One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization ability and the modern deep learning paradigm of pretraining and finetuneing. However, this feature learning process remains poorly understood from a theoretical perspective, with existing analyses largely restricted to two-layer networks. In this work we show that three-layer neural networks have provably richer feature learning capabilities than two-layer networks. We analyze the features learned by a three-layer network trained with layer-wise gradient descent, and present a general purpose theorem which upper bounds the sample complexity and width needed to achieve low test error when the target has specific hierarchical structure. We instantiate our framework in specific statistical learning settings -- single-index models and functions of quadratic features -- and show that in the latter setting three-layer networks obtain a sample complexity improvement over all existing guarantees for two-layer networks. Crucially, this sample complexity improvement relies on the ability of three-layer networks to efficiently learn nonlinear features. We then establish a concrete optimization-based depth separation by constructing a function which is efficiently learnable via gradient descent on a three-layer network, yet cannot be learned efficiently by a two-layer network. Our work makes progress towards understanding the provable benefit of three-layer neural networks over two-layer networks in the feature learning regime.

翻译：深度学习理论中的核心问题之一是理解神经网络如何学习层次化特征。深度网络提取显著特征的能力，对其卓越的泛化性能以及现代深度学习中的预训练与微调范式至关重要。然而，从理论角度而言，这一特征学习过程仍缺乏深入理解，现有分析主要局限于两层网络。本文证明，三层神经网络在特征学习能力上具有相较于两层网络可证明的显著优势。我们分析了通过逐层梯度下降训练的三层网络所学到的特征，并提出一个通用定理，该定理给出了当目标函数具有特定层次结构时，实现低测试误差所需样本复杂度和网络宽度的上界。我们将该框架应用于特定统计学习场景——单指标模型与二次特征函数——并证明在后者中，三层网络相较于现有所有两层网络保证实现了样本复杂度的提升。关键的是，这种样本复杂度的提升依赖于三层网络高效学习非线性特征的能力。随后，我们通过构造一个可由三层网络通过梯度下降高效学习、但两层网络无法高效学习的函数，建立了基于优化的具体深度分离。本文工作为理解特征学习机制中三层网络相较于两层网络的可证明优势取得了进展。