This paper studies the binary classification of unbounded data from ${\mathbb R}^d$ generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain $\unicode{x2013}$ for the first time $\unicode{x2013}$ non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension $d$, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems.

翻译：本文研究利用深度ReLU神经网络对${\mathbb R}^d$中高斯混合模型(GMMs)生成的无界数据进行二元分类问题。我们首次获得了在无模型参数限制条件下分类超额风险(超额误分类错误)的非渐近上界与收敛速度。所推导的收敛速度不依赖于维度$d$，证明深度ReLU网络能够克服分类中的维度灾难。现有大多数分类算法的泛化分析研究依赖于有界域，而我们通过利用高斯分布的解析性和快速衰减性考虑了无界域情况。为便于分析，我们给出了深度ReLU网络对一般解析函数的全新逼近误差界，该结果可能具有独立研究价值。高斯分布能很好地适应语音、图像和文本等实际应用场景中的数据建模；我们的结果从理论上验证了深度神经网络在实践分类问题中的高效性。