An investigation of the hidden structure of states in a mean field spin glass model (original) (raw)

Algorithmic Thresholds in Mean Field Spin Glasses

arXiv: Statistical Mechanics, 2020

Optimizing a high-dimensional non-convex function is, in general, computationally hard and many problems of this type are hard to solve even approximately. Complexity theory characterizes the optimal approximation ratios achievable in polynomial time in the worst case. On the other hand, when the objective function is random, worst case approximation ratios are overly pessimistic. Mean field spin glasses are canonical families of random energy functions over the discrete hypercube −1,+1N\{-1,+1\}^N−1,+1N. The near-optima of these energy landscapes are organized according to an ultrametric tree-like structure, which enjoys a high degree of universality. Recently, a precise connection has begun to emerge between this ultrametric structure and the optimal approximation ratio achievable in polynomial time in the typical case. A new approximate message passing (AMP) algorithm has been proposed that leverages this connection. The asymptotic behavior of this algorithm has been analyzed, conditional ...

A Probability Density Theory for Spin-Glass Systems

arXiv (Cornell University), 2020

Spin-glass systems are universal models for representing many-body phenomena in statistical physics and computer science. High quality solutions of NP-hard combinatorial optimization problems can be encoded into low energy states of spin-glass systems. In general, evaluating the relevant physical and computational properties of such models is difficult due to critical slowing down near a phase transition. Ideally, one could use recent advances in deep learning for characterizing the low-energy properties of these complex systems. Unfortunately, many of the most promising machine learning approaches are only valid for distributions over continuous variables and thus cannot be directly applied to discrete spin-glass models. To this end, we develop a continuous probability density theory for spin-glass systems with arbitrary dimensions, interactions, and local fields. We show how our formulation geometrically encodes key physical and computational properties of the spin-glass in an instance-wise fashion without the need for quenched disorder averaging. We show that our approach is beyond the mean-field theory and identify a transition from a convex to non-convex energy landscape as the temperature is lowered past a critical temperature. We apply our formalism to a number of spin-glass models including the Sherrington-Kirkpatrick (SK) model, spins on random Erdős-Rényi graphs, and random restricted Boltzmann machines.

On the mean-field spin glass transition

European Physical Journal B, 2008

In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and the Viana-Bray (VB) models, to show that, in the framework of the cavity method, the transition from the annealed regime to a broken replica symmetry phase can be thought of as the failure of the saturability property (detailed explained along the paper) of the overlap fluctuations which act as the order parameters of the theory. We show furthermore how this coincides with the lacking of the commutativity of the infinite volume limit with respect to a, suitably chosen, vanishing perturbing field inducing the transition as prescribed by standard statistical mechanics. This is another step towards a complete theory of disordered systems. As a well known consequence it turns out that the annealed and the replica symmetric regions must coincide, implying that the averaged overlap is zero in this phase. Within our framework the finding of the values of the critical point for the SK and line for the VB becomes available straightforwardly and the method is of a large generality and applicable to several other mean field models

Hierarchical Random Energy Model of a Spin Glass

Physical Review Letters, 2010

We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a nonmean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occurring in the usual mean field Random Energy Model. At variance with mean field, the high temperature branch of the free-energy is non-analytic at the transition point. PACS numbers: 05.10.-a,05.50.+q,75.10.Nr

Classical mutual information in mean-field spin glass models

Physical Review B, 2016

We investigate the classical Rényi entropy Sn and the associated mutual information In in the Sherrington-Kirkpatrick (S-K) model, which is the paradigm model of mean-field spin glasses. Using classical Monte Carlo simulations and analytical tools we investigate the S-K model on the n-sheets booklet. This is obtained by gluing together n independent copies of the model, and it is the main ingredient to construct the Rényi entanglementrelated quantities. We find a glassy phase at low temperature, whereas at high temperature the model exhibits paramagnetic behavior, consistent with the regular S-K model. The temperature of the paramagnetic-glassy transition depends non-trivially on the geometry of the booklet. At high-temperatures we provide the exact solution of the model by exploiting the replica symmetry. This is the permutation symmetry among the fictitious replicas that are used to perform disorder averages (via the replica trick). In the glassy phase the replica symmetry has to be broken. Using a generalization of the Parisi solution, we provide analytical results for Sn and In, and for standard thermodynamic quantities. Both Sn and In exhibit a volume law in the whole phase diagram. We characterize the behavior of the corresponding densities Sn/N, In/N , in the thermodynamic limit. Interestingly, at the critical point the mutual information does not exhibit any crossing for different system sizes, in contrast with local spin models.

A Solvable Mean Field Model of a Gaussian Spin Glass

2011

We introduce a mean field spin glass model with gaussian distribuited spins and pairwise interactions, whose couplings are drawn randomly from a normal gaussian distribution too. We completely control the main thermodynamical properties of the model (free energy, phase diagram, fluctuations theory) in the whole phase space. In particular we prove that in thermodynamic limit the free energy equals its replica symmetric expression.

Energy landscape of the finite-size spherical three-spin glass model

Physical Review E, 2013

We study the 3-spin spherical model with mean-field interactions and Gaussian random couplings. For moderate system sizes of up to 20 spins, we obtain all stationary points of the energy landscape by means of the numerical polynomial homotopy continuation method. On the basis of these stationary points, we analyze the complexity and other quantities related to the glass transition of the model and compare these finite-system quantities to their exact counterparts in the thermodynamic limit.

TOWARD A MEAN FIELD THEORY FOR SPIN GLASSES

We find an approximate solution of the Sherrington-Kirkpatrick model for spin glasses; the internal energy and the specific heat are in very good agreement with the computer simulations, the zero temperature entropy is unfortunately negative, although it is very small.

Mean-field spin glass models from the cavity-ROSt perspective

Contemporary Mathematics, 2007

The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dynamics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal structures which are discussed here in some detail. While the Parisi solution was proven to be correct, through recent works of F. Guerra and M. Talagrand, the reasons for the effectiveness of the Parisi ansatz still remain to be elucidated. We question whether this could be related to the quasi-stationarity of the special subclass of ROSts given by Ruelle's hierarchal 'random probability cascades' (also known as GREM).