Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

Gamarnik, David; Kızıldağ, Eren C.

doi:10.48550/arxiv.1810.05907

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…Finally, perhaps the most intriguing question which remains open is one regarding the genuine hardness of the problem of finding ground states in models exhibiting the OGP. While formal hardness of problems associated with spin glass models is known, in particular it is shown in [GK18] that computing the partition function of the p-spin models is hard on average even in p = 2 regime, these results are established using more "standard" average case hardness proof approaches, and do not take advantage of the intricate solution space topology, such as the one expressed by OGP. At the same time, as of now we have very compelling consistence of the presence of OGP and the apparent hardness of the associated optimization problem in many models.…”

Section: Some Open Questionsmentioning

confidence: 99%

The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models

Gamarnik¹,

Jagannath²

2019

Preprint

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We consider the algorithmic problem of finding a near ground state (near optimal solution) of a p-spin model. We show that for a class of algorithms broadly defined as Approximate Message Passing (AMP), the presence of the Overlap Gap Property (OGP), appropriately defined, is a barrier. We conjecture that when p ≥ 4 the model does indeed exhibits OGP (and prove it for the space of binary solutions). Assuming the validity of this conjecture, as an implication, the AMP fails to find near ground states in these models, per our result. We extend our result to the problem of finding pure states by means of Thouless, Anderson and Palmer (TAP) based iterations, which is yet another example of AMP type algorithms. We show that such iterations fail to find pure states approximately, subject to the conjecture that the space of pure states exhibits the OGP, appropriately stated, when p ≥ 4.

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Section: Some Open Questionsmentioning

confidence: 99%

The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models

Gamarnik¹,

Jagannath²

2019

Preprint

Self Cite

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“…The latter problem is known to be in the #P complexity class which subsumes N P . A similar reduction exists [GK18] for the problem of computing the partition function of a Sherrington-Kirkpatrick model described below, thus implying that computing partition functions for spin glass models is not possible by polynomial time algorithms unless P = #P . Another problem admitting average-case to worst-case reduction is the problem of finding a shortest vector in a lattice [Ajt96].…”

Section: In Search Of the "Right" Algorithmic Complexity Theorymentioning

confidence: 96%

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

Gamarnik

2021

Preprint

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The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means of fast algorithms is not known and often is believed not possible. At the same time the formal hardness of these problems in form of say complexity-theoretic N P -hardness is lacking.In this introductory article a new approach for algorithmic intractability in random structures is described, which is based on the topological disconnectivity property of the set of pair-wise distances of near optimal solutions, called the Overlap Gap Property. The article demonstrates how this property a) emerges in most models known to exhibit an apparent algorithmic hardness b) is consistent with the hardness/tractability phase transition for many models analyzed to the day, and importantly c) allows to mathematically rigorously rule out large classes of algorithms as potential contenders, in particular the algorithms exhibiting the input stability (insensitivity).

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“…Little is known on average-case hardness, when A is drawn from one of the random matrix distributions considered here. As an exception, Gamarnik [Gam18] proved that exact computation of the partition function Z n (β) is hard on average.…”

Section: Further Backgroundmentioning

confidence: 99%

Optimization of the Sherrington-Kirkpatrick Hamiltonian

Montanari

2019

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

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Let A ∈ R n×n be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing σ, Aσ over binary vectors σ ∈ {+1, −1} n . In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any ε > 0, outputs σ * ∈ {−1, +1} n such that σ * , Aσ * is at least (1 − ε) of the optimum value, with probability converging to one as n → ∞. The algorithm's time complexity is C(ε) n 2 . It is a message-passing algorithm, but the specific structure of its update rules is new.As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations.

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Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

Cited by 3 publications

References 11 publications

The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models

The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

Optimization of the Sherrington-Kirkpatrick Hamiltonian

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