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Distributed Learning with Discretely Observed Functional Data

Jiading Liu, Lei Shi

TL;DR

This work develops a distributed spectral learning framework for functional linear regression with discretely observed covariates, leveraging unanchored Sobolev RKHS to regularize the slope function. By combining Sobolev kernel-based operators with filter-based regularization and a distributed averaging scheme, the authors derive matching upper and lower bounds in Sobolev norm, establishing minimax optimal rates that depend on the regularity parameter $\theta$ and eigen-decay exponent $p$. The analysis handles discretely observed data and Gaussian-type noise, and provides explicit conditions on sampling density, regularization, and data partitioning that guarantee convergence at the rate $N^{-\frac{2\theta}{1+2\theta+p}}$. The results advance understanding of the trade-offs in distributed functional learning and improve upon prior literature by delivering tight rates under realistic sampling and noise models, with extensions to broader settings discussed in the paper. The approach also highlights how real interpolation and effective dimension concepts underpin the convergence analysis of kernel-based distributed estimators in functional data contexts.

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.

Distributed Learning with Discretely Observed Functional Data

TL;DR

This work develops a distributed spectral learning framework for functional linear regression with discretely observed covariates, leveraging unanchored Sobolev RKHS to regularize the slope function. By combining Sobolev kernel-based operators with filter-based regularization and a distributed averaging scheme, the authors derive matching upper and lower bounds in Sobolev norm, establishing minimax optimal rates that depend on the regularity parameter and eigen-decay exponent . The analysis handles discretely observed data and Gaussian-type noise, and provides explicit conditions on sampling density, regularization, and data partitioning that guarantee convergence at the rate . The results advance understanding of the trade-offs in distributed functional learning and improve upon prior literature by delivering tight rates under realistic sampling and noise models, with extensions to broader settings discussed in the paper. The approach also highlights how real interpolation and effective dimension concepts underpin the convergence analysis of kernel-based distributed estimators in functional data contexts.

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.
Paper Structure (18 sections, 18 theorems, 211 equations)

This paper contains 18 sections, 18 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notations
  4. Reproducing Kernel Hilbert Space
  5. Unanchored Sobolev Spaces
  6. Filter Functions
  7. Distributed Spectral Regularization Algorithms
  8. Main Results
  9. Assumptions
  10. Mini-max Lower Bounds
  11. Upper Bounds
  12. Discussions and Comparisons
  13. Discussions on Assumption \ref{['assumption: regularity condition']}
  14. Comparisons with Related Results
  15. Further Discussion
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that Assumption assumption: regularity condition is satisfied with $0\leq \theta<\infty$, Assumption assumption: epsilon2 is satisfied with $\sigma>0$ and the eigenvalues $\{\mu_{\alpha^*,j}\}_{j=1}^\infty$ satisfy $\mu_{\alpha^*,j}\asymp j^{-1/p}$ for some $0< p \leq 1$. Then there holds where the supremum is taken over all $\beta_0 \in \mathscr W^{{\alpha^*},2}(\mathscr T)$ satisfying $

Theorems & Definitions (29)

  • Definition 1: real interpolation
  • Definition 2: filter functions
  • Example 1: Tikhonov regularization
  • Example 2: iterated Tikhonov regularization
  • Example 3: gradient flow
  • Theorem 1: mini-max lower bound
  • Theorem 2: upper bound I
  • Theorem 3: upper bound II
  • Proposition 1
  • Corollary 1
  • ...and 19 more