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 $H_N:\Sigma_N\to {\mathbb R}$ontheHamminghypercube$\Sigma_N = {\pm 1}^N$, which are defined by the property that ${H_N({\boldsymbol \sigma})}_{{\boldsymbol \sigma}\in \Sigma_N}$isacenteredGaussianprocesswithcovariance${\mathbb E}{H_N({\boldsymbol \sigma}_1)H_N({\boldsymbol \sigma}_2)}$ depending only on the scalar product $\langle {\boldsymbol \sigma}1,{\boldsymbol \sigma}2\rangle$\mathrm{.}Theasymptoticvalueoftheoptimum$\max{{\boldsymbol \sigma}\in \Sigma_N}H_N({\boldsymbol \sigma})$ 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 $\mathit{\sigma }$ 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 $\epsilon >0$, the algorithm outputs $\mathit{\sigma }\in {\mathrm{\Sigma }}_N$ 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 $\epsilon$. 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.