Optimization of mean-field spin glasses

Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed $p$-spin models, namely Hamiltonians $H_N:\Sigma_N\to {\mathbb R}$ on the Hamming hypercube $\Sigma_N = \{\pm 1\}^N$, which are defined by the property that $\{H_N({\boldsymbol \sigma})\}_{{\boldsymbol \sigma}\in \Sigma_N}$ is a centered Gaussian process with covariance ${\mathbb E}\{H_N({\boldsymbol \sigma}_1)H_N({\boldsymbol \sigma}_2)\}$ depending only on the scalar product $\langle {\boldsymbol \sigma}_1,{\boldsymbol \sigma}_2\rangle$. The asymptotic value of the optimum $\max_{{\boldsymbol \sigma}\in \Sigma_N}H_N({\boldsymbol \sigma})$ was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration ${\boldsymbol \sigma}$ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of $H_N$, and characterize the typical energy value it achieves. When the $p$-spin model $H_N$ satisfies a certain no-overlap gap assumption, for any $\varepsilon>0$, the algorithm outputs ${\boldsymbol \sigma}\in\Sigma_N$ such that $H_N({\boldsymbol \sigma})\ge (1-\varepsilon)\max_{{\boldsymbol \sigma}'} H_N({\boldsymbol \sigma}')$, with high probability. The number of iterations is bounded in $N$ and depends uniquely on $\varepsilon$. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.