论文标题
通过凸优化的Wasserstein梯度方向的最佳神经网络近似
Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization
论文作者
论文摘要
Wasserstein梯度方向的计算对于后验采样问题和科学计算至关重要。 Wasserstein梯度与有限样品的近似需要解决变异问题。我们研究具有平方relu激活的两层网络家族中的变异问题，我们将获得半明确的编程（SDP）放松。该SDP可以被视为在包括两层网络的更广泛函数家族中Wasserstein梯度的近似值。通过求解凸SDP，我们获得了此类函数中Wasserstein梯度方向的最佳近似。数值实验包括COVID-19建模中的PDE受限的贝叶斯推断和参数估计，证明了该方法的有效性。
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.
