# Optimal rates for the regularized learning algorithms under general source condition

arXiv: https://arxiv.org/abs/1611.01900 · HTML: https://mlsanity.com/paper/1611.01900/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2016.

## Built on by (most cited first, 11 of 11)

- [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)
- [On the Saturation Effect of Kernel Ridge Regression](https://mlsanity.com/paper/2405.09362/) (2024)
- [Regularized least squares learning with heavy-tailed noise is minimax optimal](https://mlsanity.com/paper/2505.14214/) (2025)
- [Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition](https://mlsanity.com/paper/2204.07856/) (2022)
- [Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer](https://mlsanity.com/paper/2501.10870/) (2025)
- [On the Saturation Effects of Spectral Algorithms in Large Dimensions](https://mlsanity.com/paper/2503.00504/) (2025)
- [Generalization error of spectral algorithms](https://mlsanity.com/paper/2403.11696/) (2024)
- [Towards regularized learning from functional data with covariate shift](https://mlsanity.com/paper/2601.21019/) (2026)
- [Scalable Operator Learning via Nystr"om Approximation With Denoising Applications](https://mlsanity.com/paper/2606.26652/) (2026)
- [Statistical inverse learning and ℓ1-regularization](https://mlsanity.com/paper/2607.07468/) (2026)

## Abstract (arXiv)

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. The convergence issues are studied for general regularization schemes by using the concept of operator monotone index functions in minimax setting. Further we also address the minimum possible error for any learning algorithm.