Gram-Gauss-Newton Method: Learning Overparameterized Neural Networks for Regression Problems

Tianle Cai*, Ruiqi Gao*, Jikai Hou*, Siyu Chen, Dong Wang, Di He, Zhihua Zhang, Liwei Wang

Keywords: Deep learning, Optimization, Second-order method, Neural Tangent Kernel regression

TL;DR: A novel Gram-Gauss-Newton method to train neural networks, inspired by neural tangent kernel and Gauss-Newton method, with fast convergence speed both theoretically and experimentally.

Abstract: First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the pro- hibitive computational cost in calculating the second-order information. In this paper, we propose a novel Gram-Gauss-Newton (GGN) algorithm to train deep neural networks for regression problems with square loss. Our method draws inspiration from the connection between neural network optimization and kernel regression of neural tangent kernel (NTK). Different from typical second-order methods that have heavy computational cost in each iteration, GGN only has minor overhead compared to first-order methods such as SGD. We also give theoretical results to show that for sufficiently wide neural networks, the convergence rate of GGN is quadratic. Furthermore, we provide convergence guarantee for mini-batch GGN algorithm, which is, to our knowledge, the first convergence result for the mini-batch version of a second-order method on overparameterized neural net- works. Preliminary experiments on regression tasks demonstrate that for training standard networks, our GGN algorithm converges much faster and achieves better performance than SGD.