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

While accuracy, robustness, and calibration are all essential for reliable neural networks, they are often studied separately; developing models that satisfy all three simultaneously remains a central challenge. Lipschitz-constrained models guarantee robustness by design, yet the manual selection of the Lipschitz constraint L governs the resulting accuracy-robustness trade-off, and their calibration properties remain largely underexplored. In this work, we highlight a theoretical and empirical link between the enforced Lipschitz constraint and Temperature Scaling, a state-of-the-art calibration method. Specifically, we find that for a given training scheme, there exists a non-trivial value L* that yields an out-of-the-box calibrated network, and that calibration acts as a principled criterion to select a well-defined operating point on the accuracy-robustness Pareto front. Leveraging these insights, we introduce Lipschitz Scaling Training (LiST), a novel training paradigm that iteratively adjusts the global Lipschitz constant to reach this operating point. Through a margin parameter in the training loss, LiST further enables the construction of a fully calibrated Pareto front, allowing users to navigate the accuracy-robustness trade-off while remaining calibrated throughout. At convergence, LiST also enables the reintegration of calibration data into training, improving sample efficiency without sacrificing calibration. We validate LiST on CIFAR-10/100 and Tiny-ImageNet, demonstrating competitive accuracy and robustness against constrained and unconstrained baselines, while remaining calibrated out of the box. Code is available at GitHub.

Randomized Smoothing (RS) provides rigorous robustness guarantees for neural networks without architectural constraints, yet its adoption is limited by extreme computational costs. Standard RS requires tens of thousands of model evaluations per input and forces practitioners to commit to fixed sample sizes a priori. In this work, we present a novel meta-learning framework for anytime-valid certified robustness that adaptively deploys computational resources. By using a lightweight meta-learner to predict image-specific priors for a sequential E-process, we achieve a 20-fold reduction in sample complexity compared to traditional methods while maintaining rigorous statistical guarantees. Beyond raw efficiency, we demonstrate how anytime-validity enables adaptively allocating compute based upon application-specific risk thresholds, a form of resource triage impossible under classic certification frameworks. That this is achievable while also providing similar certification performance demonstrates that our approach provides a pathway for real-time, safety-critical certification deployments.

We investigate the problem of the robustness of a trained neural network to the perturbation of its input values. More specifically, we examine the interplay between the accuracy of the network, as measured by the mean squared error, and robustness. Accordingly, we present a robustness measure, which, with high probability, suggests an upper bound on the mean squared error of the network, with respect to an input data set, for a given perturbation of the input values of the network. The measure we propose is both simple and efficient to compute, treating the neural network as a black box. We provide experimental results on several real-world data sets showing the efficacy of the proposed method. We also introduce the concept of robustness curves, which allows us to further analyse robustness within and between data sets.