Work

The overlap gap property and approximate message passing algorithms for $p$-spin models

Year: 2021

Type: article

Abstract: 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 $p\ge 4$.

Source: The Annals of Probability

DOI: 10.1214/20-aop1448

Authors: David Gamarnik, Aukosh Jagannath

Institutions: Massachusetts Institute of Technology, University of Waterloo

Language: English

Topic: Theoretical and Computational Physics

Subfield: Condensed Matter Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: bronze

Funders: National Science Foundation, Natural Sciences and Engineering Research Council of Canada, Office of Naval Research, Office of International Science and Engineering