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

• URL: http://arxiv.org/abs/2307.04042v1
• Date: Sat, 8 Jul 2023 20:24:14 GMT
• Title: Sup-Norm Convergence of Deep Neural Network Estimator for Nonparametric Regression by Adversarial Training
• Authors: Masaaki Imaizumi
• Abstract summary: We show the sup-norm convergence of deep neural network estimators with a novel adversarial training scheme. A deep neural network estimator achieves the optimal rate in the sup-norm sense by the proposed adversarial training with correction.
• Score: 5.68558935178946
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show the sup-norm convergence of deep neural network estimators with a novel adversarial training scheme. For the nonparametric regression problem, it has been shown that an estimator using deep neural networks can achieve better performances in the sense of the $L2$-norm. In contrast, it is difficult for the neural estimator with least-squares to achieve the sup-norm convergence, due to the deep structure of neural network models. In this study, we develop an adversarial training scheme and investigate the sup-norm convergence of deep neural network estimators. First, we find that ordinary adversarial training makes neural estimators inconsistent. Second, we show that a deep neural network estimator achieves the optimal rate in the sup-norm sense by the proposed adversarial training with correction. We extend our adversarial training to general setups of a loss function and a data-generating function. Our experiments support the theoretical findings.

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