Neither Adversarial Training Nor Purification: Emergent Adversarial Robustness from Oscillatory Predictive Learning

Adversarial Training (AT) is a leading defense against adversarial examples but often suffers from Catastrophic Overfitting (CO) in efficient single-step variants, where robustness to multi-step attacks collapses despite high single-step performance. We address this failure mode with two contributions. First, we formalize Epsilon Overfitting (EO), a perspective in which fixed perturbation magnitudes and directions exacerbate CO, and show that introducing perturbation variability significantly improves robust generalization across different architectures and datasets. Second, we propose PertAlign (Perturbation Alignment), a theoretically grounded, computationally negligible metric that predicts CO onset by measuring gradient alignment across attack stages. Leveraging these insights, we introduce SORA, an adaptive step-size AT method that dynamically adjusts perturbations based on loss surface geometry. SORA consistently prevents CO, achieves state-of-the-art robustness and clean accuracy, and generalizes across datasets and architectures using a single fixed set of hyperparameters, which is essential for applicability in fast AT. Extensive experiments on diverse datasets and architectures show that SORA matches or surpasses the robustness of prior methods while delivering higher clean accuracy and superior efficiency. Code is available at https://github.com/SecondOrderAT/SORA.

Vision models have been found to be susceptible to perturbations such as motion blur induced at runtime by a shaking camera. This impedes their deployment in critical applications since phenomena such as slightly blurred vision might lead to failures, for example an object detector missing objects. While methods such as data augmentation or Adversarial Training can improve empirical robustness, they lack formal safety guarantees, making it difficult to identify and mitigate hidden vulnerabilities. We introduce a novel Certified Training approach that leverages an efficient encoding of convolutional perturbations to train provably robust models. Our method significantly outperforms Adversarial Training, achieving, for example, over 80% robust accuracy against motion blur of reasonable intensity on CIFAR10 while maintaining comparable standard accuracy.

We introduce Hybrid Convolutions with Attention Stochasticity (HyCAS), an adversarial defense that narrows the long-standing gap between provable robustness under L2 certificates and empirical robustness against strong L attacks, while preserving strong generalization across diverse imaging benchmarks. HyCAS unifies deterministic and randomized principles by coupling 1-Lipschitz, spectrally normalized convolutions with two stochastic components, spectral normalized random, projection filters and a randomized attention-noise mechanism, to realize a randomized defense. Injecting smoothing randomness inside the architecture yields an overall <= 2-Lipschitz network with formal certificates. Exten-sive experiments on diverse imaging benchmarks, including CIFAR-10/100, ImageNet-1k, NIH Chest X-ray, HAM10000, show that HyCAS surpasses prior leading certified and empirical defenses, boosting certified accuracy by up to 7.3% (on NIH Chest X-ray) and empirical robustness by up to 3.1% (on HAM10000), without sacrificing clean accuracy. These results show that a randomized Lipschitz constrained architecture can simultaneously improve both certified L2 and empirical L adversarial robustness, thereby supporting safer deployment of deep models in high-stakes applications. Code: https://github.com/misti1203/HyCAS