OPTIMAL RATES FOR AVERAGED STOCHASTIC GRA-DIENT DESCENT UNDER NEURAL TANGENT KERNEL REGIME

Abstract

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.

1. INTRODUCTION

Recent studies have revealed why a stochastic gradient descent for neural networks converges to a global minimum and why it generalizes well under the overparameterized setting in which the number of parameters is larger than the number of given training examples. One prominent approach is to map the learning dynamics for neural networks into function spaces and exploit the convexity of the loss functions with respect to the function. The neural tangent kernel (NTK) (Jacot et al., 2018) has provided such a connection between the learning process of a neural network and a kernel method in a reproducing kernel Hilbert space (RKHS) associated with an NTK. 



The global convergence of the gradient descent was demonstrated in Du et al. (2019b); Allen-Zhu et al. (2019a); Du et al. (2019a); Allen-Zhu et al. (2019b) through the development of a theory of NTK with the overparameterization. In these theories, the positivity of the NTK on the given training examples plays a crucial role in exploiting the property of the NTK. Specifically, the positivity of the Gram-matrix of the NTK leads to a rapid decay of the training loss, and thus the learning dynamics can be localized around the initial point of a neural network with the overparameterization, resulting in the equivalence between two learning dynamics for neural networks and kernel methods with the NTK through a linear approximation of neural networks. Moreover, Arora et al. (2019a) provided a generalization bound of O(T -1/2 ), where T is the number of training examples, on a gradient descent under the positivity assumption of the NTK. These studies provided the first steps in understanding the role of the NTK. 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

