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Convergence rates of kernel conjugate gradient for random design regression

• We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeledWe prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.  • premath08.pdf Author: Gilles BlanchardGND, Nicole Krämer urn:nbn:de:kobv:517-opus4-94195 2193-6943 (online) Preprints des Instituts für Mathematik der Universität Potsdam (5 (2016) 8) Universitätsverlag Potsdam Potsdam Preprint English 2016 2016 Universität Potsdam Universitätsverlag Potsdam 2016/08/08 conjugate gradient; minimax convergence rates; nonparametric regression; partial least squares; reproducing kernel Hilbert space 5 8 31 Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 62-XX STATISTICS / 62Gxx Nonparametric inference / 62G08 Nonparametric regression 62-XX STATISTICS / 62Gxx Nonparametric inference / 62G20 Asymptotic properties 62-XX STATISTICS / 62Lxx Sequential methods / 62L15 Optimal stopping [See also 60G40, 91A60] Universitätsverlag Potsdam Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2016 Keine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht