Koopman-based generalization bound: New aspect for full-rank weights
Yuka Hashimoto, Sho Sonoda, Isao Ishikawa, Atsushi Nitanda, Taiji, Suzuki
TL;DR
This paper introduces a new generalization bound for neural networks using Koopman operators, focusing on full-rank weights and providing a tighter, complementary perspective to existing bounds, especially for orthogonal matrices.
Contribution
It presents a novel Koopman-based generalization bound applicable to full-rank weight matrices, expanding understanding beyond low-rank assumptions and connecting operator theory with neural network generalization.
Findings
Bound is tighter with small condition numbers
Bound is independent of network width for orthogonal weights
Supports that low-rankness isn't the sole factor for generalization
Abstract
We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization…
Peer Reviews
Decision·ICLR 2024 poster
1. This paper proposed a new complexity bound that involves both the norm and determinant of the weight matrices. This bound is particularly useful when the condition numbers of the weight matrices are small. 2. It provides a new perspective on why networks with high-rank weights generalize well. By combining our bound with existing bounds, we can obtain a more comprehensive description of the role of each layer in the network. 3. This paper presented an operator-theoretic approach to analyzin
This paper gives the generalization error bound of neural networks from a novel perspective which sounds very interesting and introduces new tools to generalization analysis. But since I'm not familiar with dynamic-based Koopman operators, I have some concerns that I'd like to see answered by the author. 1. As the author said, Efficient learning algorithms have been proposed by describing the learning dynamics of the parameters of neural networks by Koopman operators. It seems that the author r
The proposed generalization bound is sharp and fills the theoretical gap. Specifically, benefiting from the denominator induced by the Koopman operator, the generalization bound can be sharp when the condition number of the weight matrix is small. What’s more, if the weight matrices are orthogonal, the bound reduces to 1 and is independent of the width of the network. This result explains the generalization ability of neural networks when the weight matrices are full-rank. By contrast, existing
- The authors mainly consider the neural networks with dense layers. I wonder whether these theoretical results can generalize well to neural networks with other structures such as convolution. A simple explanation is recommended. - The experimental results on MNIST validate the effectiveness of the induced regularization term. Can it boost model performance on datasets with larger scales such as CIFAR? - Besides, there are some typos. For example, - In the introduction part, "depth of the net
This is an **extremely interesting direction** which, to the best of my knowledge, is underexplored. The results presented in this paper have the potential to be of great interest to the community for further research, since it approaches the problem from an entirely different perspective: instead of controlling the function class capacity through norms of the weights or number of parameters, properties of the learned functions in terms of their smoothness over the inputs are used instead. This
The writing is quite crisp and abstract, sometimes to the detriment of precision. I think the results make sense in that bounds for the norms of the neural network are indeed given, but the interpretation in terms of asymptotics and the lack of dependence on the number of parameters don't fully make sense due to the presence of obscure quantities with unclear asymptotic behavior. To be honest, I am **not completely convinced of the correctness** of the final conclusions from a mathematical st
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Taxonomy
TopicsModel Reduction and Neural Networks · Tensor decomposition and applications · Sparse and Compressive Sensing Techniques