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Journal of Statistical Physics

, Volume 154, Issue 1–2, pp 466–490 | Cite as

Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization

  • Yan V. Fyodorov
  • Pierre Le Doussal
Article

Abstract

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N−1)-dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the “trust region subproblem” or “constraint least square problem”. When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number \(\mathcal{N}_{tot}\) of critical (stationary) points in the cost function landscape. In the first regime \(\mathcal{N}_{tot}\) remains of the order N and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.

Keywords

Random matrices Random optimization Random landscapes Large deviations Spin glasses Replica method Tracy-Widom distribution 

Notes

Acknowledgements

We are grateful to Antonio Auffinger for a useful communication related to the content of the paper [20], to Jean-Philippe Bouchaud and Satya Majumdar for lively discussions of results and encouraging interest in the subject, to Peter Forrester and Mark Mezard for bringing a few relevant references to our attention, and to Ofer Zeitouni for informing us on his forthcoming rigorous large-deviation analysis of the problem. YF was supported by EPSRC grant EP/J002763/1 “Insights into Disordered Landscapes via Random Matrix Theory and Statistical Mechanics”. PLD was supported by ANR Grant No. 09-BLAN-0097-01/2.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueen Mary University of LondonLondonUK
  2. 2.CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue LhomondParis CedexFrance

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