Potential Hessian Ascent: The Sherrington-Kirkpatrick Model

TL;DR

This paper introduces a novel iterative spectral algorithm based on Hessian ascent to find near-optimal solutions for a complex quadratic optimization problem over the hypercube, resolving a longstanding conjecture.

Contribution

It develops the first spectral algorithm utilizing free probability and eigenspace projectors to approach solutions for the Sherrington-Kirkpatrick model, advancing theoretical understanding.

Findings

Algorithm successfully approximates solutions with high probability.

Uses free probability to construct eigenspace projectors for the Hessian.

Provides groundwork for sum-of-squares certificates related to the Parisi formula.

Abstract

We present the first iterative spectral algorithm to find near-optimal solutions for a random quadratic objective over the discrete hypercube, resolving a conjecture of Subag [Subag, Communications on Pure and Applied Mathematics, 74(5), 2021]. The algorithm is a randomized Hessian ascent in the solid cube, with the objective modified by subtracting an instance-independent potential function [Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]. Using tools from free probability theory, we construct an approximate projector into the top eigenspaces of the Hessian, which serves as the covariance matrix for the random increments. With high probability, the iterates' empirical distribution approximates the solution to the primal version of the Auffinger-Chen SDE [Auffinger et al., Communications in Mathematical Physics, 335, 2015]. The per-iterate change in the modified objective is bounded via a Taylor expansion, where the derivatives are controlled through Gaussian concentration bounds and smoothness properties of a semiconcave regularization of the Fenchel-Legendre dual to the Parisi PDE. These results lay the groundwork for (possibly) demonstrating low-degree sum-of-squares certificates over high-entropy step distributions for a relaxed version of the Parisi formula [Open Question 1.8, arXiv:2401.14383].