Optimization of the Sherrington-Kirkpatrick Hamiltonian

Abstract: Let $${\boldsymbol A}\in{\mathbb R}^{n\times n}$$ be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing $$\langle{\boldsymbol \sigma},{\boldsymbol A}{\boldsymbol \sigma}\rangle$$ over binary vectors ${\boldsymbol \sigma}\in{+1,-1}<sup>n$. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any $\varepsilon&gt;0$, outputs ${\boldsymbol \sigma}<em>*\in{-1,+1}<sup>n$ such that $$\langle{\boldsymbol \sigma}</em><em>,{\boldsymbol A}{\boldsymbol \sigma}_</em>\rangle$$ is at least $(1-\varepsilon)$ of the optimum value, with probability converging to one as $n\to\infty$. The algorithm's time complexity is $C(\varepsilon)\, n<sup>2$. It is a message-passing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations.