ResearchPod Summary
Learning Boolean circuits of constant depth (AC0) is a foundational problem in computational learning theory. While efficient learning is well-understood under uniform or product distributions, extending these results to highly correlated Gibbs distributions—common in physics and network science—has been difficult. This paper addresses the challenge of learning AC0 functions under Gibbs distributions on general bounded-degree graphs, moving beyond restrictive geometric assumptions like polynomial growth.
The authors utilize a framework based on the connection between sampling and learning. They define a 'Samp-InvSamp' pair, where 'Samp' maps random bits to a configuration following the Gibbs distribution, and 'InvSamp' performs the reverse. By using a systematic-scan Glauber dynamics as the sampler, the authors demonstrate that if the dynamics can be computed locally (i.e., determining a spin value by querying only a small number of random bits), then the Gibbs distribution can be approximated by a low-degree polynomial. This allows the application of standard Fourier-analytic learning techniques.
The paper establishes that for any Gibbs distribution admitting an efficient local sampler, AC0 functions can be learned in quasipolynomial time. The authors apply this to the hard-core model and the Ising model on arbitrary bounded-degree graphs. Specifically, they show that their learning algorithm succeeds in regimes approaching the classical Dobrushin threshold. This result is significant because it demonstrates that local samplability is a sufficient condition for learning, effectively decoupling the learning problem from the global geometric structure of the underlying graph.
This work bridges the gap between the theory of local computation algorithms and supervised learning. By showing that local samplers can be used to construct low-degree approximations, the authors provide a robust toolset for learning in complex, dependent systems. This expands the reach of learning theory to a wider class of probabilistic graphical models, offering a path forward for analyzing data generated by systems with long-range dependencies.
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