Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization

Yifei Wang, Peng Chen, Mert Pilanci, Wuchen Li

Keywords: Bayesian inference, convex optimization, semi-definite programming

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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TL;DR: Wasserstein gradient descent meets neural networks and convex optimization