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Adaptive stepsize algorithms for Langevin dynamics

Alix Leroy, Benedict Leimkuhler, Jonas Latz, Desmond J. Higham

TL;DR

This work introduces an invariant-measure-preserving transformed Langevin framework that uses a state-dependent time change with a monitor function to adapt timesteps while sampling from the Gibbs measures of overdamped and underdamped dynamics. A fluctuation-dissipation correction term β^{-1} ∇ g(x) is added to maintain the correct invariant distribution under the transformed dynamics, and a suite of splitting-based numerical integrators is developed to efficiently simulate the IP-transformed SDEs. The authors establish conditions ensuring existence, uniqueness, and ergodicity of the IP-transformed processes, and demonstrate substantial efficiency gains in numerical experiments, including a Bayesian posterior problem with a steep prior and a 2D landscape with multiple pathways. Overall, the paper provides a principled design for adaptive timestepping in stochastic sampling, with practical impact for Bayesian inference and high-dimensional sampling problems where stiffness or sharp features hinder fixed-step Langevin integration.

Abstract

We discuss the design of an invariant measure-preserving transformed dynamics for the numerical treatment of Langevin dynamics based on rescaling of time, with the goal of sampling from an invariant measure. Given an appropriate monitor function which characterizes the numerical difficulty of the problem as a function of the state of the system, this method allows the stepsizes to be reduced only when necessary, facilitating efficient recovery of long-time behavior. We study both the overdamped and underdamped Langevin dynamics. We investigate how an appropriate correction term that ensures preservation of the invariant measure should be incorporated into a numerical splitting scheme. Finally, we demonstrate the use of the technique in several model systems, including a Bayesian sampling problem with a steep prior.

Adaptive stepsize algorithms for Langevin dynamics

TL;DR

This work introduces an invariant-measure-preserving transformed Langevin framework that uses a state-dependent time change with a monitor function to adapt timesteps while sampling from the Gibbs measures of overdamped and underdamped dynamics. A fluctuation-dissipation correction term β^{-1} ∇ g(x) is added to maintain the correct invariant distribution under the transformed dynamics, and a suite of splitting-based numerical integrators is developed to efficiently simulate the IP-transformed SDEs. The authors establish conditions ensuring existence, uniqueness, and ergodicity of the IP-transformed processes, and demonstrate substantial efficiency gains in numerical experiments, including a Bayesian posterior problem with a steep prior and a 2D landscape with multiple pathways. Overall, the paper provides a principled design for adaptive timestepping in stochastic sampling, with practical impact for Bayesian inference and high-dimensional sampling problems where stiffness or sharp features hinder fixed-step Langevin integration.

Abstract

We discuss the design of an invariant measure-preserving transformed dynamics for the numerical treatment of Langevin dynamics based on rescaling of time, with the goal of sampling from an invariant measure. Given an appropriate monitor function which characterizes the numerical difficulty of the problem as a function of the state of the system, this method allows the stepsizes to be reduced only when necessary, facilitating efficient recovery of long-time behavior. We study both the overdamped and underdamped Langevin dynamics. We investigate how an appropriate correction term that ensures preservation of the invariant measure should be incorporated into a numerical splitting scheme. Finally, we demonstrate the use of the technique in several model systems, including a Bayesian sampling problem with a steep prior.
Paper Structure (16 sections, 3 theorems, 76 equations, 12 figures, 4 algorithms)

This paper contains 16 sections, 3 theorems, 76 equations, 12 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Motivating example
  3. Design of monitor function
  4. Stepsize bounds
  5. Well-posedness of the IP-transformed SDE
  6. Invariant distribution and geometric ergodicity
  7. The overdamped Langevin dynamics
  8. Underdamped Langevin dynamic
  9. Numerical integrators
  10. Numerical experiments
  11. Modified harmonic potential
  12. Overdamped transformed dynamic
  13. Underdamped transformed dynamic
  14. A Bayesian example
  15. Example 2D: a system with two pathways
  16. ...and 1 more sections

Key Result

Proposition 1

\newlabelprop:uniquesol0 If $b(x) \equiv - \nabla V(x)$ in the SDE eq:original_sde satisfies condition eq:cdts_th_oksen_2_1 then, under Assumptions ass:pot_1 and ass:pot_2 and Criteria gLipschitz - VgLipschitz, the IP-transformed SDE eq:transformed_overdamped_sde satisfies condition eq:cdts_th_oks

Figures (12)

  • Figure 1: Plots of the potential, the absolute value of its derivative and an example of a monitor function $g$.
  • Figure 1: Comparison of the monitor function for different values of the parameter $r$ and $t$.
  • Figure 1: The average number of iterations over the two steps A to reach the required tolerance for the algorithms $\hat{\rm{B}}\hat{\rm{A}}\hat{\rm{O}}\hat{\rm{A}}\hat{\rm{B}}$ and $\tilde{\rm{B}}\tilde{\rm{A}}\tilde{\rm{O}}\tilde{\rm{A}}\tilde{\rm{B}}$.
  • Figure 1: Order of accuracy in the case of a modified harmonic potential.
  • Figure 2: Histograms of the invariant distribution and samples obtained using the Euler-Maruyama scheme applied (a) to the original SDE \ref{['eq:original_sde']} and (b) to the direct time-rescaled SDE \ref{['eq:naive_time_rescaled']} after $50000$ steps, with step of size $h=0.001$, temperature $\beta^{-1}=0.1$, and $10^5$ samples. The function $g$ is bounded by $M=2$, $m=0.001$. The parameters of the potential are set to $c=0.1$, $b=0.1$, $a=10$ and $x_0=0.5$.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2