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 Ising mixed p-spin models,; namely, Hamiltonians HN:ΣN→R Hamming hypercube ΣN={±1}N, which defined by property that {HN(σ)}σ∈ΣN is a centered Gaussian process with covariance E{HN(σ1)HN(σ2)} depending only scalar ⟨σ1,σ2⟩. The asymptotic value optimum maxσ∈ΣNHN(σ) was characterized in terms variational principle known as Parisi formula, first proved Talagrand and, more general setting, Panchenko. structure superlevel sets extremely rich and has been studied number authors. Here, ask whether near optimal configuration σ can be computed polynomial time. We develop message passing algorithm whose complexity per-iteration same order evaluating gradient HN, characterize typical it achieves. When model HN satisfies certain no-overlap gap assumption, for any ε>0, outputs σ∈ΣN such HN(σ)≥(1−ε)maxσ′HN(σ′), high probability. iterations bounded N depends uniquely ε. More generally, regardless assumption holds, achieved given an extended principle, generalizes formula.