Learning Sparse Compositional Functions with Norm-Constrained Neural Networks
Provides theoretical guarantees for overparameterized neural networks learning compositional functions, benefiting the machine learning theory community by extending understanding of why deep networks avoid the curse of dimensionality.
This paper develops a theoretical framework for learning sparse compositional functions with norm-constrained neural networks, establishing approximation rates and excess risk bounds that avoid the curse of dimensionality by exploiting hierarchical compositional structures represented by directed acyclic graphs.
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.