While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools from Information-based complexity, recent work by Grohs and Voigtlaender [Journal of the FoCM (2023)] developed a rigorous framework for assessing this so-called "theory-to-practice gap". More precisely, in that work it is shown that there exist functions that can be approximated by neural networks with ReLU activation function at an arbitrary rate while requiring an exponentially growing (in the input dimension) number of samples for their numerical computation. The present study extends these findings by showing analogous results for the ReQU activation function.

翻译：尽管众所周知神经网络具有优异的逼近能力，但从点样本中计算此类逼近仍是一个重大挑战。基于信息复杂度理论工具，Grohs与Voigtlaender近期工作[Journal of the FoCM (2023)]建立了一套严格框架来评估这一所谓的"理论-实践鸿沟"。具体而言，该研究证明存在一类可通过带有ReLU激活函数的神经网络以任意速率逼近的函数，但其数值计算所需的样本数量随输入维度呈指数增长。本研究通过展示ReQU激活函数的类似结论，拓展了上述发现。