Principles of Lipschitz continuity in neural networks

2026-02-03

AI Summary

This work addresses the well-known limitations of neural networks in robustness and generalization when exposed to small perturbations or out-of-distribution data. It proposes a unified theoretical framework centered on Lipschitz continuity, uniquely integrating dual perspectives: internal training dynamics and external modulation of signal propagation in the frequency domain. By synthesizing Lipschitz theory, dynamical analysis of training processes, and frequency-domain modeling, the study systematically uncovers the intrinsic mechanisms through which Lipschitz continuity governs model robustness and generalization. This approach transcends prior reliance on empirical regularization techniques and offers principled guidance for designing more reliable neural architectures.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Natural Language Processing: Safety and Robustness
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).

Problem

• Lipschitz continuity
• robustness
• generalization
• training dynamics
• frequency signal propagation

Innovation

• Lipschitz continuity
• training dynamics
• frequency signal propagation
• robustness
• generalization