A core principle in statistical learning is that smoothness of target functions allows to break the curse of dimensionality. However, learning a smooth function through Taylor expansions requires enough samples close to one another to get meaningful estimate of high-order derivatives, which seems hard in machine learning problems where the ratio between number of data and input dimension is relatively small. Should we really hope to break the curse of dimensionality based on Taylor expansion estimation? What happens if Taylor expansions are replaced by Fourier or wavelet expansions? By deriving a new lower bound on the generalization error, this paper investigates the role of constants and transitory regimes which are usually not depicted beyond classical learning theory statements while that play a dominant role in practice.

翻译：统计学习的一个核心原则是目标函数的平滑性能够打破维度灾难。然而，通过泰勒展开学习平滑函数需要足够多的邻近样本，才能获得高阶导数的有意义的估计，这在数据数量与输入维度之比较小的机器学习问题中似乎难以实现。我们是否真的应该寄希望于基于泰勒展开估计来打破维度灾难？如果泰勒展开被傅里叶展开或小波展开取代，情况又会如何？通过推导泛化误差的一个新下界，本文研究了常数和暂态过程的作用——这些因素通常超出了经典学习理论的论述范围，但在实践中却起着主导作用。