# Disordered high-dimensional optimal control

arXiv: https://arxiv.org/abs/2101.00920 · HTML: https://mlsanity.com/paper/2101.00920/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2021.

## Built on by (most cited first, 2 of 2)

- [Optimal Protocols for Continual Learning via Statistical Physics and Control Theory](https://mlsanity.com/paper/2409.18061/) (2024)
- [Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methods](https://mlsanity.com/paper/2606.07247/) (2026)

## Abstract (arXiv)

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→ ∞, 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.