Distributed Learning of Conditional Quantiles in the Reproducing Kernel Hilbert Space

We study distributed learning of nonparametric conditional quantiles with Tikhonov regularization in a reproducing kernel Hilbert space (RKHS). Although distributed parametric quantile regression has been investigated in several existing works, the current nonparametric quantile setting poses different challenges and is still unexplored. The difficulty lies in the illusive explicit bias-variance decomposition in the quantile RKHS setting as in the regularized least squares regression. For the simple divide-and-conquer approach that partitions the data set into multiple parts and then takes an arithmetic average of the individual outputs, we establish the risk bounds using a novel second-order empirical process for quantile risk.