2008/02/01 by Andrew R. Barron, Albert Cohen, Wolfgang Dahmen +2 · 4 citations
Computer Science · Engineering · Mathematics · #Distributed Sensor Networks and Detection Algorithms #Machine Learning and Algorithms #Sparse and Compressive Sensing Techniques #math.ST #msc:41A46 #msc:41A63 #msc:46N30 #msc:62G07 #stat.TH
paper · pdf · doi:10.1214/009053607000000631
published as Annals of Statistics 2008, Vol. 36, No. 1, 64-94 · Published in at http://dx.doi.org/10.1214/009053607000000631 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/02/01 · arxiv created 2008/03/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of approximating a given element f from a Hilbert space H by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118–2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.