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

TL;DR

The paper introduces Gram-Gauss-Newton, a second-order training method for overparameterized neural networks in regression tasks, offering faster convergence and better performance with minimal additional computational cost.

Contribution

It presents a novel GGN algorithm that combines second-order benefits with low overhead, along with theoretical convergence guarantees for wide and mini-batch networks.

Findings

GGN converges quadratically for sufficiently wide networks.

Mini-batch GGN has proven convergence guarantees.

Experiments show GGN outperforms SGD in speed and accuracy.

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 \emph{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 networks, our GGN algorithm converges much faster and achieves better performance than SGD.