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arXiv · 2206.10217

Optimization of random high-dimensional functions: Structure and algorithms

Antonio Auffinger·Andrea Montanari·Eliran Subag
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Abstract

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.

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Antonio Auffinger, Andrea Montanari, Eliran Subag. 2022-06-21. Optimization of random high-dimensional functions: Structure and algorithms. https://arxiv.org/abs/2206.10217

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