Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target class. This accuracy is typically measured in terms of a generalization error, that is, an expected value of a given loss function. However, for several applications -- for example in a security-critical context or for problems in the computational sciences -- accuracy in this sense is not sufficient. In such cases, one would like to have guarantees for high accuracy on every input value, that is, with respect to the uniform norm. In this paper we precisely quantify the number of training samples needed for any conceivable training algorithm to guarantee a given uniform accuracy on any learning problem formulated over target classes containing (or consisting of) ReLU neural networks of a prescribed architecture. We prove that, under very general assumptions, the minimal number of training samples for this task scales exponentially both in the depth and the input dimension of the network architecture.

翻译：统计学习理论为在给定目标类上定义的机器学习问题中达到预设精度所需的最少训练样本数提供了理论界限。这种精度通常通过泛化误差来衡量，即某种损失函数的期望值。然而，对于某些应用——例如安全关键领域或计算科学问题——这种意义上的精度并不足够。在此类情形中，人们需要确保每个输入值都能达到高精度，即关于一致范数的精度保证。本文精确量化了任何可设想的训练算法为在包含（或由）特定架构的ReLU神经网络构成的目标类上定义的任何学习问题中保证给定均匀精度所需的训练样本数量。我们证明，在非常一般的假设下，实现此任务所需的最少训练样本数随网络架构的深度和输入维度呈指数级增长。