Paper: A qualitative theory of dynamical systems for assessing stability in ResNets (ICLR-2024)

Reviewer comment (Writing):
The presentation in this paper is poor. There are way too many typos, grammatical errors, and poorly written sentences; often making it difficult to understand what the authors are trying to convey. The poor writing often leads to sentences that come off as grandiose (whether intended or not), such as: _"What sets this study apart from previous research is the solid theoretical framework bridging deep learning and dynamical systems."_

Author response:
We apologize for the deviation from our original intent, which was to emphasize that "this study aims to establish a framework for deep learning and dynamical systems based on the solid theorem".

Reviewer comment (Novelty):
The related work is incomplete— it is limited to only closely related papers. For example, there's a lot of work connecting the training of deep nets to potential (algorithmic) stability/robustness properties, e.g. https://arxiv.org/pdf/1509.01240.pdf, http://proceedings.mlr.press/v97/du19c/du19c.pdf.

Author response:
Thank you for introducing related works.  
We reviewed papers roughly with colleagues. Looking at the two papers, we found that they were contributing by researching stability based on qualitative or quantitative characteristics and focusing on optimization issues. In particular, We realized that the paper [1] is philosophically similar to the optimization perspective of our paper. We will update to related papers.

Reviewer comment (Experiments):
The Experiments section is weak. The impact of experiments based on MNIST and CIFAR-10 needs to be higher every year to justify using such overused datasets, and I don't think the authors demonstrated that impact here.

Author response:
What you pointed out is correct. We acknowledge that our applied research lacks impact as we only provided trend analyses in Figures 2 and 3 and showcased examples exclusively from CIFAR-10 in Table 3.  

However, we believe that attempting to analyze the complex phenomena occurring while passing through residual blocks will significantly enhance the impact of MNIST and CIFAR-10 datasets for others.  

As a result of Table 3, inspired by the $g$-expansive constant, it is evident that when a feature is not captured well, it can be identified as the same image at some point, even they are different classes image. In the case of MNIST in Table 5, a stable image (handwritten digit 0) was identified through the $g$-shadowing property, demonstrating solid classification as it passes through the residual blocks. Except for this case, the rest of the Table 4 and 5 shows vulnerabilities in a sort of adversarial attacks.  

To show that $g$-expansive is one of the deserve property to analyze, we will include a detailed analysis of MNIST related to $g$-expansivity (we already have all necessary components).

Reviewer comment (Presentation):
The figures in the Applications section (specifically 2 and 3) are poorly constructed and hard to parse, undermining what should be the proof of the ideas laid out in the theory section. These figures do not do a good job of establishing why any of the preceding mathematical formalism is needed and/or what value it brings.

Author response:
In Figures 2 and 3, the $x$-axis represents the corresponding class, and the $y$-axis represents the value of the $g$-expansive constant, respectively. A round dot in the picture indicates the presence of an image of the class represented by a sort of $g$-expansive constant.  

For example, in the bottom-left case (ResNets-18 on CIFAR-10) in Figure 2, you can see that a gray circle representing the class ``Ship" is observed on the first line (class: Airplane). When comparing the distances between pairs of ship images and airplane images, as well as between pairs of airplane images, it indicates that there are instances where the distances for pairs of ship images and airplane images are closer on average than when comparing the distances between airplane images.  

In Figure 3, in the bottom-left case (ResNets-18 on CIFAR-10), the moment when Bird and Deer were close was captured of the existence. Therefore, Table 3 specifically examines blocks and how closely they were distinguished.  

In conclusion, Figure 2 was intended to capture the tendency of whether classes were classified as close overall while passing through the blocks for different class images, and Figure 3 was intended to capture which pair of images were close in any one block.

Reviewer comment (Novelty):
There is uncertainty about whether the presented concepts are indeed new. For example, the proposed concept of "$g$-expansivity": if there is an $\epsilon$ such that
$$
g(x) \neq g(x') \implies d(g(\varphi^n(x)), g(\varphi^n(x')))\geq \epsilon \text{ for some } n\.
$$
The reviewer is skeptical that this definition is new to dynamical systems.

