Super-fast rates of convergence for Neural Networks Classifiers under the Hard Margin Condition

• URL: http://arxiv.org/abs/2505.08262v1
• Date: Tue, 13 May 2025 06:26:04 GMT
• Title: Super-fast rates of convergence for Neural Networks Classifiers under the Hard Margin Condition
• Authors: Nathanael Tepakbong, Ding-Xuan Zhou, Xiang Zhou,
• Abstract summary: We show that DNNs which minimize the empirical risk with square loss surrogate and $ell_p$ penalty can achieve finite-sample excess risk of order $mathcalOleft(n-alpharight)$ for arbitrarily large $alpha>0$ under the hard-margin condition.<n>The proof relies on a novel decomposition of the excess risk which might be of independent interest.
• Score: 9.993044620455338
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess risk bounds of order $\mathcal{O}\left(n^{-\alpha}\right)$ for arbitrarily large $\alpha>0$ under the hard-margin condition, provided that the regression function $\eta$ is sufficiently smooth. The proof relies on a novel decomposition of the excess risk which might be of independent interest.

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