Gram-Gauss-Newton Method: Learning Overparameterized Neural Networks for Regression Problems
Tianle Cai, Ruiqi Gao, Jikai Hou, Siyu Chen, Dong Wang, Di He {caitianle1998, grq, 1600010681, siyuchen, wangdongcis, Peking University equal contribution Zhihua Zhang Peking University Liwei Wang Peking University
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 prohibitive 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 networks. Preliminary experiments on regression tasks demonstrate that for training standard network
中文速览
训练深度神经网络时，随机梯度下降（SGD）等一阶方法虽然计算高效，但收敛速度慢；而传统二阶方法虽然收敛快，却因为需要计算和存储规模为"参数数量×参数数量"的巨型矩阵而完全无法落地。为此，研究者提出了一种名为"Gram高斯-牛顿法"（Gram-Gauss-Newton，GGN）的新算法，其核心思路是：利用神经切核（Neural Tangent Kernel，NTK）将网络优化转化为核回归问题，并在每一步直接求解该核回归的闭式解，而非再用梯度下降去迭代逼近它——这样只需操作一个"样本数×样本数"（mini-batch时进一步缩减为"批大小×批大小"）的小矩阵，彻底绕开了巨型参数矩阵的计算瓶颈。理论上，对于足够宽的过参数化网络，GGN被证明具有二次收敛速度（远优于梯度下降的线性收敛），且论文给出了mini-batch版本二阶方法在过参数化网络上的首个收敛保证；实验也表明，在真实回归任务上用ResNet-32等标准网络时，GGN比SGD收敛更快、性能更好，而每次迭代的额外计算开销与SGD相比却微乎其微。
原文 arXiv:1905.11675;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1905.11675v2