标题： 通过深度ReLU网络的成对二元分类的平滑边界统计可学习性 显示英文标题

标题： Statistical learnability of smooth boundaries via pairwise binary classification with deep ReLU networks

摘要： 非参数估计光滑边界的问题在传统设置中得到了广泛研究，其中观察到的是单个协变量和响应变量的配对数据。 然而，这种传统设置通常会受到数据收集成本的困扰。 近年来，二分类问题的学习算法得到了持续的发展，其中可以观察到协变量对和表示协变量之间统计关系的二元变量。 在本工作中，我们从理论上研究了在考虑成对二分类设置的情况下，是否可以学习多个光滑边界的问题。 我们通过成对协变量的统计依赖性来研究这个问题，以开发一种使用向量值函数的学习算法。 主要定理表明，在深度ReLU网络的一个类中存在一个经验风险最小化算法，该算法可以为由光滑边界定义的指示函数生成一致估计量。 我们还讨论了成对二分类设置与传统设置的不同之处，重点在于函数类的结构条件。 作为副产品，我们将主要定理应用于一个多类非参数分类问题，其中估计性能通过误分类的额外风险来衡量。 显示英文摘要

摘要： The topic of nonparametric estimation of smooth boundaries is extensively studied in the conventional setting where pairs of single covariate and response variable are observed. However, this traditional setting often suffers from the cost of data collection. Recent years have witnessed the consistent development of learning algorithms for binary classification problems where one can instead observe paired covariates and binary variable representing the statistical relationship between the covariates. In this work, we theoretically study the question of whether multiple smooth boundaries are learnable if the pairwise binary classification setting is considered. We investigate the question with the statistical dependence of paired covariates to develop a learning algorithm using vector-valued functions. The main theorem shows that there is an empirical risk minimization algorithm in a class of deep ReLU networks such that it produces a consistent estimator for indicator functions defined with smooth boundaries. We also discuss how the pairwise binary classification setting is different from the conventional settings, focusing on the structural condition of function classes. As a by-product, we apply the main theorem to a multiclass nonparametric classification problem where the estimation performance is measured by the excess risk in terms of misclassification.