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January 2021 The overlap gap property and approximate message passing algorithms for $p$-spin models
David Gamarnik, Aukosh Jagannath
Ann. Probab. 49(1): 180-205 (January 2021). DOI: 10.1214/20-AOP1448


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 $p\ge 4$, the model does indeed exhibit 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 $p\ge 4$.


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David Gamarnik. Aukosh Jagannath. "The overlap gap property and approximate message passing algorithms for $p$-spin models." Ann. Probab. 49 (1) 180 - 205, January 2021.


Received: 1 December 2019; Revised: 1 May 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1448

Primary: 60K35 , 68Q87
Secondary: 90C26

Keywords: approximate message passing , overlap gap property , Spin glasses

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 1 • January 2021
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