Physics of Local Optimization in Complex Disordered Systems

Limited resources motivate decomposing large-scale problems into smaller, "local" subsystems and stitching together the so-found solutions. We explore the physics underlying this approach and discuss the concept of "local hardness," i.e., the complexity of predicting local properties of the solution from local information, for the ground-state problem of both P- and NP-hard spin glasses and related frustrated spin systems. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem. These errors are intimately connected to global critical threshold instabilities, characterized by gapless, avalanchelike excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the computationally much more expensive global minimization. We leverage these findings to introduce a heuristic contraction-based algorithm for globally studying spin-glass ground states. The local solvers further display sharp imprints of the phase transition from the spin-glass to the ferromagnetic phase as the distribution of spin-glass couplings is shifted, as well as characteristic differences for the infinite-range model, implying the existence of specific classes of local hardness. Our findings shed light on how nature may operate solely through local actions at her disposal.