Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target functions. We analyze their learnability using an explicit three-layer fitting model trained via layer-wise spectral estimators. Although the target is globally a high-degree polynomial, its compositional structure allows learning to proceed in stages: an intermediate representation reveals structure that is inaccessible at the input level. This reduces learning to simpler spectral estimation problems, well studied in the context of multi-index models, whereas any shallow estimator must resolve all components simultaneously. Our analysis relies on Gaussian universality, leading to sharp separations in sample complexity between two and three-layer learning strategies.

翻译：深度学习方法为何在计算上真正优于浅层方法，这仍然是学习理论中的一个核心开放性问题。我们在受控的高斯高维环境中研究这一问题，重点关注组合目标函数。我们通过使用逐层谱估计器训练的三层显式拟合模型来分析其可学习性。尽管目标函数整体上是高次多项式，但其组合结构允许学习分阶段进行：中间表示揭示了在输入层面无法获取的结构。这将学习简化为更简单的谱估计问题，该问题在多索引模型背景下已有深入研究，而任何浅层估计器都必须同时解析所有分量。我们的分析依赖于高斯普适性，从而在两层与三层学习策略之间实现了样本复杂度的显著分离。