用范数约束神经网络学习稀疏复合函数
Learning Sparse Compositional Functions with Norm-Constrained Neural Networks
Shuo Huang, Lorenzo Fiorito, Lorenzo Rosasco, Tomaso Poggio
AI 速读 基于摘要
该文以参数范数衡量复杂度,为 DAG 表示的稀疏复合函数建立逼近速率与超额风险界,论证深度网络可借层级结构规避维数灾难。
stat.ML · cs.LG
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The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.