Complexity of Functional Learning on Some Classes of Multivariate Functions

We study the error of the functional learning on anisotropic Sobolev classes W<sub>p</sub> <sup>r</sup> (Id) and Holder-Nikolskiiclasses H<sub>p</sub> <sup>r</sup>(Id) with respect to the worst case randomized methods and the average case deterministic methods, where 1 les p les infin. Our results show that if p ges 2 then the stochastic and average error bounds are essentially smaller than the deterministic ones. Quantitatively the improvement amounts to the factor n<sup>-1/2</sup>.

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