Tight Lipschitz hardness for optimizing mean field spin glasses

Abstract

AbstractWe study the problem of algorithmically optimizing the Hamiltonian of a spherical or Ising mixed ‐spin glass. The maximum asymptotic value of is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing (AMP) algorithms efficiently optimize up to a value given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property (OGP). However, can also occur, and no efficient algorithm producing an objective value exceeding is known. We prove that for mixed even ‐spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than with non‐negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of . It encompasses natural formulations of gradient descent, AMP, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving mentioned above. To prove this result, we substantially generalize the OGP framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.