Optimization of Mean-field Spin Glasses [article]

Ahmed El Alaoui, Andrea Montanari, Mark Sellke
2020 arXiv   pre-print
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:Σ_N→R on the Hamming hypercube Σ_N = {± 1}^N, which are defined by the property that {H_N(σ)}_σ∈Σ_N is a centered Gaussian process with covariance E{H_N(σ_1)H_N(σ_2)} depending only on the scalar product 〈σ_1,σ_2〉. The asymptotic value of the optimum max_σ∈Σ_NH_N(σ) was characterized in terms
more » ... 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 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 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 ε>0, the algorithm outputs σ∈Σ_N such that H_N(σ)> (1-ε)max_σ' H_N(σ'), with high probability. The number of iterations is bounded in N and depends uniquely on ε. 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.
arXiv:2001.00904v1 fatcat:kzbnx4m57zbgxmf7xpl5sdweri