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Fisher information dissipation for time inhomogeneous stochastic differential equations

Qi Feng, Xinzhe Zuo, Wuchen Li

TL;DR

This work develops a time-dependent Lyapunov framework for the convergence of time-inhomogeneous SDEs to a reference distribution by leveraging a time-varying KL divergence and Fisher information. Through a Hessian-curve bound, it derives decay estimates for the combined Fisher information $I_{a,z}$ and provides a Gamma-calculus based toolkit to quantify dissipation in both gradient and non-gradient directions. The authors apply the theory to reversible, irreversible, and underdamped Langevin dynamics, obtaining explicit rates such as $I_a(t)\lesssim 1/t$ for time-dependent overdamped Langevin dynamics and demonstrating speed-ups with irreversible drift. Numerical experiments in 1D and 2D confirm the predicted decay behavior and illustrate the practical relevance for simulated annealing and sampling in nonconvex settings. The results offer a principled way to design time-dependent diffusion and drift to accelerate convergence in sampling and global optimization tasks.

Abstract

We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations (SDEs). Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first formula the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback-Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distribution. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of the SDE. We verify the proposed conditions for several time-inhomogeneous Langevin dynamics. For the overdamped Langevin dynamics, we prove the $O(t^{-1/2})$ convergence in $L^1$ distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped Langevin dynamics. Numerical examples demonstrate the convergence results for the time-dependent Langevin dynamics.

Fisher information dissipation for time inhomogeneous stochastic differential equations

TL;DR

This work develops a time-dependent Lyapunov framework for the convergence of time-inhomogeneous SDEs to a reference distribution by leveraging a time-varying KL divergence and Fisher information. Through a Hessian-curve bound, it derives decay estimates for the combined Fisher information and provides a Gamma-calculus based toolkit to quantify dissipation in both gradient and non-gradient directions. The authors apply the theory to reversible, irreversible, and underdamped Langevin dynamics, obtaining explicit rates such as for time-dependent overdamped Langevin dynamics and demonstrating speed-ups with irreversible drift. Numerical experiments in 1D and 2D confirm the predicted decay behavior and illustrate the practical relevance for simulated annealing and sampling in nonconvex settings. The results offer a principled way to design time-dependent diffusion and drift to accelerate convergence in sampling and global optimization tasks.

Abstract

We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations (SDEs). Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first formula the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback-Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distribution. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of the SDE. We verify the proposed conditions for several time-inhomogeneous Langevin dynamics. For the overdamped Langevin dynamics, we prove the convergence in distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped Langevin dynamics. Numerical examples demonstrate the convergence results for the time-dependent Langevin dynamics.
Paper Structure (19 sections, 19 theorems, 154 equations, 9 figures)

This paper contains 19 sections, 19 theorems, 154 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. General setting
  4. Time dependent Gradient and Non-gradient decompositions
  5. Lyapunov functionals
  6. Time dependent Fisher information decay
  7. Fisher information decay
  8. Information Gamma calculus
  9. Dissipation of $\mathrm I_{a}({p} \|\pi)$
  10. Dissipation of auxillary Fisher information
  11. Example I: reversible SDE
  12. Convergence analysis
  13. Time dependent overdamped Langevin dynamics
  14. Numerics
  15. Example II: non-degenerate, non-reversible SDEs
  16. ...and 4 more sections

Key Result

Proposition 1

For the Fokker-Planck equation FPE and a reference distribution density function $\pi(t,x)$, we define a non-gradient vector field $\gamma(t, x): \mathbb R_+\times \mathbb R^{n+m}\rightarrow \mathbb R^{n+m}$ as Then the Fokker-Planck equation FPE is equivalent to the following equation:

Figures (9)

  • Figure 1: Convergence rate of two strongly convex functions in one-dimension. Here y-axis represents the logarithm of the KL divergence between the empirical distribution and the invariant measure $\pi(t,x)$ given by \ref{['defn: pi']}. And the x-axis is $\log(t)$. We have also added a dotted line representing $t^{-1}$ on a logarithmic scale for comparison.
  • Figure 2: Convergence rate of two non-convex functions in one-dimension.
  • Figure 3: Convergence rate of two strongly convex functions (A) and (B), and a non-convex function (C) in two dimensions.
  • Figure 4: Eigenvalue comparison between $V(x)$ and $\mathfrak R(x)$ for $x\in [-0.5,0.5]\times[-0.5,0.5]$. The parameters are chosen as in Example \ref{['ex:new J']}. The yellow surface represents the smallest eigenvalue of $\mathfrak R(x)$ and the purple surface represents the smallest eigenvalue of $V(x)$. The global minimum of $V$ is marked with a blue asterisk. As shown in the figure, the smallest eigenvalue of $\mathfrak R$ is larger than that of $V$ near the global minimum.
  • Figure 5: Convergence comparison between \ref{['eq:langevin']} and \ref{['eq:new_J']} in Example \ref{['ex:new J']}. $x$-axis represents time. $y$-axis represents the average distance of the particles to the global minimum using the two SDEs.
  • ...and 4 more figures

Theorems & Definitions (44)

  • Proposition 1: Decomposition
  • proof
  • Remark 1
  • Definition 1: Fisher information functionals
  • Proposition 2
  • Remark 2
  • proof
  • Lemma 1
  • proof
  • Theorem 2: Fisher decay
  • ...and 34 more