Optimization for ML and AI Seminar: (De)regularized Wasserstein Gradient Flows via Reproducing Kernels

Speaker: Bharath Sriperumbudur, Pennsylvania State University

Abstract: Wasserstein gradient flows have become a popular tool in machine learning with applications in sampling, variational inference, generative modeling, and reinforcement learning, among others. The Wasserstein gradient flow (WGF) involves minimizing a probability functional over the Wasserstein space (by taking into account the intrinsic geometry of the Wasserstein space). In this work, we introduce approximate/regularized Wasserstein gradient flows in two different settings: (a) approximate the probability functional and (b) approximate the Wasserstein geometry. In (a), we consider the probability functional to be chi^2-divergence, whose WGF is difficult to implement. To this end, we propose a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) as an approximation of chi^2-divergence and develop an approximate WGF, which is easy to implement and has applications in generative modeling. On the other hand, in the setting of (b), we use Kullback-Leibler divergence as the probability functional and develop an approximation to the Wassertein geometry, which allows for an efficient implementation than that of the exact WGF, with applications in sampling. In both settings, we present a variety of theoretical results that relate the approximate flow to the exact flow and demonstrate the superiority of the approximate flows via numerical simulations.

Bio: Bharath Sriperumbudur is a professor in the Department of Statistics (with a courtesy appointment in the Department of Mathematics) at the Pennsylvania State University. His research interests include non-parametric statistics, machine learning, statistical learning theory, optimal transport and gradient flows, regularization and inverse problems, reproducing kernel spaces in probability and statistics, functional and topological data analysis.

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