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A Dynamical System View of Langevin-Based Non-Convex Sampling

Mohammad Reza Karimi, Ya-Ping Hsieh, Andreas Krause

TL;DR

This work addresses the challenge of achieving last-iterate convergence for Langevin-based sampling in fully non-convex settings by linking discrete sampling updates to their continuous-time Langevin counterparts through a Picard-process-inspired framework. By introducing Wasserstein asymptotic pseudotrajectories (WAPT) and a general Langevin--Robbins--Monro (LRM) template, the authors show that many advanced discretizations (proximal, randomized mid-point, Runge–Kutta, Mirror-Langevin) inherit last-iterate convergence guarantees from the continuous dynamics. The key technical contribution is developing the Picard process and leveraging dynamical-systems tools to prove that the interpolated iterates are a WAPT of the Langevin SDE, which, under dissipativity-type conditions, yields convergence in Wasserstein distance to the target distribution $\pi \propto e^{-f}$. The framework supports design of more efficient schemes with the same rigorous guarantees and highlights a path toward broader applicability beyond Euler–Maruyama, while outlining limitations and future extensions to non-Langevin dynamics such as Metropolis–Hastings.

Abstract

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important challenges: Existing guarantees (1) typically only hold for the averaged iterates rather than the more desirable last iterates, (2) lack convergence metrics that capture the scales of the variables such as Wasserstein distances, and (3) mainly apply to elementary schemes such as stochastic gradient Langevin dynamics. In this paper, we develop a new framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as proximal, randomized mid-point, and Runge-Kutta integrators. Beyond existing methods, our framework also motivates more efficient schemes that enjoy the same rigorous guarantees.

A Dynamical System View of Langevin-Based Non-Convex Sampling

TL;DR

This work addresses the challenge of achieving last-iterate convergence for Langevin-based sampling in fully non-convex settings by linking discrete sampling updates to their continuous-time Langevin counterparts through a Picard-process-inspired framework. By introducing Wasserstein asymptotic pseudotrajectories (WAPT) and a general Langevin--Robbins--Monro (LRM) template, the authors show that many advanced discretizations (proximal, randomized mid-point, Runge–Kutta, Mirror-Langevin) inherit last-iterate convergence guarantees from the continuous dynamics. The key technical contribution is developing the Picard process and leveraging dynamical-systems tools to prove that the interpolated iterates are a WAPT of the Langevin SDE, which, under dissipativity-type conditions, yields convergence in Wasserstein distance to the target distribution . The framework supports design of more efficient schemes with the same rigorous guarantees and highlights a path toward broader applicability beyond Euler–Maruyama, while outlining limitations and future extensions to non-Langevin dynamics such as Metropolis–Hastings.

Abstract

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important challenges: Existing guarantees (1) typically only hold for the averaged iterates rather than the more desirable last iterates, (2) lack convergence metrics that capture the scales of the variables such as Wasserstein distances, and (3) mainly apply to elementary schemes such as stochastic gradient Langevin dynamics. In this paper, we develop a new framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as proximal, randomized mid-point, and Runge-Kutta integrators. Beyond existing methods, our framework also motivates more efficient schemes that enjoy the same rigorous guarantees.
Paper Structure (40 sections, 13 theorems, 106 equations, 1 figure, 1 table)

This paper contains 40 sections, 13 theorems, 106 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. § Contributions and Approaches.
  3. § Related work.
  4. The Langevin--Robbins--Monro Template
  5. Technique Overview: A Dynamical System Perspective
  6. § Langevin Diffusion as Gradient Flows.
  7. § Wasserstein Asymptotic Pseudotrajectories.
  8. § LRM Schemes.
  9. § Framework overview.
  10. The Dynamics of Langevin--Robbins--Monro Schemes
  11. Technical Assumptions and Requirements
  12. From Discrete to Continuous: LRM Schemes and WAPTs
  13. § Sketch of the Proof for \ref{['thm:APT']}.
  14. § Step 1: The Picard Process.
  15. § Step 2: Accumulated Noise and Bias.
  16. ...and 25 more sections

Key Result

Theorem 1

Suppose we discretize eq:LD as with step-sizes $\{\gamma_k\}_{k\in\mathbb{N}}$ and i.i.d. standard Gaussians $\{\xi_{k}\}_{k\in\mathbb{N}}$. Then, under an easy-to-verify condition on the bias (see eq:bias-assumption in ass:noise-bias), $\{x_k\}_{k\in\mathbb{N}}$ converges in Wasserstein distance to $\pi$. In addition, these cond

Figures (1)

  • Figure 1: High-level overview of the two components of the dynamical perspective.

Theorems & Definitions (26)

  • Theorem : Informal
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Definition 1: WAPT
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • ...and 16 more