Author response:
Answer: The concept of expansivity goes back to Utz's paper [2] mentioned in the references.  
There are several generalizations of this concept: Positive expansivity (dealing with not homeomorphism but continuous maps), expansive flows and group actions, expansive random systems, and so on. A common point in those definitions is that they all deal with dynamical systems in a given space $X$ (say), actually these definitions need this constraint. The term $g$-expansive appears in the terminal situation where we have a dynamical system on a space $X$ subject to measurement represented by the terminal map $g$ from $X$ to another space $Y$. This definition is new and the closest some other work in the literature have been to ours is Achigar et al paper [3].  
 
We will include this little history into the revised version.

Reviewer comment (Theory):
Distance \(d\) has been used in Definition 1 as a distance between vectors and the same distance \(d\) has been used in Definition 3 as a distance between functions. What distances do authors use?

Author response:
We utilized the supremum distance of continuous maps, as mentioned in the Appendix:  
$$d_{C^0}(\varphi,\hat{\varphi}) = \sup_{x \in X} d(\varphi(x), \hat{\varphi}(x))$$  

This term represents the distance between the solutions of trained ResNets and their perturbed counterparts.

Reviewer comment (Writing):
The statement ``In this context, we attempted to explore compositions of functions within the domain of H, gaining some insights into their behavior at each processing block, even if it diverges from the conventional ResNet structure'' — in what way is this diverging from the ResNet structure?

Author response:
As demonstrated in studies on ODE and ResNet, the theory "Differential Dynamical Systems" (focus on vector fields) is crucial for understanding the conventional ResNet structure. However, applying this theory directly to ResNets can be challenging. As an alternative approach, our aim is to initially explore topological dynamics to gain insights into their behavior at each processing block. In this sense, we expressed that it diverges from the conventional ResNets structure.

Reviewer comment (Reproducibility):
For \(g\)-expansive constant in Definition 1, the definition says that there exists \(n \geq 0\) such that \(d(g(\phi^n(x)), g(\phi^n(x'))) \geq \varepsilon\). How do authors find this \(n\) to calculate \(\varepsilon\) in Table 2?

Author response:
For every test image, we calculate all values across every block and its fully connected layer. So, we can determine $n$ the number of residual blocks, and its corresponding minimum distance, denoted as $\epsilon$.  

In other words, we calculate all distance for pairs of images with layers from $$d(g_{\theta_1} \circ \varphi_{\theta_1}(x), g_{\theta_1} \circ \varphi_{\theta_1}(y))$$ to $$d(g_{\theta_T} \circ \varphi_{\theta_T} \circ \cdots \circ \varphi_{\theta_1}(x), g_{\theta_T} \circ \varphi_{\theta_T} \circ \cdots \circ \varphi_{\theta_1}(y))$$

Reviewer comment (Writing):
Not sure I understand the definition \(Sh_g(\phi,\epsilon)=\sup (\delta : \delta \in Sh_g(\phi,\epsilon))\). Same with \(Top_g\).

Author response:
For $g \in \mathcal{G}$, let $Sh_g : \Phi \times [0, \infty) \to 2^{\mathbb{R^+}}$ be a map defined by $Sh_g(\varphi, \epsilon)$. We can consider $Sh_g(\varphi)$ with the limit inferior for $\epsilon$.  

In the case of experiments where $X$ is a finite set, the definition can be replaced by using minimum and maximum values. First, fix $\epsilon>0$, and we can measure the range of $\delta$ in all calculation and select the maximum value of $\delta$ for $\epsilon$. Finally, we can find $\delta$ with the minimum value of $\epsilon$.

Reviewer comment (Theory):
In Definition 3, Identity map \(Id_X:X\rightarrow X\), but \(H:X\rightarrow 2^X\). How do authors compute distance \(d(H, Id_X)?\)

Author response:
The distance between point and a set can be measured by $$d(H, Id_X) = d(H(x), x) = \inf_{a \in H} d(a,x).$$  

This term signifies the degree of perturbation between the solutions of trained ResNets.  

However, we did not calculate it, and insights can only be gained when the value of topological stability is known based on Theorem 1, considering the shadowing property and Lipschitz constant.

Reviewer comment (Writing):
In the statement of Theorem 1, theorem statements should not include a "we". Also, what is \(Y\) in “g is bounded to bounded”? Also, what does it mean for a hypothesis space to have a Lipschitz constant?

