# Fisher information dissipation for time inhomogeneous stochastic differential equations

arXiv: https://arxiv.org/abs/2402.01036 · HTML: https://mlsanity.com/paper/2402.01036/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 59 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Builds on (most cited first, 15 of 15)

- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo](https://mlsanity.com/paper/1710.02736/) (2017)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [First-order optimization algorithms via inertial systems with Hessian driven damping](https://mlsanity.com/paper/1907.10536/) (2019)
- [State-Dependent Temperature Control for Langevin Diffusions](https://mlsanity.com/paper/2011.07456/) (2020)
- [Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation](https://mlsanity.com/paper/1701.04247/) (2017)
- [Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law](https://mlsanity.com/paper/2102.02339/) (2021)
- [Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics](https://mlsanity.com/paper/2108.08247/) (2021)
- [Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping](https://mlsanity.com/paper/2107.05943/) (2021)
- [Primal-Dual Damping algorithms for optimization](https://mlsanity.com/paper/2304.14574/) (2023)

## Built on by (most cited first, 2 of 2)

- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Continuous Policy and Value Iteration for Relaxed Stochastic Control Problems and Its Convergence](https://mlsanity.com/paper/2506.08121/) (2025)

## Abstract (arXiv)

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 L1 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.