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In this study, we tackle the challenge of outlier-robust predictive modelingusing highly expressive neural networks. Our approach integrates two keycomponents: (1) a transformed trimmed loss (TTL), a computationally efficientvariant of the classical trimmed loss, and (2) higher-order variationregularization (HOVR), which imposes smoothness constraints on the predictionfunction. While traditional robust statistics typically assume low-complexitymodels such as linear and kernel models, applying TTL alone to modern neuralnetworks may fail to ensure robustness, as their high expressive power allowsthem to fit both inliers and outliers, even when a robust loss is used. Toaddress this, we revisit the traditional notion of breakdown point and adapt itto the nonlinear function setting, introducing a regularization scheme via HOVRthat controls the model's capacity and suppresses overfitting to outliers. Wetheoretically establish that our training procedure retains a high functionalbreakdown point, thereby ensuring robustness to outlier contamination. Wedevelop a stochastic optimization algorithm tailored to this framework andprovide a theoretical guarantee of its convergence.

Outlier-robust neural network training: variation regularization meets trimmed loss to prevent functional breakdown