A Probability Density Theory for Spin-Glass Systems
This work addresses the challenge of applying machine learning to discrete spin-glass models in statistical physics and computer science, offering a novel theoretical framework that could enhance optimization and analysis in these domains.
The paper tackles the difficulty of evaluating physical and computational properties of discrete spin-glass systems, which are hindered by critical slowing down near phase transitions, by developing a continuous probability density theory that geometrically encodes these properties without quenched disorder averaging, showing a transition from convex to non-convex energy landscapes at a critical temperature and applying it to models like the SK model and random restricted Boltzmann machines.
Spin-glass systems are universal models for representing many-body phenomena in statistical physics and computer science. High quality solutions of NP-hard combinatorial optimization problems can be encoded into low energy states of spin-glass systems. In general, evaluating the relevant physical and computational properties of such models is difficult due to critical slowing down near a phase transition. Ideally, one could use recent advances in deep learning for characterizing the low-energy properties of these complex systems. Unfortunately, many of the most promising machine learning approaches are only valid for distributions over continuous variables and thus cannot be directly applied to discrete spin-glass models. To this end, we develop a continuous probability density theory for spin-glass systems with arbitrary dimensions, interactions, and local fields. We show how our formulation geometrically encodes key physical and computational properties of the spin-glass in an instance-wise fashion without the need for quenched disorder averaging. We show that our approach is beyond the mean-field theory and identify a transition from a convex to non-convex energy landscape as the temperature is lowered past a critical temperature. We apply our formalism to a number of spin-glass models including the Sherrington-Kirkpatrick (SK) model, spins on random Erdős-Rényi graphs, and random restricted Boltzmann machines.