for both vanilla and kinetic Langevin Monte Carlo under constraints, improving upon existing theoretical results.

Randomized Midpoint Method for Log-Concave Sampling under Constraints

TL;DR

This work addresses constrained log-concave sampling by pairing a proximal, projection-based framework with randomized midpoint discretizations of Langevin dynamics to sample from on a convex set . A two-stage approach first constructs smooth surrogate densities via projections (Euclidean, Bregman, or Gauge) and then analyzes the Wasserstein distance between and the target , establishing both upper and lower bounds. The authors introduce CRLMC and CRKLMC algorithms for constrained vanilla and kinetic Langevin diffusion, provide sharp non-asymptotic convergence guarantees in and , and then derive improved error bounds for CLMC and CKLMC that leverage tighter surrogate-target gaps; empirical results in 2D corroborate the theoretical advantages and show midpoint methods outperform Euler-based schemes. Collectively, the paper advances constrained sampling with a flexible projection framework and faster, provably accurate midpoint-based diffusion samplers, enhancing practicality for high-dimensional, constrained log-concave targets.

Abstract

In this paper, we study the problem of sampling from log-concave distributions supported on convex, compact sets, with a particular focus on the randomized midpoint discretization of both vanilla and kinetic Langevin diffusions in this constrained setting. We propose a unified proximal framework for handling constraints via a broad class of projection operators, including Euclidean, Bregman, and Gauge projections. Within this framework, we establish non-asymptotic bounds in both and distances, providing precise complexity guarantees and performance comparisons. In addition, our analysis leads to sharper convergence guarantees for both vanilla and kinetic Langevin Monte Carlo under constraints, improving upon existing theoretical results.
Paper Structure (25 sections, 17 theorems, 205 equations, 4 figures, 1 table)

This paper contains 25 sections, 17 theorems, 205 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Smooth Approximation for the Target Density
  4. Wasserstein Distance Bounds Between Distributions $\nu^\lambda$ and $\nu$
  5. Randomized Midpoint Method for Langevin Diffusion under Constraints
  6. Randomized Midpoint Method for Kinetic Langevin Diffusion under Constraints
  7. Improved Error Bounds for Constrained Langevin and Kinetic Langevin Monte Carlo
  8. Numerical Experiments
  9. Discussion
  10. Acknowledgments
  11. Relation to Works brosse2017sampling and gurbuzbalaban2022penalized
  12. Comparison with brosse2017sampling
  13. Comparison with the concurrent work gurbuzbalaban2022penalized
  14. Summary
  15. Proofs of Section \ref{['sec:approx_target']}
  16. ...and 10 more sections

Key Result

Proposition 1

Under Assumptions asm:radius-asm:dK, for any $q\geqslant 1$ and when $\lambda=o(r/p),$ it holds that

Figures (4)

  • Figure 1: Scatter plots of samples generated by $\{\text{CL,CKL,CRL,CRKL}\}\text{MC}$ algorithms.
  • Figure 2: Scatter plots of samples generated by $\{\text{CL,CKL,CRL,CRKL}\}\text{MC}$ algorithms.
  • Figure 3: Scatter plots of samples generated by $\{\text{CL,CKL,CRL,CRKL}\}\text{MC}$ algorithms.
  • Figure 4: Scatter plots of samples generated by $\{\text{CL,CKL,CRL,CRKL}\}\text{MC}$ algorithms.

Theorems & Definitions (19)

  • Example 2.1
  • Example 2.2
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • ...and 9 more