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We show the sup-norm convergence of deep neural network estimators with anovel adversarial training scheme. For the nonparametric regression problem, ithas been shown that an estimator using deep neural networks can achieve betterperformances in the sense of the $L2$-norm. In contrast, it is difficult forthe neural estimator with least-squares to achieve the sup-norm convergence,due to the deep structure of neural network models. In this study, we developan adversarial training scheme and investigate the sup-norm convergence of deepneural network estimators. First, we find that ordinary adversarial trainingmakes neural estimators inconsistent. Second, we show that a deep neuralnetwork estimator achieves the optimal rate in the sup-norm sense by theproposed adversarial training with correction. We extend our adversarialtraining to general setups of a loss function and a data-generating function.Our experiments support the theoretical findings.

Sup-Norm Convergence of Deep Neural Network Estimator for Nonparametric Regression by Adversarial Training