We study distributed learning with the least squares regularization scheme in
a reproducing kernel Hilbert space (RKHS). By a divide-and-conquer approach,
the algorithm partitions a data set into disjoint data subsets, applies the
least squares regularization scheme to each data subset to produce an output
function, and then takes an average of the individual output functions as a
final global estimator or predictor. We show with error bounds in expectation
in both the $L^2$-metric and RKHS-metric that the global output function of
this distributed learning is a good approximation to the algorithm processing
the whole data in one single machine. Our error bounds are sharp and stated in
a general setting without any eigenfunction assumption. The analysis is
achieved by a novel second order decomposition of operator differences in our
integral operator approach. Even for the classical least squares regularization
scheme in the RKHS associated with a general kernel, we give the best learning
rate in the literature.

使用分布式学习和最小二乘正则化方案，在再生核希尔伯特空间（RKHS）中对分块数据子集应用最小二乘正则化方案生成输出函数，以其平均值作为全局估计器或预测器，具有良好的 $L^2$ 度量和 RKHS 度量误差界限。在我们的积分算子方法中，通过运算符差分的新型二阶分解实现了分析，即使对于与一般核相关联的 RKHS 中的经典最小二乘正则化方案，我们也可以在文献中给出最佳的学习速率。