Tight Low Degree Hardness for Optimizing Pure Spherical Spin Glasses
Mark Sellke
TL;DR
This work analyzes the difficulty of optimizing the pure spherical $p$-spin Hamiltonian on the high-dimensional sphere, showing that constant-degree polynomial algorithms cannot surpass the sharp threshold $\mathsf{ALG}(p)=2\sqrt{\frac{p-1}{p}}$ predicted by physics. The authors introduce a state-following reduction that translates hardness for Lipschitz algorithms into hardness for a broader class of stable algorithms, including low-degree polynomials, by tracking a moving well along a correlated Gaussian ensemble. Central to the argument is the identification of wells as approximate stationary points with a locally concave Hessian and the demonstration that stable well-finding can be converted into a globally Lipschitz procedure, even in the presence of Hessian outlier modes. The result unifies Lipschitz and low-degree hardness within this spin-glass landscape, providing a rigorous barrier at the algorithmic threshold and aligning with physical predictions about the intractability of surpassing $\mathsf{ALG}(p)$ in polynomial time. The techniques—state following on correlated ensembles, geometric control of well-subspaces, and a careful Lipschitz extension—also shed light on algorithmic limitations in related high-dimensional random optimization problems.
Abstract
We prove constant degree polynomial algorithms cannot optimize pure spherical $p$-spin Hamiltonians beyond the algorithmic threshold $\mathsf{ALG}(p)=2\sqrt{\frac{p-1}{p}}$. The proof goes by transforming any hypothetical such algorithm into a Lipschitz one, for which hardness was shown previously by the author and B. Huang.