Learning $\mathsf{AC}^0$ under Locally Sampleable Graphical Models

2026-07-09

AI Summary

This work investigates the efficient learnability of constant-depth circuits ($\mathsf{AC}^0$) under distributions induced by non-uniform graphical models that satisfy a local samplability condition. By combining truncated Glauber dynamics with low-degree Fourier approximation, the authors propose a quasipolynomial-time learning algorithm. This approach overcomes previous restrictions requiring graphical models to exhibit polynomial growth, thereby extending learnability to two-spin systems—including the hard-core and Ising models—on arbitrary bounded-degree graphs. Notably, the algorithm remains effective even in parameter regimes approaching the sampling threshold of these models.

Abstract

The problem of learning constant-depth circuits holds profound implications for computational learning theory. In a seminal result, by introducing the low-degree algorithm, Linial, Mansour, and Nisan (J. ACM 1993) presented a quasipolynomial-time learner for $\mathsf{AC}^0$ under the uniform distribution. However, obtaining comparable learning guarantees for broader classes of correlated distributions has remained a longstanding challenge. Recently, Chandrasekaran, Gaitonde, Moitra, and Vasilyan (arXiv 2026) extended these guarantees to Gibbs distributions on bounded-degree graphical models with both strong spatial mixing and polynomial growth. In this paper, we give a quasipolynomial-time learner for $\mathsf{AC}^0$ under graphical models that admit efficient local samplers, circumventing the polynomial-growth requirement in prior work. The key ingredient is a new low-degree approximation for Gibbs distributions, established by simulating and suitably truncating the classical Glauber dynamics. As applications, this framework yields learners for two-spin systems, including the hard-core model and Ising model, on arbitrary bounded-degree graphs, in regimes approaching their respective sampling thresholds.

Problem

• AC^0 learning
• graphical models
• locally sampleable distributions
• correlated distributions
• computational learning theory

Innovation

• AC^0 learning
• graphical models
• local sampling
• Glauber dynamics
• low-degree approximation