Beyond Lipschitz: Data-Driven Robustness via Discrete Modulus of Continuity

2026-05-27

AI Summary

This work addresses the limitations of traditional Lipschitz-constant-based robustness measures, which are often overly conservative and fail to capture data-dependent, fine-grained model behavior. The authors propose an architecture-agnostic, data-driven framework for robustness evaluation by introducing the Discrete Modulus of Continuity (DMOC) as a nonlinear generalization of Lipschitz continuity, shifting the analytical focus from the model to the data. Requiring no access to internal model structure, the method provides tight robustness bounds via data-adaptive seminorms and establishes theoretical convergence rates. A minibatch approximation algorithm is developed to substantially reduce computational complexity. Experiments demonstrate that DMOC effectively discriminates between training states, identifies underfitting and overfitting, scales efficiently to large-scale datasets such as ImageNet, and achieves state-of-the-art accuracy in Lipschitz constant estimation.

Abstract

Robustness of neural networks is commonly quantified via local or global Lipschitz constants. However, Lipschitz continuity can be overly coarse or overly restrictive as global robustness measure, failing to capture nuanced, data-dependent behavior. We propose a data-driven, architecture-agnostic framework based on the discrete modulus of continuity (DMOC), a non linear generalization of Lipschitz continuity that provides a finer notion of robustness. Unlike many existing approaches, DMOC does not require access to model internals and instead evaluates regularity relative to the data distribution. This shifts the focus from the model to the data, which provide a data-driven baseline of regularity against which the network's robustness is assessed. We establish convergence results for DMOC-induced seminorms with explicit data-driven rates in terms of the separation distance, and introduce a scalable minibatch algorithm that reduces the quadratic cost of exact computation, enabling application to large-scale data sets such as ImageNet. Empirically, DMOC serves as an architecture independent diagnostic: it distinguishes trained from untrained networks, reveals underfitting and overfitting regimes, and yields, as a special case, tight Lipschitz estimates comparable to state-of-the-art method such as ECLipsE and ECLipsE-fast.

Problem

• robustness
• Lipschitz continuity
• data-driven
• modulus of continuity
• neural networks

Innovation

• Discrete Modulus of Continuity
• Data-Driven Robustness
• Lipschitz Constant
• Model-Agnostic
• Scalable Minibatch Algorithm