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Approximation and learning by greedy algorithms. (English) Zbl 1138.62019

Summary: 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 of W. S. Lee et al. [IEEE Trans. Inf. Theory 42, No. 6, 2118–2132 (1996; Zbl 0874.68253)] 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.

##### MSC:
 62G08 Nonparametric regression 68T05 Learning and adaptive systems 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A63 Multidimensional approximation problems 46N30 Applications of functional analysis in probability theory and statistics 65C60 Computational problems in statistics