2020/09/30 by Alberto Bietti, Francis Bach, Bietti, Alberto +1 · 3 citations
Computer Science · Physics and Astronomy · #Advanced Graph Neural Networks #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2009.14397
openalex publication_date 2020/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show that for ReLU activations, the kernels derived from deep fully-connected networks have essentially the same approximation properties as their shallow two-layer counterpart, namely the same eigenvalue decay for the corresponding integral operator. This highlights the limitations of the kernel framework for understanding the benefits of such deep architectures. Our main theoretical result relies on characterizing such eigenvalue decays through differentiability properties of the kernel function, which also easily applies to the study of other kernels defined on the sphere.