Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss
It provides theoretical guidance for designing neural network architectures in machine learning, particularly for solving partial differential equations, but is incremental as it builds on existing analysis of generalization error.
This paper investigates whether deeper or wider neural networks achieve better optimal generalization error with Sobolev losses, finding that more parameters favor wider networks while more sample points and higher loss regularity favor deeper networks.
Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks.