Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization

• URL: http://arxiv.org/abs/2205.13098v1
• Date: Thu, 26 May 2022 00:51:12 GMT
• Title: Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization
• Authors: Yifei Wang, Peng Chen, Mert Pilanci, Wuchen Li
• Abstract summary: The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. We study the variational problem in the family of two-layer networks with squared-ReLU activations, towards which we derive a semi-definite programming (SDP) relaxation. This SDP can be viewed as an approximation of the Wasserstein gradient in a broader function family including two-layer networks.
• Score: 43.6961980403682
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving a variational problem. We study the variational problem in the family of two-layer networks with squared-ReLU activations, towards which we derive a semi-definite programming (SDP) relaxation. This SDP can be viewed as an approximation of the Wasserstein gradient in a broader function family including two-layer networks. By solving the convex SDP, we obtain the optimal approximation of the Wasserstein gradient direction in this class of functions. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation in COVID-19 modeling demonstrate the effectiveness of the proposed method.

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