Statistical and Algorithmic Thresholds in Spin Glasses

Abstract

This thesis studies spin glasses, disordered complex systems originating in statistical physics. Such systems model optimization, sampling, and inference problems from probability and statistics, which are of fundamental importance to modern data science. In particular, spin glasses provide natural examples of random, high-dimensional, and often highly non-convex cost or log-likelihood functions, making them an excellent testing ground for such questions. Part I of this thesis studies statistical properties of these models. Chapter 2 identifies the storage capacity of the Ising perceptron, a simple model of a neural network, subject to a numerical condition. This gives a conditional proof of a 1989 conjecture of Krauth and M´ezard. Chapter 3 gives a new proof of the celebrated Parisi formula for the free energy of the spherical mean-field spin glass, which was first proved by Talagrand and in more generality by Panchenko. Our proof takes a simpler modular approach, drawing on recent advances in spin glass free energy landscapes due to Subag. Chapter 4 characterizes the topology trivialization phase transition of multi-species spherical spin glasses and shows that lowtemperature Langevin dynamics finds the ground state in the topologically trivial regime; the latter result is new even in the single-species setting. Part II of this thesis concerns algorithms for optimization and sampling problems on spin glasses. Chapter 5 studies the problem of optimizing the Hamiltonian of a multi-species spherical spin glass. Our main result exactly characterizes the maximum value attainable by a class of algorithms that are suitably Lipschitz in the disorder. This class includes gradient-based algorithms and Langevin dynamics on constant time scales, and in particular includes the best algorithm known for this problem. This chapter is part of a series of works where we establish exact algorithmic thresholds using the branching overlap gap property (OGP), a landscape property introduced in our earlier work (which appears in our S.M. thesis). In this chapter, we develop a more robust way to establish the branching OGP that does not require Guerra’s interpolation; this allows our method to be applied to models well beyond the (single-species) mean-field spin glass we previously considered. Chapters 6 and 7 study sampling from the Gibbs measure of a spherical mean-field spin glass. Chapter 6 develops a sampling algorithm based on simulating Eldan’s stochastic localization scheme, while Chapter 7 analyzes simulated annealing of Langevin dynamics. We prove both algorithms succeed for inverse temperatures up to a stochastic localization threshold. Chapter 6 gives the first stochastic localization-based sampler with a guarantee of vanishing total variation error, improving on earlier algorithms with vanishing Wasserstein error. Chapter 7 provides the first provable guarantees for a Markov chain in this model beyond the uniqueness threshold, where mixing from worst-case initialization is provably slow.