Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime

We analyze the convergence of the averaged stochastic gradient descent for over-parameterized two-layer neural networks for regression problems. It was recently found that, under the neural tangent kernel (NTK) regime, where the learning dynamics for overparameterized neural networks can be mostly characterized by that for the associated reproducing kernel Hilbert space (RKHS), an NTK plays an important role in revealing the global convergence of gradient-based methods. However, there is still room for a convergence rate analysis in the NTK regime. In this study, we show the global convergence of the averaged stochastic gradient descent and derive the optimal convergence rate by exploiting the complexities of the target function and the RKHS associated with the NTK. Moreover, we show that the target function specified by the NTK of a ReLU network can be learned at the optimal convergence rate through a smooth approximation of ReLU networks under certain conditions.