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

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.