Tight Lipschitz hardness for optimizing mean field spin glasses

Abstract

We study the problem of algorithmically optimizing the Hamiltonian $H_N$ of a spherical or Ising mixed $p$-spin glass. The maximum asymptotic value $\mathrm{OPT}$ of $H_N/N$ 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 $H_N/N$ up to a value $\mathrm{ALG}$ 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, $\mathrm{ALG}<\mathrm{OPT}$ can also occur, and no efficient algorithm producing an objective value exceeding $\mathrm{ALG}$ is known. We prove that for mixed even $p$-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than $\mathrm{ALG}$ with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of $H_N$. It encompasses natural formulations of gradient descent, AMP, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving $\mathrm{ALG}$ mentioned above. To prove this result, we substantially generalize the OGP framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.