Optimization-Induced Dynamics of Lipschitz Continuity in Neural Networks

2025-06-23

AI Summary

Existing theoretical frameworks lack a principled characterization of the interplay between deterministic and stochastic dynamics in Lipschitz constant evolution during SGD training of neural networks. Method: We propose the first “optimization-induced Lipschitz evolution” framework, modeling Lipschitz dynamics via stochastic differential equations and orthogonally decomposing them into gradient-flow and gradient-noise components—projected onto the Jacobian operator norm and Hessian matrix. Contribution/Results: Our framework quantitatively uncovers how key factors—including noise supervision and parameter initialization—regulate the evolution of input robustness. Theoretical predictions align closely with large-scale empirical validation across diverse batch sizes, initialization schemes, and training trajectories. This advances both the interpretability and fundamental understanding of the dynamic mechanisms underlying generalization and stability in deep learning.

Technology Category

• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

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: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Lipschitz continuity characterizes the worst-case sensitivity of neural networks to small input perturbations; yet its dynamics (i.e. temporal evolution) during training remains under-explored. We present a rigorous mathematical framework to model the temporal evolution of Lipschitz continuity during training with stochastic gradient descent (SGD). This framework leverages a system of stochastic differential equations (SDEs) to capture both deterministic and stochastic forces. Our theoretical analysis identifies three principal factors driving the evolution: (i) the projection of gradient flows, induced by the optimization dynamics, onto the operator-norm Jacobian of parameter matrices; (ii) the projection of gradient noise, arising from the randomness in mini-batch sampling, onto the operator-norm Jacobian; and (iii) the projection of the gradient noise onto the operator-norm Hessian of parameter matrices. Furthermore, our theoretical framework sheds light on such as how noisy supervision, parameter initialization, batch size, and mini-batch sampling trajectories, among other factors, shape the evolution of the Lipschitz continuity of neural networks. Our experimental results demonstrate strong agreement between the theoretical implications and the observed behaviors.

Problem

• Modeling temporal evolution of Lipschitz continuity during SGD training
• Identifying key factors driving Lipschitz continuity dynamics
• Exploring impact of noise and hyperparameters on Lipschitz evolution

Innovation

• SDEs model Lipschitz continuity dynamics
• Gradient flows and noise projections analyzed
• Theoretical framework explains training factors impact