A Controlled Particle Filter for Global Optimization

TL;DR

This paper introduces a novel controlled particle filter for global optimization that leverages variational principles and gradient flows, avoiding resampling and potentially reducing variance.

Contribution

It presents a new particle filter framework based on optimal control and density transport, differing from classical methods and offering practical advantages.

Findings

The particle filter is formulated as a mean-field optimal control problem.

Density transport follows a gradient flow for the optimal value function.

Numerical examples demonstrate the method's effectiveness.

Abstract

A particle filter is introduced to numerically approximate a solution of the global optimization problem. The theoretical significance of this work comes from its variational aspects: (i) the proposed particle filter is a controlled interacting particle system where the control input represents the solution of a mean-field type optimal control problem; and (ii) the associated density transport is shown to be a gradient flow (steepest descent) for the optimal value function, with respect to the Kullback--Leibler divergence. The optimal control construction of the particle filter is a significant departure from the classical importance sampling-resampling based approaches. There are several practical advantages: (i) resampling, reproduction, death or birth of particles is avoided; (ii) simulation variance can potentially be reduced by applying feedback control principles; and (iii) the parametric approximation naturally arises as a special case. The latter also suggests systematic approaches for numerical approximation of the optimal control law. The theoretical results are illustrated with numerical examples.