Distributed Learning with Discretely Observed Functional Data

TL;DR

This paper develops distributed spectral algorithms using Sobolev kernels for functional linear regression with discretely observed data, providing tight convergence bounds and advancing theoretical understanding.

Contribution

It introduces a novel combination of distributed spectral algorithms with Sobolev kernels for functional linear regression, with rigorous convergence analysis under discrete observations.

Findings

Established matching upper and lower bounds for convergence in Sobolev norm.

Demonstrated the effectiveness of Sobolev kernels in functional regression.

Enhanced existing theoretical results with new analytical techniques.

Abstract

By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms with Sobolev kernels to tackle the functional linear regression problem. The design and mathematical analysis of the algorithms require only that the functional covariates are observed at discrete sample points. Furthermore, the hypothesis function spaces of the algorithms are the Sobolev spaces generated by the Sobolev kernels, optimizing both approximation capability and flexibility. Through the establishment of regularity conditions for the target function and functional covariate, we derive matching upper and lower bounds for the convergence of the distributed spectral algorithms in the Sobolev norm. This demonstrates that the proposed regularity conditions are reasonable and that the convergence analysis under these conditions is tight, capturing the essential characteristics of functional linear regression. The analytical techniques and estimates developed in this paper also enhance existing results in the previous literature.