# 2007 OptimalRatesfortheRegularizedLe

- (Caponnetto & De Vito, 2007) ⇒ Andrea Caponnetto, and Ernesto De Vito. (2007). “Optimal Rates for the Regularized Least-Squares Algorithm.” In: Foundations of Computational Mathematics Journal, 7(3). doi:10.1007/s10208-006-0196-8

**Subject Headings:** Regularized Least-Square Algorithm, Supervised Learning Algorithm, Vector-Valued Function.

## Notes

- It proposes a Regression Algorithm.

## Cited By

- http://scholar.google.com/scholar?q=%222007%22+Optimal+Rates+for+the+Regularized+Least-Squares+Algorithm
- http://dl.acm.org/citation.cfm?id=1290530.1290534&preflayout=flat#citedby

### 2008

- (Argyriou et al., 2008) ⇒ Andreas Argyriou, Theodoros Evgeniou, and Massimiliano Pontil. (2008). “Convex Multi-task Feature Learning.” In: Machine Learning Journal, 73(3). doi:10.1007/s10994-007-5040-8

## Quotes

### Abstract

We develop a theoretical analysis of the performance of the regularized least-square algorithm on a reproducing kernel Hilbert space in the supervised learning setting. The presented results hold in the general framework of vector-valued functions; therefore they can be applied to multitask problems. In particular, we observe that the concept of effective dimension plays a central role in the definition of a criterion for the choice of the regularization parameter as a function of the number of samples. Moreover, a complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel. Finally, we give an improved lower rate result describing worst asymptotic behavior on individual probability measures rather than over classes of priors.

### Introduction

In this paper we investigate the estimation properties of the regularized least-squares (RLS) algorithm on a reproducing kernel Hilbert space (RKHS) in the regression setting. Following the general scheme of supervised statistical learning theory, the available input-output samples are assumed to be drawn i.i.d. according to an unknown probability distribution. The aim of a regression algorithm is estimating a particular invariant of the unknown distribution: the *regression function*, using only the available empirical samples.

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## References

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Author | volume | Date Value | title | type | journal | titleUrl | doi | note | year | |
---|---|---|---|---|---|---|---|---|---|---|

2007 OptimalRatesfortheRegularizedLe | Andrea Caponnetto Ernesto De Vito | Optimal Rates for the Regularized Least-Squares Algorithm | 10.1007/s10208-006-0196-8 | 2007 |

Author | Andrea Caponnetto + and Ernesto De Vito + |

doi | 10.1007/s10208-006-0196-8 + |

title | Optimal Rates for the Regularized Least-Squares Algorithm + |

year | 2007 + |