Complexity of Deep Neural Networks from the Perspective of Functional Equivalence

In this paper, we investigate the complexity of feed-forward neural networks by examining the concept of functional equivalence, which suggests that different network parameterizations can lead to the same function. We utilize the permutation invariance property to derive a novel covering number bound for the class of feedforward neural networks, which reveals that the complexity of a neural network can be reduced by exploiting this property. We discuss the extensions to convolutional neural networks, residual networks, and attention-based models. We demonstrate that functional equivalence benefits optimization, as overparameterized networks tend to be easier to train since increasing network width leads to a diminishing volume of the effective parameter space. Our findings offer new insights into overparameterization and have significant implications for understanding generalization and optimization in deep learning.