Lipschitz Continuity in Deep Learning: A Systematic Review of Theoretical Foundations, Estimation Methods, Regularization Approaches, and Certifiable Robustness

2026-07-15

AI Summary

Research on Lipschitz continuity in deep learning has long been fragmented, lacking a unified framework and thereby hindering a systematic understanding of its role in robustness, generalization, and optimization. This work presents the first comprehensive survey from the perspective of Lipschitz continuity, systematically organizing foundational theories, methods for estimating Lipschitz constants, regularization techniques, and approaches to certifiable robustness analysis. By synthesizing results across diverse domains, the paper constructs a cohesive knowledge体系 encompassing theory, algorithms, and applications. It clarifies the intrinsic connections between Lipschitz properties and core challenges in deep learning, offering a unified reference framework and a solid foundation for future theoretical investigations and practical advancements.

Abstract

Lipschitz continuity is a fundamental property of neural networks that characterizes their sensitivity to input perturbations. It plays a pivotal role in deep learning, governing \textbf{robustness}, \textbf{generalization} and \textbf{optimization dynamics}. Despite its importance, research on Lipschitz continuity is scattered across various domains, lacking a unified perspective. This paper addresses this gap by providing a systematic review of Lipschitz continuity in deep learning. We explore its \textbf{theoretical foundations}, \textbf{estimation methods}, \textbf{regularization approaches}, and \textbf{certifiable robustness}. By reviewing existing research through the lens of Lipschitz continuity, this survey serves as a comprehensive reference for researchers and practitioners seeking a deeper understanding of Lipschitz continuity and its implications in deep learning.

Problem

• Lipschitz continuity
• deep learning
• robustness
• generalization
• regularization

Innovation

• Lipschitz continuity
• systematic review
• certifiable robustness
• regularization
• theoretical foundations