Optimal Convergence Rates of Deep Neural Network Classifiers

In this paper, we study the binary classification problem on $[0,1]^d$ under the Tsybakov noise condition (with exponent $s \in [0,\infty]$) and the compositional assumption. This assumption requires the conditional class probability function of the data distribution to be the composition of $q+1$ vector-valued multivariate functions, where each component function is either a maximum value function or a H\"{o}lder-$\beta$ smooth function that depends only on $d_*$ of its input variables. Notably, $d_*$ can be significantly smaller than the input dimension $d$. We prove that, under these conditions, the optimal convergence rate for the excess 0-1 risk of classifiers is $\left( \frac{1}{n} \right)^{\frac{\beta\cdot(1\wedge\beta)^q}{{\frac{d_*}{s+1}+(1+\frac{1}{s+1})\cdot\beta\cdot(1\wedge\beta)^q}}}$, which is independent of the input dimension $d$. Additionally, we demonstrate that ReLU deep neural networks (DNNs) trained with hinge loss can achieve this optimal convergence rate up to a logarithmic factor. This result provides theoretical justification for the excellent performance of ReLU DNNs in practical classification tasks, particularly in high-dimensional settings. The generalized approach is of independent interest.