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 HN:SigmaN→ mathbb R on the Hamming hypercube SigmaN = ± 1N, which are defined by the property that HN(boldsymbol σ)_boldsymbol σ∈ SigmaN is a centered Gaussian process with covariance mathbb EHN(boldsymbol σ1)HN(boldsymbol σ2) depending only on the scalar product langle boldsymbol σ1,boldsymbol σ2rangle. The asymptotic value of the optimum max_boldsymbol σ∈ SigmaNHN(boldsymbol σ) 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 σ 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 HN, and characterize the typical energy value it achieves. When the p-spin model HN satisfies a certain no-overlap gap assumption, for any varepsilon>0, the algorithm outputs boldsymbol σ∈SigmaN such that HN(boldsymbol σ)≥ (1-varepsilon)max_boldsymbol σ' HN(boldsymbol σ'), 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.