Generalization bounds for neural ordinary differential equations and deep residual networks

Summary

【Post-rebuttal Comments】 I thank the authors for the discussions after the authors' rebuttal. My questions are appropriately answered. So, I want to keep my score and vote for acceptance. 【Original Comments】 This paper evaluates the generalization performance of the class of functions represented as solutions of time-dependent parametrized ODEs (including time-dependent Neural ODEs) and their discretized variants, that is, Residual Network, that smoothly changes weight parameters with respect to layer index. The obtained generalization bound is $O(1/n^4)$ for the general case and $O(1/n^2)$ for the time-independent case (for sample size $n$.) Numerical experiments were conducted to examine the relationship between the smoothness of weights between successive layers and generalization performance. Also, a learning method was proposed to add the difference in weights between successive layers as a regularization.

Strengths

- The derived results are consistent with classical statistical learning theory. Specifically, setting $K_\Theta=0$ yields the classical $O(1/n^2)$ rate. - Numerical experiments verify the relationship between the smoothness of weights and generalization performance. These results align with the theory. - Comparisons with existing results (Bartlett et al. (2017), Golowich et al. (2018)) are carefully considered. In particular, this study is novel in that it provides generalization bounds for a class of depth-independent models to which Golowich et al. (2018) are not applicable. - The paper is well-written. The organization and mathematical description of the paper are appropriate, and it was easy to understand the paper's main point.

Weaknesses

- To the best of my knowledge, it is rare to impose smoothness between the weights of successive layers in a discretized ResNet model. In addition, the prediction accuracy of those models has yet to be confirmed to be sufficient. Therefore, it is difficult to say that this paper provides a theoretical guarantee for practical models.

Questions

In l.84--91, this paper refers to existing studies of the generalization bound of continuous-time NNs. However, their relationship with this study has yet to be discussed extensively. Is the novelty of this compared with them that the former is a time-independent ODE while the latter is a time-dependent one? Also, is there any study that dealt with time-independent ODEs?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

Limitations are discussed in the experiment section (l.338) and conclusion section (l.346.)

Summary

The authors present a generalization bound for a large class of ODEs in this work. They connect ODEs to residual architectures to control the generalization with the differences between weight matrices.

Strengths

The authors present a theoretical bound, which is something worth highlighting when everybody is just building ad hoc pipelines and then testing them against some benchmark. From this point of view, this work represents a clear advance over other results.

Weaknesses

Even though the results are strong, the consequences and how they can be used are not done in a way that can help the reader to fully understand the applications of these results.

Questions

Have you measured how the bounds fare in time series problems?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

3 good

Contribution

4 excellent

Limitations

None.

Summary

The paper explores the generalization ability of neural ordinary differential equations and deep residual networks through Lipschitz-based complexity arguments. The bound, specifically for the discretized version, involves the maximum magnitude of the difference between successive weight matrices, which is not commonly seen. The paper also uses numerical experiments to investigate how this quantity relates to the generalization capability of deep residual networks. The generalization gap is smaller when the quantity is smaller, but adding the quantity as a penalization term did not improve the prediction performance.

Strengths

[originality] - To the best of my knowledge, this paper is the first to present a generalization error bound for neural ODEs. The proof techniques used in this work are rather standard, but they are combined in new ways to derive the results. - The potential importance of the Lipschitzness of the weights (i.e., the maximal magnitude of the differences between successive weight matrices) is newly proposed. [quality] - The analysis and results seem reasonable, and the mathematical setup is adequately explained. [clarity] - The paper is well-organized and articulately written. Its concise abstract enhances the paper's accessibility and ease of comprehension. [significance] - Neural ODEs and deep residual networks are now among the standard tools in the machine learning community. Therefore, theoretical results that advance our understanding of the behavior of these models are relevant and can improve their reliability for use in various applications. - The analysis in Theorem 1 reveals a term whose convergence rate is O(n^{-1/4}). Corollary 1 exemplifies that the slow-rate term can be eliminated by making the coefficient parameter time-independent in the infinite-dimensional case, partly elucidating the consequences of having time-dependent component in the model.

Weaknesses

[originality] - None in particular [quality] - The paper would benefit from additional citations, particularly where named theorems are mentioned in the main text. For instance, it would be helpful to add citations to the Picard-Lindelof theorem at Line 59 and to Gronwall's inequality at Line 166 for the convenience of the reader. [clarity] - None in particular [significance] - None in particular

Questions

Have you conducted any experiments of Figure 1b using the maximum max-norm of the weights as a penalty term (or a sum of the max-norm of the weights) instead of the squared sum of Frobenius norms? While I acknowledge the equivalence between matrix norms, and although it may not be strictly necessary for the paper, I believe that readers (including myself) would be interested in seeing whether the behavior is different when the quantity that appeared directly in deriving the theoretical guarantee is used. [minor suggestions] - Line 112 “takes values into” → “takes values in” - Line 285 “Corollary 2 (of Theorem 1.1 of Bartlett et al. (2017))” → “Corollary 2 (corollary of Theorem 1.1 of Bartlett et al. (2017))” - Line 319 “a.k.a.” → “i.e.,” may be more appropriate?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

3 good

Contribution

3 good

Limitations

None in particular

Summary

The paper provides a generalization bound for the large class of time-dependent and time-independent neural ODEs. In addition, by leveraging on the connection between neural ODEs and deep residual networks, the paper provides a depth-independent generalization bound for the class of deep residual networks. The bound is compared with some earlier results, showing its novelty. The paper introduces a novel way of controlling the statistical complexity of neural networks and the bound depends on the magnitude of the difference between successive weight matrices. Numerical experiments are provided to show the relationship between this bound and the generalization ability of neural networks.

Strengths

The paper provides a generalization bound for the large class of time-dependent, time-independent neural ODEs and deep residual networks. The bound is compared with some earlier results, showing its novelty. The paper introduces a novel way of controlling the statistical complexity of neural networks and the bound depends on the magnitude of the difference between successive weight matrices. Numerical experiments are provided to show the relationship between this bound and the generalization ability of neural networks.

Weaknesses

Some expressions are not very clear, and some results have too strong conditions. Please see Questions below.

Questions

1. In Proposition 2 and Theorem 1, what do $R_X$ and $R_Y$ mean? The reviewer guesses that $R_X$ is the bound of the set of initial values $x$, but these two notations as well as $X$ and $Y$ do not have clear definitions. 2. A clear definition of $n$ in Theorem 1 is missing. If the data is any $n$ samples, the conclusion of Theorem 1 is obviously wrong. 3. The condition in Corollary 1 is too strong. The reviewer think Neural ODE should contains at least one hidden layer. And the reviewer think that the generalization of Neural ODE with one hidden layer can be obtained by using Theorem 1, by studying the Lipschitz constant of $\sigma(Wx)$. 4. The reviewer thinks that some existing definitions or symbols, such as covering number, subGaussian, should be briefly explained. 5. The numerical experiments show the correlation between the generalization gap and the maximum Lipschitz constant of the weight. However, the generalization gap proved in this paper mainly depends on the number of samples $n$. Could the paper provide experiments to show the correlation between these two quantities as well as the convergence order? 6. Some abnormal result points can be found in the experimental results, such as the point in the bottom left corner of Figure 1. Is there any explanation for this？ 7. Theory and numerical experiments show small Lipschitz constant of the weights reduce the generalization gap. Does this indicate that time independent coefficients ($K_{\Theta}=0$?) have better generalization?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

The authors have adequately addressed the limitations.

Thank you for the response

I have read the authors' responses, and points have been noted. I have no further questions, and my evaluation of the paper did not change.