Optimization of Random High-Dimensional Functions: Structure and Algorithms

Replica symmetry breaking postulates that near optima of spin glass Hamiltonians have an ultrametric structure. Namely, near optima can be associated with leaves of a tree, and the Euclidean distance between them corresponds to the distance along this tree. We survey recent progress toward a rigorous proof of this picture in the context of mixed p-spin spin glass models. We focus in particular on the following topics: (i) The structure of critical points of the Hamiltonian; (ii) The realization of the ultrametric tree as near optima of a suitable TAP free energy; (iii) The construction of an efficient optimization algorithm that exploits this picture.