Error bounds for approximations with deep ReLU networks
This work addresses the theoretical understanding of neural network efficiency for researchers in approximation theory and machine learning, providing incremental insights into depth advantages.
The paper tackles the problem of approximating smooth functions with deep ReLU networks, establishing rigorous upper and lower bounds on network complexity in Sobolev spaces, and proves that deep networks are more efficient than shallow ones, with specific results for 1D Lipschitz functions using adaptive depth-6 architectures.
We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In particular, we prove that deep ReLU networks more efficiently approximate smooth functions than shallow networks. In the case of approximations of 1D Lipschitz functions we describe adaptive depth-6 network architectures more efficient than the standard shallow architecture.