Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

TL;DR

This paper develops an importance sampling method for rare event statistics in weakly interacting diffusions, leveraging Hamilton-Jacobi-Bellman equations on Wasserstein space, and demonstrates its asymptotic optimality and accuracy.

Contribution

It introduces a novel importance sampling scheme based on HJB equations on Wasserstein space for weakly interacting diffusions, connecting large deviations, mean-field control, and optimal importance sampling.

Findings

Scheme is asymptotically optimal under certain conditions.

Numerical evidence shows vanishing relative error with sufficient regularity.

Method outperforms standard Monte Carlo in high-dimensional particle systems.

Abstract

We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB Equation on Wasserstein Space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB Equation, our scheme can have vanishingly small relative error in the many particle limit.