Nonlinear Weighted Directed Acyclic Graph and A Priori Estimates for Neural Networks
Yuqing Li, Tao Luo, Chao Ma · SIAM Journal on Mathematics of Data Science · 2022
In an attempt to better understand structural benefits and generalization power of deep neural networks, we first present a novel graph theoretical formulation of neural network models, including fully connected, residual network (ResNet) and densely connected networks (DenseNet). Second, we extend the error analysis of the population risk for a two-layer network [W. E., C. Ma, and L. Wu, Commun. Math. Sci., 17 (2019), pp. 1407--1425] and ResNet [W. E., C. Ma, and Q. Wang, Commun. Math. Sci., 18 (2020), pp. 1755--1774] to DenseNet, and show further that for neural networks satisfying certain mild conditions, similar estimates can be obtained. These estimates are a priori in nature since they depend solely on the information prior to the training process, in particular, the bounds for the estimation errors do not suffer from the curse of dimensionality.