# Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression

arXiv: https://arxiv.org/abs/2211.10968 · HTML: https://mlsanity.com/paper/2211.10968/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 20 Nov 2022.

## Builds on (most cited first, 16 of 16)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes](https://mlsanity.com/paper/1805.10074/) (2018)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates](https://mlsanity.com/paper/1305.5029/) (2013)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Distributed learning with regularized least squares](https://mlsanity.com/paper/1608.03339/) (2016)
- [Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model](https://mlsanity.com/paper/2006.08212/) (2020)
- [Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms](https://mlsanity.com/paper/1801.07226/) (2018)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Kernel Truncated Randomized Ridge Regression: Optimal Rates and Low Noise Acceleration](https://mlsanity.com/paper/1905.10680/) (2019)
- [Capacity dependent analysis for functional online learning algorithms](https://mlsanity.com/paper/2209.12198/) (2022)
- [Optimal Rates of Distributed Regression with Imperfect Kernels](https://mlsanity.com/paper/2006.16744/) (2020)
- [Coefficient-based Regularized Distribution Regression](https://mlsanity.com/paper/2208.12427/) (2022)
- [Optimal prediction for kernel-based semi-functional linear regression](https://mlsanity.com/paper/2110.15536/) (2021)

## Built on by (most cited first, 8 of 8)

- [On the Optimality of Misspecified Spectral Algorithms](https://mlsanity.com/paper/2303.14942/) (2023)
- [On the Optimality of Misspecified Kernel Ridge Regression](https://mlsanity.com/paper/2305.07241/) (2023)
- [Learning Curves of Stochastic Gradient Descent in Kernel Regression](https://mlsanity.com/paper/2505.22048/) (2025)
- [Distributed Gradient Descent for Functional Learning](https://mlsanity.com/paper/2305.07408/) (2023)
- [On regularized polynomial functional regression](https://mlsanity.com/paper/2311.03036/) (2023)
- [Spectral Algorithms on Manifolds through Diffusion](https://mlsanity.com/paper/2403.03669/) (2024)
- [Distributed Learning with Discretely Observed Functional Data](https://mlsanity.com/paper/2410.02376/) (2024)
- [Theory of Decentralized Robust Kernel-Based Learning](https://mlsanity.com/paper/2506.05215/) (2025)

## Abstract (arXiv)

Previous analysis of regularized functional linear regression in a reproducing kernel Hilbert space (RKHS) typically requires the target function to be contained in this kernel space. This paper studies the convergence performance of divide-and-conquer estimators in the scenario that the target function does not necessarily reside in the underlying RKHS. As a decomposition-based scalable approach, the divide-and-conquer estimators of functional linear regression can substantially reduce the algorithmic complexities in time and memory. We develop an integral operator approach to establish sharp finite sample upper bounds for prediction with divide-and-conquer estimators under various regularity conditions of explanatory variables and target function. We also prove the asymptotic optimality of the derived rates by building the mini-max lower bounds. Finally, we consider the convergence of noiseless estimators and show that the rates can be arbitrarily fast under mild conditions.