The Asymptotics of the Clustering Transition for Random Constraint Satisfaction Problems

Budzynski, Louise; Semerjian, Guilhem

doi:10.1007/s10955-020-02635-8

Cited by 4 publications

(3 citation statements)

References 31 publications

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“…Our main interest is in investigating the possibility of finding solutions efficiently when κ > κ SAT (α). Mézard and Krauth [25] showed in their seminal work using the non-rigorous replica method [31] that the solution landscape of the one-sided perceptron (where there is no absolute value in the constraints (1)) is dominated by isolated solutions lying at large mutual Hamming distances, a structure sometimes called 'frozen replica symmetry breaking' [17,22,23,29,40]. From the mathematics point of view, the frozen replica symmetry breaking prediction was proven true for the SBP in works by Perkins and Xu [34] and Abbé et al [1], who showed that for all κ > κ SAT (α), a solution drawn uniformly at random from S(G, κ) is isolated with high probability, in the sense that it is separated from any other solution by a Hamming distance linear in N.…”

Section: Background and Motivationmentioning

confidence: 99%

“…In the existing statistical physics literature, using the replica method on the one-step replica symmetry breaking level, researchers so far have not found clusters of solutions of extensive entropy in the binary perceptron. This is a point of concern as this method is supposed to count all clusters of solutions corresponding to the TAP/AMP fixed points, including the rare nonequilibrium ones [17,26,30,32,38]. This a priori casts doubt on the efficacy of the replica method and the validity of its predictions for the number of clusters of a given size, since the method misses a large part of the phase space (unless some explicit conditioning is done as in [9,10]).…”

Section: The 1rsb Computation Of the Complexity Curvementioning

confidence: 99%

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On the atypical solutions of the symmetric binary perceptron

Barbier,

El Alaoui,

Krzakala

et al. 2024

J. Phys. A: Math. Theor.

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We study the random binary symmetric perceptron problem, focusing on the behavior of rare high-margin solutions. While most solutions are isolated, we demonstrate that these rare solutions are part of clusters of extensive entropy, heuristically corresponding to non-trivial fixed points of an approximate message-passing algorithm. We enumerate these clusters via a local entropy, defined as a Franz-Parisi potential, which we rigorously evaluate using the first and second moment methods in the limit of a small constraint density $\alpha$ (corresponding to vanishing margin $\kappa$) under a certain assumption on the concentration of the entropy. This examination unveils several intriguing phenomena:i) We demonstrate that these clusters have an entropic barrier in the sense that the entropy as a function of the distance from the reference high-margin solution is non-monotone when $\knew \le 1.429 \sqrt{-\alpha/\log{\alpha}}$, while it is monotone otherwise, and that they have an energetic barrier in the sense that there are no solutions at an intermediate distance from the reference solution when $\knew \le 1.239 \sqrt{-\alpha/ \log{\alpha}}$. The critical scaling of the margin $\kappa$ in $\sqrt{-\alpha/\log\alpha}$ corresponds to the one obtained from the earlier work of Gamarnik et al. for the overlap-gap property, a phenomenon known to present a barrier to certain efficient algorithms. ii) We establish using the replica method that the complexity (the logarithm of the number of clusters of such solutions) versus entropy (the logarithm of the number of solutions in the clusters) curves are partly non-concave and correspond to very large values of the Parisi parameter, with the equilibrium being reached when the Parisi parameter diverges.

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Section: Background and Motivationmentioning

confidence: 99%

Section: The 1rsb Computation Of the Complexity Curvementioning

confidence: 99%

On the atypical solutions of the symmetric binary perceptron

Barbier,

El Alaoui,

Krzakala

et al. 2024

J. Phys. A: Math. Theor.

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“…Statistical physics models on random hypergraphs is another growing area of research, see e.g. [70][71][72][73][74].…”

Section: Introductionmentioning

confidence: 99%

Random recursive hypergraphs

Krapivsky

2023

J. Phys. A: Math. Theor.

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Random recursive hypergraphs grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat expressions via harmonic numbers, Bernoulli numbers, Eulerian numbers, and Stirling numbers of the first kind. Natural deformations of random recursive hypergraphs give rise to fascinating models of growing random hypergraphs.

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Learning from survey propagation: a neural network for MAX-E-3-SAT

Marino

2021

Mach. Learn.: Sci. Technol.

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Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, it suffices to get reasonably good solutions for all (or even most) instances in practice. This paper presents a new algorithm for computing approximate solutions in Θ(N) for the maximum exact 3-satisfiability (MAX-E-3-SAT) problem by using supervised learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random conjunctive normal form instances of the MAX-E-3-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art maximum satisfiability solvers, it can solve substantially larger and more complicated problems than it ever saw during training.

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The Asymptotics of the Clustering Transition for Random Constraint Satisfaction Problems

Cited by 4 publications

References 31 publications

On the atypical solutions of the symmetric binary perceptron

On the atypical solutions of the symmetric binary perceptron

Random recursive hypergraphs

Learning from survey propagation: a neural network for MAX-E-3-SAT

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