Worst-case Nonlinear Regression with Error Bounds

Abstract: This paper proposes an active-learning approach to worst-case nonlinear regression with deterministic error guarantees. Given a known nonlinear function defined over a compact set, we compute a surrogate model, such as a feedforward neural network, by minimizing the maximum absolute approximation error. To address the nonsmooth nature of the resulting minimax problem, we introduce a smooth approximation of the $L_\infty$-type loss that enables efficient gradient-based training. We iteratively enrich the training set by actively learning points of largest approximation error through global optimization. The resulting models admit certified worst-case error bounds, either constant or input-dependent, over the entire input domain. The approach is demonstrated through approximations of nonlinear functions and nonconvex sets, as well as through the derivation of uncertain models of more complex nonlinear dynamics within a given model class, and the approximation of explicit model predictive control laws.