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Long-time reverse transportation inequalities for non-globally-dissipative Langevin dynamics

Jianfeng Lu, Yuliang Wang

TL;DR

The paper tackles the problem of quantifying how close distributions of Langevin dynamics with non-convex potentials are when started from different points, by establishing a long-time, dimension-free reverse transportation inequality in KL divergence and extending it to Rényi divergences. It introduces a novel coupling that combines a shifted (drifted) coupling with noise reflection and a carefully designed Lyapunov function to obtain uniform-in-time contraction estimates. The main results yield exponential decay rates in KL and Rényi divergences and imply dual log- and power-Harnack inequalities, providing a powerful framework for non-convex sampling and potential extensions to discretized algorithms and differential privacy in stochastic optimization. These contributions advance understanding of non-log-concave Langevin dynamics by delivering rigorous, dimension-free contraction results that bridge information-theoretic and functional-analytic perspectives.

Abstract

We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the Rényi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting.

Long-time reverse transportation inequalities for non-globally-dissipative Langevin dynamics

TL;DR

The paper tackles the problem of quantifying how close distributions of Langevin dynamics with non-convex potentials are when started from different points, by establishing a long-time, dimension-free reverse transportation inequality in KL divergence and extending it to Rényi divergences. It introduces a novel coupling that combines a shifted (drifted) coupling with noise reflection and a carefully designed Lyapunov function to obtain uniform-in-time contraction estimates. The main results yield exponential decay rates in KL and Rényi divergences and imply dual log- and power-Harnack inequalities, providing a powerful framework for non-convex sampling and potential extensions to discretized algorithms and differential privacy in stochastic optimization. These contributions advance understanding of non-log-concave Langevin dynamics by delivering rigorous, dimension-free contraction results that bridge information-theoretic and functional-analytic perspectives.

Abstract

We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the Rényi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting.

Paper Structure

This paper contains 16 sections, 5 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature review.
  3. Main contributions.
  4. Organization of the paper.
  5. Construction of coupling and Lyapunov function
  6. Basic assumptions
  7. Auxiliary coupled dynamics with noise reflection
  8. Choice of Lyapunov function
  9. Some useful lemmas
  10. Reverse transportation inequalities
  11. Reverse transportation inequality under KL divergence
  12. Reverse transportation inequality under Rényi divergence
  13. Asymptotic uniform Harnack inequalities
  14. Conclusion and discussions
  15. Acknowledgment.
  16. ...and 1 more sections

Key Result

Lemma 2.1

Under Assumption ass:farfield, if $\eta_t \geq 0, \; \forall \, t\geq 0$, then it holds that

Figures (1)

  • Figure 1: An illustration of the coupling \ref{['eq:coupling']}.$X$ and $X'$ are two realizations of the overdamped Langevin equation \ref{['eq:overdamped']}. $X"$ is an auxiliray dynamics that interpolates $X$ and $X'$. The KL or Rényi divergence between $X'$ and $X"$ is then given by Girsanov transform.

Theorems & Definitions (11)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Remark 3.1
  • proof : Proof of Theorem \ref{['thm:overdamped']}
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm:renyi']}
  • ...and 1 more