Disordered high-dimensional optimal control

TL;DR

This paper studies high-dimensional optimal control problems with disordered interactions among many agents, deriving a reduced set of stochastic PDEs to analyze the average minimal cost in the large system limit.

Contribution

It introduces a model for disordered high-dimensional control and reduces the complex problem to solvable stochastic PDEs, enabling analysis of typical costs.

Findings

Dimensional reduction to stochastic PDEs

Self-consistent computation of stochastic terms

Analysis of average minimal cost in large systems

Abstract

Mean field optimal control problems are a class of optimization problems that arise from optimal control when applied to the many body setting. In the noisy case one has a set of controllable stochastic processes and a cost function that is a functional of their trajectories. The goal of the optimization is to minimize this cost over the control variables. Here we consider the case in which we have $N$ stochastic processes, or agents, with the associated control variables, which interact in a disordered way so that the resulting cost function is random. The goal is to find the average minimal cost for $N\to \infty$, when a typical realization of the quenched random interactions is considered. We introduce a simple model and show how to perform a dimensional reduction from the infinite dimensional case to a set of one dimensional stochastic partial differential equations of the Hamilton-Jacobi-Bellman and Fokker-Planck type. The statistical properties of the corresponding stochastic terms must be computed self-consistently, as we show explicitly.