Optimization of the Sherrington-Kirkpatrick Hamiltonian

Montanari, Andrea

doi:10.1109/focs.2019.00087

Cited by 52 publications

(32 citation statements)

References 46 publications

(61 reference statements)

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“…Another algorithmic approach worth mentioning is a message passing algorithm developed by Montanari for optimization of the Sherrington-Kirkpatrick Hamiltonian (Montanari, 2018). The optimization of the Sherrington-Kirkpatrick (SK) Hamiltonian can be viewed as a "gaussian" analogue of max-cut on random graphs; and it was its study that allowed for the asymptotic understanding of max-cut on random d-regular graphs (Dembo et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

“…The optimization of the Sherrington-Kirkpatrick (SK) Hamiltonian can be viewed as a "gaussian" analogue of max-cut on random graphs; and it was its study that allowed for the asymptotic understanding of max-cut on random d-regular graphs (Dembo et al, 2017). This algorithm is proved (Montanari, 2018) (conditioned on a mild statistical physics conjecture) to be asymptotic optimal in the SK setting. While it would be fascinating to understand the performance of an analogue of the algorithm in Montanari (2018) to the max-cut problem in random sparse graphs, this is outside of the scope of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…This algorithm is proved (Montanari, 2018) (conditioned on a mild statistical physics conjecture) to be asymptotic optimal in the SK setting. While it would be fascinating to understand the performance of an analogue of the algorithm in Montanari (2018) to the max-cut problem in random sparse graphs, this is outside of the scope of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Experimental performance of graph neural networks on random instances of max-cut

Yao¹,

Bandeira²,

Villar³

2019

Wavelets and Sparsity XVIII

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This note explores the applicability of unsupervised machine learning techniques towards hard optimization problems on random inputs. In particular we consider Graph Neural Networks (GNNs) -a class of neural networks designed to learn functions on graphs -and we apply them to the max-cut problem on random regular graphs. We focus on the max-cut problem on random regular graphs because it is a fundamental problem that has been widely studied. In particular, even though there is no known explicit solution to compare the output of our algorithm to, we can leverage the known asymptotics of the optimal max-cut value in order to evaluate the performance of the GNNs.In order to put the performance of the GNNs in context, we compare it with the classical semidefinite relaxation approach by Goemans and Williamson (SDP), and with extremal optimization, which is a local optimization heuristic from the statistical physics literature. The numerical results we obtain indicate that, surprisingly, Graph Neural Networks attain comparable performance to the Goemans and Williamson SDP. We also observe that extremal optimization consistently outperforms the other two methods. Furthermore, the performances of the three methods present similar patterns, that is, for sparser, and for larger graphs, the size of the found cuts are closer to the asymptotic optimal max-cut value.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Experimental performance of graph neural networks on random instances of max-cut

Yao¹,

Bandeira²,

Villar³

2019

Wavelets and Sparsity XVIII

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“…[9,38]) and exact [22,27,28]. Although it was recently proven that ground state energies of the S-K model can be approximated efficiently [31], there is no known efficient (i.e. polynomial-time in n) method to compute them exactly.…”

Section: Return "No Solution"mentioning

confidence: 99%

Quantum speedup of branch-and-bound algorithms

Montanaro

2020

Phys. Rev. Research

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Branch-and-bound is a widely used technique for solving combinatorial optimisation problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that determines a lower bound on the cost of any solution in a given subset. Here we describe a quantum algorithm that can accelerate classical branch-and-bound algorithms near-quadratically in a very general setting. We show that the quantum algorithm can find exact ground states for most instances of the Sherrington-Kirkpatrick model in time O(2 0.226n ), which is substantially more efficient than Grover's algorithm.

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“…by analogy with the zero-temperature Sherrington-Kirkpatrick model [ACZ17]. A stronger version of has been shown in several recent works (in mean-eld se ings) to have algorithmic implications [AM18,Sub18,Mon18]. By contrast, results near the satis ability threshold [DSS16,SSZ16] are consistent only with "one-step replica symmetry breaking" ( ).…”

mentioning

confidence: 92%

Breaking of 1RSB in Random Regular MAX-NAE-SAT

Bartha

Sun

Zhang

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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A. For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satis ability transition occurs. For random k-and related models it happens at clause density α α sat 2 k . Just below the threshold, further results suggest that the solution space has a " " structure of a large bounded number of near-orthogonal clusters inside {0, 1} N .In the unsatis able regime α > α sat , it is natural to consider the problem of max-satis ability: violating the least number of constraints.is is a combinatorial optimization problem on the random energy landscape de ned by the problem instance. For a simpli ed variant, the strong refutation problem, there is strong evidence that an algorithmic transition occurs around α N k/2−1 . For α bounded in N, a very precise estimate of the max-sat value was obtained by Achlioptas, Naor, and Peres (2007), but it is not sharp enough to indicate the nature of the energy landscape. Later work (Sen, 2016; Panchenko, 2016) shows that for α very large (roughly, Ω(64 k )) the max-sat value approaches the mean-eld (complete graph) limit: this is conjectured to have an " " structure where near-optimal con gurations form clusters within clusters, in an ultrametric hierarchy of in nite depth inside {0, 1} N . A stronger form of was shown in several recent works to have algorithmic implications (again, in complete graphs). Consequently we nd it of interest to understand how the model transitions from near α sat , to (conjecturally) for large α. In this paper we show that in the random regular kmodel, the description breaks down already above α 4 k /k 3 . is is proved by an explicit perturbation in the parameter space. e choice of perturbation is inspired by the "bug proliferation" mechanism proposed by physicists (Montanari and Ricci-Tersenghi, 2003;Krzakala, Pagnani, and Weigt, 2004), corresponding roughly to a percolation-like threshold for a subgraph of dependent variables.

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Optimization of the Sherrington-Kirkpatrick Hamiltonian

Cited by 52 publications

References 46 publications

Experimental performance of graph neural networks on random instances of max-cut

Experimental performance of graph neural networks on random instances of max-cut

Quantum speedup of branch-and-bound algorithms

Breaking of 1RSB in Random Regular MAX-NAE-SAT

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