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

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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.

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Notes

  1. Note that the value of C(N,ν′,ν) given in Eq. (5.2) of [20] misses the factor \(\frac{\nu'}{\sqrt{2\nu''}}\).

  2. Note that though the treatment of [20] was formally restricted to covariances of the form ν(y)=y 2+⋯, one can check that inclusion of the linear term in the expansion does not invalidate their formalism.

  3. Note that s ∗ vanishes at the typical energy and becomes negative for e typ <e<e c , i.e. that region is controlled by negative number of replica.

  4. The actual situation in the vicinity of \(\mathcal{E}_{c}\) may appear to be even more complicated, see a note about the announced recent rigorous analysis of the problem by Dembo and Zeitouni in the Conclusion section.

  5. For σ=0 the same calculation can be easily extended to any T and the corresponding tail of the free energy distribution f (coming from the large deviation regime) is found to be \(\sim e^{- \frac{2}{3} (1-T)^{3} (-2 y)^{3/2}}\) for T<1 with f=f typ+Jy, cf. [26]. It would be interesting to investigate how the small deviation distribution of f relates to the Tracy-Widom in the whole phase T<1.

  6. We use that \(\prod_{k=0}^{n-1} \varGamma((n-k)/2) = G(\frac{N+1}{2}) G(1+ \frac {N}{2})/(G(\frac{N-n+1}{2}) G(1+ \frac{N-n}{2}))\) in terms of the Barnes function G(x).

  7. Of course, as noted above the exponent term is correct, i.e. \(\mathcal{L}_{0}(\mathcal{E}) = \psi _{+}(s)\) there.

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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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  1. School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK

    Yan V. Fyodorov

  2. CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231, Paris Cedex, France

    Pierre Le Doussal

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  1. Yan V. Fyodorov
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Correspondence to Yan V. Fyodorov.

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Fyodorov, Y.V., Le Doussal, P. Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization. J Stat Phys 154, 466–490 (2014). https://doi.org/10.1007/s10955-013-0838-1

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