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：深度学习理论的核心问题之一在于理解神经网络如何学习层级化特征。深度网络提取显著特征的能力对其卓越的泛化能力以及现代深度学习范式中的预训练与微调机制至关重要。然而，从理论视角来看，这种特征学习过程仍未被充分理解，现有分析主要局限于两层网络。本研究证明，三层神经网络相比两层网络具有可证明的更丰富特征学习能力。我们分析了通过逐层梯度下降训练的三层网络所学习的特征，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级结构时，实现低测试误差所需样本复杂度与网络宽度的上界。我们在特定的统计学习场景中（单指标模型与二次特征函数）实例化该框架，并证明在后者中，三层网络在样本复杂度上优于所有现有针对两层网络的保证。关键在于，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。我们进一步通过构造一类可在三层网络上通过梯度下降高效学习，却无法被两层网络高效学习的函数，确立了基于优化方法的明确深度分离机制。本研究在理解特征学习机制中三层神经网络相对于两层网络的可证明优势方面取得了进展。