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Numerous invariant (or equivariant) neural networks have succeeded inhandling invariant data such as point clouds and graphs. However, ageneralization theory for the neural networks has not been well developed,because several essential factors for the theory, such as network size andmargin distribution, are not deeply connected to the invariance andequivariance. In this study, we develop a novel generalization error bound forinvariant and equivariant deep neural networks. To describe the effect ofinvariance and equivariance on generalization, we develop a notion of a\textit{quotient feature space}, which measures the effect of group actions forthe properties. Our main result proves that the volume of quotient featurespaces can describe the generalization error. Furthermore, the bound shows thatthe invariance and equivariance significantly improve the leading term of thebound. We apply our result to specific invariant and equivariant networks, suchas DeepSets (Zaheer et al. (2017)), and show that their generalization bound isconsiderably improved by $\sqrt{n!}$, where $n!$ is the number of permutations.We also discuss the expressive power of invariant DNNs and show that they canachieve an optimal approximation rate. Our experimental result supports ourtheoretical claims.

Improved Generalization Bounds of Group Invariant / Equivariant Deep Networks via Quotient Feature Spaces