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

Atsushi Nitanda, Taiji Suzuki

Keywords: stochastic gradient descent, two-layer neural network, over-parameterization, neural tangent kernel

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.

One-sentence Summary: This is the first paper to overcome technical challenges of achieving the optimal convergence rate under the NTK regime.

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