Author response:
A function from one metric space into another is bounded-to-bounded if the preimage of any bounded subset of the range is a bounded subset of the domain. And the expression "Given the hypothesis space $H$ with Lipschitz constant" means that $g$ is Lipschitz. We will rewrite the statement as "If $g$ is Lipschitz and bounded-to-bounded, then ...".

Reviewer comment (Theory):
I think that \(g\)-constants and Lipschitz constants do not depend on the dataset, since it is a property of function \(g\) and \(\phi\). Why do authors separate these constants for different datasets in Table 2?

Author response:
As shown the third assumption for experiments, trained residual blocks with a number of $T>0$ can be considered with the parameted residual blocks as the family of continuous maps with parametrized fully connected layer $g_{\theta_n}$ for $n = 0, \cdots, T$, $$[ (g_{\theta_1} \circ \varphi_{\theta_1}), \cdots, (g_{\theta_T} \circ \varphi_{\theta_T} \circ \cdots \circ \varphi_{\theta_1})].$$  

For example, ResNet-18 and ResNet-50 have $T= 8, 16$, respectively.  

In conclusion, the parameters of the function will vary from model to model, and if trained on different datasets, each parameter will also differ. Thus, all constants, including the Lipschitz constant, depend on the dataset and model.  

However, we present $\varphi$ and $g$ as one parameter for theoretical proof.

Reviewer comment (Theory):
Page 3 — I don't think that "Note that if \(\delta \in Sh_g(\phi, \varepsilon)\) and \(\delta < \delta'\), then \(\delta' \in Sh_g(\phi, \varepsilon)\)." is quite right. It should be \(\delta' \leq \delta\).

Author response:
We appreciate your valuable comments regarding the manuscript's content, font, figures and the typos you pointed out. In particular, we gratitude your observation on our significant mistake, as indicated by $\delta' < \delta$.  

We will make the necessary updates soon.

Reviewer comment (Writing):
Page 6 — typo "Figure 3and Table 3" → "s in Figure 3 and Table 3".

Author response:
We appreciate your valuable comments regarding the manuscript's content, font, figures and the typos you pointed out. In particular, we gratitude your observation on our significant mistake, as indicated by $\delta' < \delta$.  

We will make the necessary updates soon.

Reviewer comment (Writing):
Page 9 — typo "initia l condition" → "initial condition".

Author response:
We appreciate your valuable comments regarding the manuscript's content, font, figures and the typos you pointed out. In particular, we gratitude your observation on our significant mistake, as indicated by $\delta' < \delta$.  

We will make the necessary updates soon.

Reviewer comment (Writing):
The content is very dense, and some parts require to be gone over more than once.

Author response:
We completely agree with the weaknesses mentioned by the reviewer. As you pointed out, the paper should have been written so that we could more easily understand the results. We don't forget your valuable comments during our update of the manuscript. Thank you again.

Reviewer comment (Evaluation):
The experimental results are merely illustrative, and it is hard to understand whether one might prefer this model of stability to the other in existence.

Author response:
We completely agree with the weaknesses mentioned by the reviewer. As you pointed out, the paper should have been written so that we could more easily understand the results. We don't forget your valuable comments during our update of the manuscript. Thank you again.

Reviewer comment (Experiments):
The main question would be about why this model of stability is chosen. There are many others proposed in the literature, and while your experiments illustrate some interesting information about the dataset obtained by this model, it is not clear whether similar information cannot easily be obtained from other models.

Author response:
In previous work, numerous studies have been conducted on ODEs and ResNets. To connect with prior research with the classical theory of Dynamical Systems, we need to establish a direct link between ResNets and Differentiable Dynamical Systems, which involves the study of vector fields. However, this study is yet to be established, and (unexpected) difficulties may arise due to intricacies that require a deeper mathematical rigor and thorough analysis.

In this situation, we tried to venture into exploring compositions of functions within the realm of $ \mathcal{H} $, thus gaining (slightly) insights into their behavior at each processing block, even if it is far from exact ResNet analysis. 

In particular, the reason we focused on ResNets induced by ODEs, as mentioned in the global response, is that a successful exploration of Topological Dynamical Systems and ResNets, which maintains dimensional consistency in passing through blocks, can establish a foundation for research in Differential Dynamical Systems (or dynamics on vector fields).