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In this talk, we first propose a new class of metrics and show that under such metrics, the convergence of empirical measures in high dimensions is free of the curse of dimensionality, in contrast to Wasserstein distance. Proposed metrics are the integral probability metrics, where we propose criteria for test function spaces. Examples include RKHS, Barron space, and flow-induced function spaces. One application studies the construction of Nash equilibrium for the homogeneous n-player game by its mean-field limit (mean-field game). Another application is to show the ability to overcome curves of dimensionality of deep learning algorithms, for example, in solving Mckean-Vlasov forward-backward stochastic differential equations with general distribution dependence. This is joint work with Jiequn Han and Jihao Long

Biography: Ruimeng Hu is an assistant professor jointly appointed by the Department of Mathematics, and Department of Applied Probability and Statistics, at the University of California, Santa Barbara (UCSB), USA. Her research includes machine learning, financial mathematics, game theory, portfolio optimization, and sequential experimental design. Her research is currently supported by an NSF grant as PI, the Regents' Junior Faculty Award, Early Career Faculty Acceleration Funding by UCSB. She has submitted and published 20+ papers in top journals including Mathematical Finance, Notices of AMS, ICML, SIAM Journal on Control and Optimization, and SIAM Journal on Financial Mathematics. Ruimeng Hu obtained her Ph.D. at the Department of Applied Probability and Statistics at UCSB in 2018, and B.S. in Pure and Applied Mathematics at Peking University, China in 2012.