Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

TL;DR

This paper develops a feedback maximum principle for controlling ensembles of particles modeled by local continuity equations, with an application to supervised machine learning, providing a new optimality condition and iterative control algorithm.

Contribution

It introduces a novel feedback maximum principle for ensemble control of continuity equations and applies it to supervised machine learning tasks.

Findings

Derived a necessary optimality condition using Pontryagin's Maximum Principle.

Formulated an iterative algorithm for optimal control based on the optimality condition.

Applied the method to a supervised machine learning problem with promising results.

Abstract

We consider an optimal control problem for a system of local continuity equations on a space of probability measures. Such systems can be viewed as macroscopic models of ensembles of non-interacting particles or homotypic individuals, representing several different ``populations''. For the stated problem, we propose a necessary optimality condition, which involves feedback controls inherent to the extremal structure, designed via the standard Pontryagin's Maximum Principle conditions. This optimality condition admits a realization as an iterative algorithm for optimal control. As a motivating case, we discuss an application of the derived optimality condition and the consequent numeric method to a problem of supervised machine learning via dynamic systems.