VC dimensions for deep neural networks with bounded-rank weight matrices
Primary Area: learning theory
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Keywords: VC dimension, bounded-rank weight matrices network
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Abstract: Deep neural networks (DNNs) have seen immense success in the past decade, yet their lack of interpretability remains a challenge. Recent research on the VC (Vapnik-Chervonenkis) dimension of DNNs has provided valuable insights into the underlying mechanisms of deep learning's powerful generalization capabilities. Understanding the VC dimension offers a promising path toward unraveling the enigma of deep learning, ultimately leading to more interpretable and trustworthy AI systems. In this paper, we study the VC dimensions for DNNs with piecewise polynomial activations and bounded-rank weight matrices. Our main results show that the VC dimensions for DNNs with weight matrices that have bounded rank $r$ are at most $\mathcal{O}(nrL^2\log (nrL))$, where $n$ is the width of the network, and $L$ is the depth of the network. We also construct a ReLU DNN with bounded rank $r$ that can achieve the VC dimension $\Omega(nr)$, which confirms that the upper bound we obtain is nearly tight for large $n$. Based on these bounds, we compare the generalization power in terms of VC dimensions for various different DNN architectures.
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Submission Number: 893
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