  We study high-dimensional stochastic optimal control problems in which many
agents cooperate to minimize a convex cost functional. We consider both the
full-information problem, in which each agent observes the states of all other
agents, and the distributed problem, in which each agent observes only its own
state. Our main results are sharp non-asymptotic bounds on the gap between
these two problems, measured both in terms of their value functions and optimal
states. Along the way, we develop theory for distributed optimal stochastic
control in parallel with the classical setting, by characterizing optimizers in
terms of an associated stochastic maximum principle and a Hamilton-Jacobi-type
equation. By specializing these results to the setting of mean field control,
in which costs are (symmetric) functions of the empirical distribution of
states, we derive the optimal rate for the convergence problem in the
displacement convex regime.
