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. Furthermore, based on the symmetric structure of parameter space, we demonstrate that an appropriate strategy of random parameter initialization can increase the probability of convergence for optimization. We found that overparameterized networks tend to be easier to train in the sense that increasing the width of neural networks leads to a vanishing 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.

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