Distributed least squares prediction for functional linear regression* *This work was partially supported by the National Natural Science Foundation of China (Grant No. 11871438).

Abstract

To cope with the challenges of memory bottleneck and algorithmic scalability when massive data sets are involved, we propose a distributed least squares procedure in the framework of functional linear model and reproducing kernel Hilbert space. This approach divides the big data set into multiple subsets, applies regularized least squares regression on each of them, and then averages the individual outputs as a final prediction. We establish the non-asymptotic prediction error bounds for the proposed learning strategy under some regularity conditions. When the target function only has weak regularity, we also introduce some unlabelled data to construct a semi-supervised approach to enlarge the number of the partitioned subsets. Results in present paper provide a theoretical guarantee that the distributed algorithm can achieve the optimal rate of convergence while allowing the whole data set to be partitioned into a large number of subsets for parallel processing.