This was part of Machine Learning and Mean-Field Games

## Mean field approximations via log-concavity, and a non-asymptotic perspective on mean field control

Daniel Lacker, Columbia University
Thursday, May 26, 2022

Abstract: We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $R^n$ with density proportional to $e^{f(x)}$, for $f$ uniformly concave. The main application discussed in this talk is to a class of stochastic control problems in which a large number of players cooperatively choose their drifts to maximize an expected terminal reward minus a quadratic running cost. For a broad class of potentially asymmetric rewards, we show that there exist approximately optimal controls which are decentralized, in the sense that each player's control depends only on its own state and not the states of the other players. Moreover, the optimal decentralized controls can be constructed non-asymptotically, without reference to any mean field limit. The broader framework is inspired by the nonlinear large deviation theory of Chatterjee-Dembo, for which we offer an efficient new perspective in log-concave settings based on functional inequalities. As time permits, additional implications will be discussed in the context of continuous Gibbs measures on large graphs and high-dimensional Bayesian linear regression. Joint work with Sumit Mukherjee and Lane Chun Yeung.