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.

翻译：在一项里程碑式的研究中，Linial、Mansour和Nisan（J. ACM 1993）提出了一种拟多项式时间算法，用于在均匀分布下从独立同分布标注样本中学习恒定深度电路。他们的工作对计算学习理论产生了深远且持久的影响，尤其引入了$\textit{低度算法}$。然而，该领域许多结果与技术面临的一个重要批评是对乘积结构的依赖，而这种结构在现实场景中难以成立。如何在更自然的关联分布下获得类似的学习保证，已成为该领域长期存在的挑战。本文中，我们提出了一种拟多项式时间算法，用于在远超乘积设置的环境中学习$\mathsf{AC}^0$，此时输入来自任何具有强空间混合性且呈多项式增长特性的图模型。主要技术难点在于绕开傅里叶分析，我们通过展示新型采样算法如何将均匀设置下的低度多项式近似结论迁移至图模型来突破这一瓶颈。该方法具有足够普适性，可扩展至其他被广泛研究的函数类，如单调函数和半空间函数。