A Mean-Field Analysis of Neural Gradient Descent-Ascent: Applications to Functional Conditional Moment Equations

We study minimax optimization problems defined over infinite-dimensional function classes. In particular, we restrict the functions to the class of overparameterized two-layer neural networks and study (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural network. As an initial step, we consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space. For this problem, we establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $\mathcal{O}(T^{-1} + \alpha^{-1} ) $ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural network. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $\mathcal{O}(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, and asset pricing.