Wild Refitting for Model-Free Excess Risk Evaluation of Opaque ML/AI Models under Bregman Loss

Abstract: We study the problem of evaluating the excess risk of classical penalized empirical risk minimization (ERM) with Bregman losses. We show that by leveraging the recently proposed wild refitting procedure (Wainwright, 2025), one can efficiently upper bound the excess risk through the so-called "wild optimism," without relying on the global structure of the underlying function class. This property makes our approach inherently model-free. Unlike conventional analyses, our framework operates with just one dataset and black-box access to the training procedure. The method involves randomized vector-valued symmetrization with an appropriate scaling of the prediction residues and constructing artificially modified outcomes, upon which we retrain a second predictor for excess risk estimation. We establish high-probability performance guarantees both under the fixed design setting and the random design setting, demonstrating that wild refitting under Bregman losses, with an appropriately chosen wild noise scale, yields a valid upper bound on the excess risk. This work thus is promising for theoretically evaluating modern opaque ML and AI models such as deep neural networks and LLMs, where the model class is too complex for classical learning theory and empirical process techniques to apply.