insight - Neural network optimization - # Structure-guided Gauss-Newton method for shallow ReLU neural networks

Core Concepts

The authors propose a structure-guided Gauss-Newton (SgGN) method that effectively utilizes both the least squares structure and the neural network structure of the objective function to solve optimization problems involving shallow ReLU neural networks.

Abstract

The paper introduces a structure-guided Gauss-Newton (SgGN) method for solving least squares optimization problems using shallow ReLU neural networks. The method categorizes the neural network parameters into linear and nonlinear parameters, and iterates back and forth between updating these two sets of parameters.
For the nonlinear parameters, the method uses a damped Gauss-Newton method with a specially derived form of the Gauss-Newton matrix that exploits the neural network structure. This Gauss-Newton matrix is shown to be symmetric and positive definite under reasonable assumptions, eliminating the need for additional techniques like shifting to ensure invertibility.
The linear parameters are updated by solving a linear system involving a mass matrix that is also symmetric and positive definite. The authors demonstrate the convergence and accuracy of the SgGN method numerically for various function approximation problems, especially those with discontinuities or sharp transition layers that pose challenges for commonly used training algorithms. The SgGN method is also extended to discrete least-squares optimization problems.

Stats

The authors use several key metrics and figures to support their analysis:
The loss curves for the test problems clearly show that the SgGN method significantly outperforms BFGS, KFRA, and Adam in terms of convergence and accuracy.
The authors examine the ability of the methods in effectively moving the breaking hyperplanes (points for 1D and lines for 2D) to capture the discontinuities or sharp transitions in the target functions.
For the data science application, the authors test two shallow networks with 40 and 80 neurons, and report the least squares loss values achieved by the different optimization methods.

Quotes

"The SgGN method provides an innovative way to effectively take advantage of both the quadratic structure and the NN structure in least-squares optimization problems arising from shallow ReLU NN approximations."
"A significant distinction between the SgGN method and the usual Gauss-Newton method is that there is no need to use additional techniques like shifting in the Levenberg-Marquardt method to achieve invertibility of the Gauss-Newton matrix."

Key Insights Distilled From

by Zhiqiang Cai... at **arxiv.org** 04-09-2024

Deeper Inquiries

The SgGN method can be extended to deeper neural network architectures by adapting the block iterative process to handle the additional layers. In a deeper neural network, the parameters would be categorized into linear and nonlinear parameters for each layer. The method would iterate back and forth between updating the linear parameters using a linear solver and the nonlinear parameters using the Gauss-Newton method for each layer. This process would continue through the layers of the network, allowing for efficient optimization of the parameters in deeper architectures.

The SgGN method offers theoretical convergence guarantees that are significant for optimization in neural networks. The method leverages the structure of the neural network and the least squares problem to efficiently update the parameters. The convergence properties of SgGN are demonstrated through the symmetric and positive definite nature of the Gauss-Newton matrix, ensuring a reliable and effective optimization process. Compared to other Gauss-Newton-based optimization techniques, SgGN stands out for its ability to handle discontinuities and sharp transitions in the target functions, leading to faster convergence and higher accuracy in the optimization process.

The insights gained from the structure-guided approach in the SgGN method can be applied to improve the optimization of other types of neural network models, such as convolutional neural networks (CNNs) or recurrent neural networks (RNNs). By categorizing the parameters into linear and nonlinear components and utilizing efficient iterative methods for updating these parameters, the SgGN approach can enhance the optimization process for complex neural network architectures. This structured approach can help address challenges like vanishing gradients in RNNs or optimizing the convolutional layers in CNNs, leading to improved convergence and accuracy in training these models.

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