This paper quantifies the approximation capability of radial basis function networks (RBFNs) and their applications in machine learning theory. The target is to deduce almost optimal rates of approximation and learning by RBFNs. For approximation, we show that for large classes of functions, the convergence rate of approximation by RBFNs is not slower than that of multivariate algebraic polynomials. For learning, we prove that, using the classical empirical risk minimization, the RBFNs estimator can theoretically realize the almost optimal learning rate. The obtained results underlie the successful application of RBFNs in various machine learning problems.
In the practice of machine learning, one often encounters problems in which noisy data are abundant while the learning targets are imprecise and elusive. To these challenges, most of the traditional learning algorithms employ hypothesis spaces of large capacity. This has inevitably led to high computational burdens and caused considerable machine sluggishness. Utilizing greedy algorithms in this kind of learning environment has greatly improved machine performance. The best existing learning rate of various greedy algorithms is proved to achieve the order of (m/log m)(-1/2), where m is the sample size. In this paper, we provide a relaxed greedy algorithm and study its learning capability. We prove that the learning rate of the new relaxed greedy algorithm is faster than the order m(-1/2). Unlike many other greedy algorithms, which are often indecisive issuing a stopping order to the iteration process, our algorithm has a clearly established stopping criteria.
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