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Presented By: Department of Mathematics

Financial/Actuarial Mathematics Seminar

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

We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. 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 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. Our framework is inspired by the recent theory of nonlinear large deviations of Chatterjee-Dembo, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. If time allows, we discuss additional implications for continuous Gibbs measures on large graphs. Joint work with Daniel Lacker and Sumit Mukherjee. Speaker(s): Lane Yeung (Columbia University)

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