Renormalised Steepest Descent in Hilbert Space Converges to a Two-Point Attractor |
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Authors: | Luc Pronzato Henry P Wynn Anatoly A Zhigljavsky |
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Institution: | (1) Laboratoire I3S, CNRS/Université de Nice-Sophia Antipolis, bât. Euclide, Les Algorithmes, 2000 route des Lucioles, BP 121, 06903 Sophia-Antipolis Cedex, France;(2) Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK;(3) School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, Wales, UK |
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Abstract: | The result that for quadratic functions the classical steepest descent algorithm in R
d
converges locally to a two-point attractor was proved by Akaike. In this paper this result is proved for bounded quadratic operators in Hilbert space. The asymptotic rate of convergence is shown to depend on the starting point while, as expected, confirming the Kantorovich bounds. The introduction of a relaxation coefficient in the steepest-descent algorithm completely changes its behaviour, which may become chaotic. Different attractors are presented. We show that relaxation allows a significantly improved rate of convergence. |
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Keywords: | asymptotic behaviour steepest descent gradient Hilbert space quadratic operator period-2 cycles |
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