Doubly Wild Refitting: Model-Free Evaluation of High Dimensional Black-Box Predictions under Convex Losses

Abstract: We study the problem of excess risk evaluation for empirical risk minimization (ERM) under general convex loss functions. Our contribution is an efficient refitting procedure that computes the excess risk and provides high-probability upper bounds under the fixed-design setting. Assuming only black-box access to the training algorithm and a single dataset, we begin by generating two sets of artificially modified pseudo-outcomes termed wild response, created by stochastically perturbing the gradient vectors with carefully chosen scaling. Using these two pseudo-labeled datasets, we then refit the black-box procedure twice to obtain two corresponding wild predictors. Finally, leveraging the original predictor, the two wild predictors, and the constructed wild responses, we derive an efficient excess risk upper bound. A key feature of our analysis is that it requires no prior knowledge of the complexity of the underlying function class. As a result, the method is essentially model-free and holds significant promise for theoretically evaluating modern opaque machine learning system--such as deep nerral networks and generative model--where traditional capacity-based learning theory becomes infeasible due to the extreme complexity of the hypothesis class.