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Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions

Massimo Fornasier, Lukang Sun, Rachel Ward

TL;DR

This work analyzes overdamped Langevin dynamics for minimizing a general objective $\mathcal{L}$, establishing well-posedness and regularity of the law $\rho_t$ and a detailed large-time behavior that depends on whether the Gibbs density $\pi=e^{-\mathcal{L}/\sigma}$ is integrable. It shows a two-phase dynamic under Polyak-Łojasiewicz conditions: an initial exponential contraction toward the global minimizer set $\mathcal{W}^*$, followed by diffusion along that set at a rate of $\mathcal{O}(1/t)$, with exponential contractivity under global or local PL. The paper provides new a priori estimates, including higher-order time-derivative bounds, to rigorously justify well-posedness of the Fokker-Planck equation and to characterize the large-time limits even in non-integrable Gibbs regimes. These results bridge PL-type optimization guarantees with Langevin sampling in nonconvex landscapes, offering theoretical support for noisy gradient methods' ability to explore flat minima and identify multiple quasi-optimal solutions. The findings have broad implications for nonconvex optimization and stochastic sampling in high-dimensional settings, including deep learning.

Abstract

In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions $\mathcal{L}$. We establish well-posedness and regularity of the law $ρ_t$ of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of $ρ_t$ under truly minimal assumptions on $\mathcal{L}$. In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is $\mathcal{O}(1/t)$. Under a Polyak-Lojasiewicz (PL) condition on $\mathcal{L}$, we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate $\mathcal{O}(1/t)$.

Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions

TL;DR

This work analyzes overdamped Langevin dynamics for minimizing a general objective , establishing well-posedness and regularity of the law and a detailed large-time behavior that depends on whether the Gibbs density is integrable. It shows a two-phase dynamic under Polyak-Łojasiewicz conditions: an initial exponential contraction toward the global minimizer set , followed by diffusion along that set at a rate of , with exponential contractivity under global or local PL. The paper provides new a priori estimates, including higher-order time-derivative bounds, to rigorously justify well-posedness of the Fokker-Planck equation and to characterize the large-time limits even in non-integrable Gibbs regimes. These results bridge PL-type optimization guarantees with Langevin sampling in nonconvex landscapes, offering theoretical support for noisy gradient methods' ability to explore flat minima and identify multiple quasi-optimal solutions. The findings have broad implications for nonconvex optimization and stochastic sampling in high-dimensional settings, including deep learning.

Abstract

In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions . We establish well-posedness and regularity of the law of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of under truly minimal assumptions on . In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is . Under a Polyak-Lojasiewicz (PL) condition on , we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate .

Paper Structure

This paper contains 23 sections, 5 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The strongly convex regime
  3. The PL inequality regime
  4. Our Contribution
  5. Convergence to minimizers under a global Polyak-Łojasiewicz condition
  6. Assumptions
  7. Mass Concentration
  8. Convergence to minimizers under a local Polyak-Łojasiewicz condition
  9. Mathematical Context
  10. Gradient descent and Polyak-Łojasiewicz (PL) conditions.
  11. Langevin dynamics under a local PL assumption
  12. Well-posedness and regularity
  13. Formal a priori estimates and asymptotics
  14. First Order Time Derivative.
  15. Novel higher order a priori estimates
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that $\mathcal{L} \in C^{1,1}(\mathbb{R}^d)$ and $\mathfrak{L}\phi:=\sigma\Delta \phi-\mathopen{}\mathclose{\left< \nabla\mathcal{L} , \nabla\phi \right>$ fulfills conditions A or B reported in formulae eq:a36 and eq:a36. Then $\rho_t=\operatorname{Law}(w_t)$ is the unique smooth solution of here $\phi_t=\rho_t(w)/\pi(w)$. In particular $\rho_t(Z) \to \phi_\infty \pi(Z)$ for $t\to \infty$,

Theorems & Definitions (11)

  • Theorem 1
  • Remark 3
  • Remark 4
  • Lemma 5
  • Proposition 6
  • Remark 7
  • Remark 9
  • Proposition 10
  • Remark 11
  • Theorem 12
  • ...and 1 more