通过深度ReLU网络进行成对二分类的平滑边界的统计可学习性
Statistical learnability of smooth boundaries via pairwise binary classification with deep ReLU networks
AI总结：
研究在成对二分类设置下，通过深度ReLU网络学习有序多平滑边界的可能性，解决非识别问题以提升学习能力。
摘要：
非参数估计平滑边界的主题在传统设置中广泛研究，该设置观察到单个协变量和响应变量的配对。然而，这种传统设置常面临数据收集成本的问题。近年来，二分类学习算法的发展使得可以观察到配对协变量和表示协变量之间统计关系的二元变量。本文在成对二分类设置下，理论研究了有序多平滑边界的可学习性。其中一个挑战性问题是平滑子集顺序的非识别性，导致配对二分类设置下平滑边界的泛化能力和可学习性之间存在差距。为了解决由于这种非识别性直接带来的挑战，我们开发了一种使用给定向量值函数类的局部化论证的证明方法。 Consequently, 我们证明了通过定义为局部化深度ReLU网络类的成对二分类算法，某些有序多平滑边界是可学习的。
英文摘要：
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 learnability of ordered multiple smooth boundaries under a pairwise binary classification setting. One of the challenging problems is the non-identifiability issue on the order of smooth subsets, which yields the gap between the generalizability and the learnability of smooth boundaries in the pairwise binary classification setting. To deal with the challenges due to this non-identifiability directly, we develop a proof method using a localization argument of the given vector-valued function class. Consequently, we prove that some ordered multiple smooth boundaries are learnable via a pairwise binary classification algorithm defined with a localized class of deep ReLU networks.
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