Optimization of random high-dimensional functions: Structure and algorithms

Understand

Replica symmetry breaking postulates that near optima of spin glass Hamiltonians have an ultrametric structure.

• Namely, near optima can be associated to leaves of a tree, and the Euclidean distance between them corresponds to the distance along this tree.
• We survey recent progress towards 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 efficient optimization algorithm that exploits this picture.