Approximately optimal distributed controls for high-dimensional stochastic systems with pairwise interaction through controls
Elise Devey
TL;DR
This work analyzes large-population stochastic control with pairwise interaction through controls and derives a nonasymptotic bound comparing full-information and distributed-control value functions. By lifting the problem to the space of probability measures and reformulating as Hamilton-Jacobi equations, the authors couple the two problems via their Hamiltonians and quantify the gap between $V^N$ and $V^N_{dist}$. Under convexity and smoothness assumptions on the interaction costs $f^N$ and the terminal cost $g^N$, they obtain a rate $0 \,\le \, V^N_{dist}-V^N \,\le \, M/\,\sqrt{N}$ when a certain cross-derivative bound on $g^N$ holds, and a more general bound in terms of time-dependent functions $K_f(t)$ and $K_g(t)$. The results justify using distributed controls as near-optimal for large $N$ and provide explicit, quantitative convergence rates for mean-field-type control problems with control interactions.
Abstract
This paper investigates large-population stochastic control problems in which agents share their state information and cooperate to minimize a convex cost functional. The latter is decomposed into individual and coupling costs, with the distinctive feature that the coupling term is a pairwise interaction function between the controls. To address this setting, we follow closely (Jackson & Lacker, 2025): we introduce a related problem where each agent observes only its own state. We then establish a quantitative bound on the difference between the value functions associated with these two problems. We obtain this result by reformulating the problems analytically as Hamilton-Jacobi type equations and comparing their associated Hamiltonians. The main difficulty of our approach lies in establishing a precise comparison between the distributions of the corresponding optimal controls.