In this paper, we prove the universal consistency of wide and deep ReLU neural network classifiers trained on the logistic loss. We also give sufficient conditions for a class of probability measures for which classifiers based on neural networks achieve minimax optimal rates of convergence. The result applies to a wide range of known function classes. In particular, while most previous works impose explicit smoothness assumptions on the regression function, our framework encompasses more general settings. The proposed neural networks are either the minimizers of the logistic loss or the $0$-$1$ loss. In the former case, they are interpolating classifiers that exhibit a benign overfitting behavior.

翻译：本文证明了在逻辑损失函数下训练的宽深ReLU神经网络分类器具有普适一致性。我们还给出了一类概率测度的充分条件，使得基于神经网络的分类器能够达到极小极大最优收敛速度。该结果适用于广泛的已知函数类。特别地，虽然以往大多数研究对回归函数施加了明确的平滑性假设，但我们的框架涵盖了更一般的设定。所提出的神经网络要么是逻辑损失函数的最小化器，要么是0-1损失函数的最小化器。在前一种情况下，它们表现为插值分类器，展现出良性的过拟合行为。