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Risk-neutral limit of adaptive importance sampling of random stopping times

Carsten Hartmann, Annika Jöster

Abstract

We discuss importance sampling of exit problems that involve unbounded stopping times; examples are mean first passage times, transition rates or committor probabilities in molecular dynamics. The naive application of variance minimization techniques can lead to pathologies here, including proposal measures that are not absolutely continuous to the reference measure or importance sampling estimators that formally have zero variance, but that produce infinitely long trajectories. We illustrate these issues with simple examples and discuss a possible solution that is based on a risk-sensitive optimal control framework of importance sampling.

Risk-neutral limit of adaptive importance sampling of random stopping times

Abstract

We discuss importance sampling of exit problems that involve unbounded stopping times; examples are mean first passage times, transition rates or committor probabilities in molecular dynamics. The naive application of variance minimization techniques can lead to pathologies here, including proposal measures that are not absolutely continuous to the reference measure or importance sampling estimators that formally have zero variance, but that produce infinitely long trajectories. We illustrate these issues with simple examples and discuss a possible solution that is based on a risk-sensitive optimal control framework of importance sampling.
Paper Structure (11 sections, 43 equations, 2 figures)

This paper contains 11 sections, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Adaptive importance sampling
  3. Zero-variance AIS and the Feynman-Kac formula
  4. Issues for unbounded stopping times
  5. Control of moments
  6. A certainty equivalence principle
  7. Extracting the mean
  8. Numerical illustration
  9. 100-dimensional Brownian motion
  10. 100-dimensional Ornstein-Uhlenbeck process
  11. Discussion

Figures (2)

  • Figure 1: MFET estimates and their 95% confidence intervals: exact solution that agrees with the zero-variance CoV estimate (dashed blue ), standard MC (solid red) and PCoV with perturbation $\delta\sin(x)$ for $\delta=0.25$ (dotted green). Simulations were done using the Euler-Maruyama scheme with steps size $\Delta t=10^{-4}$.
  • Figure 2: MFET estimates and their 95% confidence intervals: standard MC (red) and CoV with a suboptimal CoV based on the driftless control variate based on (\ref{['mfet100']}). Simulations were done using the Euler-Maruyama scheme with steps size $\Delta t=10^{-3}$.

Theorems & Definitions (2)

  • Example 1
  • Example 2