# Online Regularized Learning Algorithm for Functional Data

arXiv: https://arxiv.org/abs/2211.13549 · HTML: https://mlsanity.com/paper/2211.13549/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 24 Nov 2022.

## Builds on (most cited first, 4 of 4)

- [Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes](https://mlsanity.com/paper/1805.10074/) (2018)
- [Beating SGD Saturation with Tail-Averaging and Minibatching](https://mlsanity.com/paper/1902.08668/) (2019)
- [Capacity dependent analysis for functional online learning algorithms](https://mlsanity.com/paper/2209.12198/) (2022)
- [Fast and Strong Convergence of Online Learning Algorithms](https://mlsanity.com/paper/1710.03600/) (2017)

## Built on by (most cited first, 2 of 2)

- [Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decay](https://mlsanity.com/paper/2602.06797/) (2026)
- [Distributed Gradient Descent for Functional Learning](https://mlsanity.com/paper/2305.07408/) (2023)

## Abstract (arXiv)

In recent years, functional linear models have attracted growing attention in statistics and machine learning, with the aim of recovering the slope function or its functional predictor. This paper considers online regularized learning algorithm for functional linear models in reproducing kernel Hilbert spaces. Convergence analysis of excess prediction error and estimation error are provided with polynomially decaying step-size and constant step-size, respectively. Fast convergence rates can be derived via a capacity dependent analysis. By introducing an explicit regularization term, we uplift the saturation boundary of unregularized online learning algorithms when the step-size decays polynomially, and establish fast convergence rates of estimation error without capacity assumption. However, it remains an open problem to obtain capacity independent convergence rates for the estimation error of the unregularized online learning algorithm with decaying step-size. It also shows that convergence rates of both prediction error and estimation error with constant step-size are competitive with those in the literature.