Learning $\mathsf{AC}^0$ Under Graphical Models

2026-04-07

AI Summary

This work addresses the long-standing limitation in efficiently learning constant-depth circuits ($\mathsf{AC}^0$) under non-i.i.d. and correlated input distributions, which has traditionally relied on product distributions. The paper presents the first quasipolynomial-time learning algorithm for $\mathsf{AC}^0$ circuits under strongly spatially mixing graphical models with polynomial growth. Departing from conventional Fourier-analytic approaches, the method integrates graphical model sampling, low-degree polynomial approximation, and spatial mixing properties to transfer approximation guarantees from the uniform distribution to correlated settings. This framework not only enables efficient learning of $\mathsf{AC}^0$ circuits under such structured dependencies but also extends to broader function classes—including monotone functions and halfspaces—significantly broadening the theoretical scope of learnability under non-product distributions.

Abstract

In a landmark result, Linial, Mansour and Nisan (J. ACM 1993) gave a quasipolynomial-time algorithm for learning constant-depth circuits given labeled i.i.d. samples under the uniform distribution. Their work has had a deep and lasting legacy in computational learning theory, in particular introducing the $\textit{low-degree algorithm}$. However, an important critique of many results and techniques in the area is the reliance on product structure, which is unlikely to hold in realistic settings. Obtaining similar learning guarantees for more natural correlated distributions has been a longstanding challenge in the field. In particular, we give quasipolynomial-time algorithms for learning $\mathsf{AC}^0$ substantially beyond the product setting, when the inputs come from any graphical model with polynomial growth that exhibits strong spatial mixing. The main technical challenge is in giving a workaround to Fourier analysis, which we do by showing how new sampling algorithms allow us to transfer statements about low-degree polynomial approximation under the uniform setting to graphical models. Our approach is general enough to extend to other well-studied function classes, like monotone functions and halfspaces.

Problem

• AC^0
• graphical models
• correlated distributions
• computational learning theory
• spatial mixing

Innovation

• AC^0 learning
• graphical models
• spatial mixing
• low-degree approximation
• correlated distributions