Kernel Distributionally Robust Optimization

• URL: http://arxiv.org/abs/2006.06981v3
• Date: Sat, 27 Feb 2021 16:39:42 GMT
• Title: Kernel Distributionally Robust Optimization
• Authors: Jia-Jie Zhu, Wittawat Jitkrittum, Moritz Diehl, Bernhard Sch\"olkopf
• Abstract summary: We propose kernel distributionally robust optimization ( Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses kernel kernel (RKHS) to construct a wide range of convex ambiguity, which can be generalized to sets based on probability metrics and finite-order moment sets. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as losses and knowledge of the Lipschitz constant.
• Score: 17.909696462645023
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization with respect to measures in DRO into the dual program that searches for RKHS functions. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as polynomial losses and knowledge of the Lipschitz constant. We then establish a connection between DRO and stochastic optimization with expectation constraints. Finally, we propose practical algorithms based on both batch convex solvers and stochastic functional gradient, which apply to general optimization and machine learning tasks.

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