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

We analyze the convergence of the averaged stochastic gradient descent for overparameterized two-layer neural networks for regression problems. It was recently found that a neural tangent kernel (NTK) plays an important role in showing the global convergence of gradient-based methods under the NTK regime, where the learning dynamics for overparameterized neural networks can be almost characterized by that for the associated reproducing kernel Hilbert space (RKHS). However, there is still room for a convergence rate analysis in the NTK regime. In this study, we show that the averaged stochastic gradient descent can achieve the minimax optimal convergence rate, with the global convergence guarantee, 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 a ReLU network under certain conditions. However, the eigenvalues of the NTK converge to zero as the number of examples increases, as shown in Su & Yang (2019) (also see Figure 1 ), resulting in the degeneration of the NTK. This phenomenon indicates that the convergence rates in previous studies in terms of generalization are generally slower than O(T -1/2 ) owing to the dependence on the minimum eigenvalue. Moreover, Bietti & Mairal (2019); Ronen et al. (2019); Cao et al. (2019) also supported this observation by providing a precise