Approximation and learning by greedy algorithms

TL;DR

This paper advances the theoretical understanding of greedy algorithms for approximation in Hilbert spaces and demonstrates their effectiveness and efficiency in statistical learning tasks, including universal consistency and convergence rates.

Contribution

It improves convergence rate theory for various greedy algorithms and develops a new framework for their performance in learning, emphasizing computational efficiency.

Findings

Enhanced convergence rate bounds for greedy algorithms.

Construction of universally consistent greedy learning algorithms.

Provable convergence rates for large function classes.

Abstract

We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ 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.