  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.
