Previous analysis of regularized functional linear regression in a reproducing kernel Hilbert space (RKHS) typically requires the target function to be contained in this kernel space. This paper studies the convergence performance of divide-and-conquer estimators in the scenario that the target function does not necessarily reside in the underlying RKHS. As a decomposition-based scalable approach, the divide-and-conquer estimators of functional linear regression can substantially reduce the algorithmic complexities in time and memory. We develop an integral operator approach to establish sharp finite sample upper bounds for prediction with divide-and-conquer estimators under various regularity conditions of explanatory variables and target function. We also prove the asymptotic optimality of the derived rates by building the mini-max lower bounds. Finally, we consider the convergence of noiseless estimators and show that the rates can be arbitrarily fast under mild conditions.

翻译：先前对再生核希尔伯特空间中正则化函数型线性回归的分析通常要求目标函数包含在此核空间中。本文研究当目标函数未必属于底层再生核希尔伯特空间时，分治估计量的收敛性能。作为基于分解的可扩展方法，函数型线性回归的分治估计量能够显著降低时间和内存的算法复杂度。我们开发了积分算子方法，在解释变量和目标函数满足不同正则性条件下，建立了分治估计量预测的精确有限样本上界。通过构建极小极大下界，我们进一步证明了所得收敛速度的渐近最优性。最后，考虑无噪声估计量的收敛性，证明在温和条件下收敛速度可任意快。