The Physics of Local Optimization in Complex Disordered Systems

Abstract: 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., complexity from the local solver perspective, in determining the ground states of both P- and NP-hard spin-glasses and related 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 stem from global critical threshold instabilities, characterized by gapless, avalanche-like excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the more computationally intensive global minimization. These findings shed light on how Nature may operate solely through local actions at her disposal.