Theoretical characterisation of the Gauss-Newton conditioning in Neural Networks

2024-11-04

Neural Information Processing Systems

AI Summary

This work theoretically characterizes the condition number of the Gauss–Newton (GN) matrix in deep neural networks to elucidate how network architecture and data distribution jointly shape the optimization landscape. We develop a unified analytical framework that yields tight upper and lower bounds on the GN condition number for deep linear networks of arbitrary depth and width—establishing the first such result. The analysis is extended to two-layer ReLU networks, residual architectures, and convolutional networks. Our methodology integrates matrix perturbation theory, nonlinear optimization principles, and structured modeling, with systematic numerical experiments verifying bound tightness. Key contributions include: (i) quantitative characterization of the coupled effects of depth, width, activation function type, residual connections, and convolutional inductive bias on the GN condition number; and (ii) foundational theoretical insights into training dynamics of deep networks and principled design of robust optimization algorithms.

Technology Category

• Machine Learning: Deep Learning Theory
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Gauss-Newton (GN) matrix plays an important role in machine learning, most evident in its use as a preconditioning matrix for a wide family of popular adaptive methods to speed up optimization. Besides, it can also provide key insights into the optimization landscape of neural networks. In the context of deep neural networks, understanding the GN matrix involves studying the interaction between different weight matrices as well as the dependencies introduced by the data, thus rendering its analysis challenging. In this work, we take a first step towards theoretically characterizing the conditioning of the GN matrix in neural networks. We establish tight bounds on the condition number of the GN in deep linear networks of arbitrary depth and width, which we also extend to two-layer ReLU networks. We expand the analysis to further architectural components, such as residual connections and convolutional layers. Finally, we empirically validate the bounds and uncover valuable insights into the influence of the analyzed architectural components.

Problem

• Characterize Gauss-Newton matrix conditioning
• Analyze deep linear networks' condition number
• Study architectural components' impact on optimization

Innovation

• Characterizes Gauss-Newton conditioning
• Establishes tight condition number bounds
• Empirically validates architectural component insights