What training reveals about neural network complexity

Abstract: This work explores the Benevolent Training Hypothesis (BTH) which argues that the complexity of the function a deep neural network (NN) is learning can be deduced by its training dynamics. Our analysis provides evidence for BTH by relating the NN's Lipschitz constant at different regions of the input space with the behavior of the stochastic training procedure.  We first observe that the Lipschitz constant close to the training data affects various aspects of the parameter trajectory, with more complex networks having a longer trajectory, bigger variance, and often veering further from their initialization. We then show that NNs whose 1st layer bias is trained more steadily (i.e., slowly and with little variation) have bounded complexity even in regions of the input space that are far from any training point. Finally, we find that steady training with Dropout implies a training- and data-dependent generalization bound that grows poly-logarithmically with the number of parameters. Overall, our results support the intuition that good training behavior can be a useful bias towards good generalization.


Reviews and author responses:

The paper studies the relation between the distribution of Lipschitz constants in a neural network during training, the trajectory of gradient descent and the generalization of the network. The authors relate the change in the bias of the first layer to the Lipschitz constant of the network around the training data. Then, it is shown that when if the bias changes slowly enough, this implies that the network has small Lipschitz constant, which can then be used to derive generalization results, assuming the data has small covering number. The theoretical results are complemented with some experiments.  Exploring the relation between the training dynamics and various properties of the network seems like a worthy direction to study, and the setup studies in this work seems highly relevant for this effort. It seems that the authors have some positive and novel results that can be utilized to advance our understanding of generalization in neural networks.

That said, I have some concerns about the results presented in the paper:

- The main result shows the relation between the change in the bias of the first layer and the average Lipschitz constant of the network, and all other theoretical results build on this simple observation. However, it seems to me that there is nothing specifically interesting in how the bias of the first layer behaves, beyond the fact that it captures the Lipschitz-ness of the network. This makes all the arguments seem somewhat circular - you define "steady learner" using a bound on the change of the bias in order to enforce a bound on the Lipschitz constant, and then prove that this means that the Lipschitz constant is bounded. Why is it more natural to assume a "steady learner" the way you define it? Why not simply give your assumptions on the Lipschitz constant? 
To put things another way, suppose we don't use a bias term in the first layer, or for other layers along the network. Does this mean your analysis is vacuous?
It seems to me that studying the generalization through the Lipschitz-ness of the network is a very promising direction, but using the change in the bias seems unnecessary.

- Generalization bound for learners with "dropout" (section 5.2): I found it hard to follow the results and specifically to validate how "sane" are the assumptions of Theorem 3. Generally, any function that is L-Lipschitz around a large enough sample will generalize well if the support of the distribution has a small r-covering number (depending on L and r), because you are likely to sample every r-ball of the covering, and the Lipschitzness guarantees that the function is well-behaved inside each ball. Then, why does this generic argument not apply to the case studied in the paper? Why is the dropout necessary for the generalization argument?

- It seems that the theoretical analysis focuses on the complexity (i.e., Lipschitz constant) of the function being learned, while the experiments focus on the complexity of the target function. What is the connection between the theoretical and empirical results? Do you assume that a network trained on a target function with large Lipschitz constant will also be non-Lipschitz?

Minor comments:
- In line 45, you write: "NNs that are everywhere Lipschitz continuous can generalize". All NNs are typically everywhere Lipschitz continuous, I guess you should add "with a bounded Lipschitz constant".
- You refer to the "length of the optimization trajectory" in various places, but it is not properly defined.
- In Line 70, you write that the sample complexity of neural networks needs to grow exponentially with depth, however this is not true for the basic VC dimension sample complexity, as the VC dimension grows only linearly with depth.
-  You write "with high probability" in several places, I assume you mean with probability at least 1-delta.
- Line 207: "MSE loss can find near solutions...", meaning - solutions near the initialization?


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Reading the author feedback, I still think that the fact that the analysis relies heavily on the "bias of the first layer" is a major limitation of the paper. My concern is that the fact that the result breaks when removing the bias of the first layer hints that the overall result is very fragile. The observation that the bias of the first layer corresponds to the gradient w.r.t. the input is not very surprising, and the fact that the analysis of the whole paper relies on this observation makes me feel that the results are not very strong. Therefore, I maintain my score. Yes

This paper studies the connection between the network’s complexity, its weight’s trajectory during training, and its generalization capabilities. Theorem 1 shows that the trajectory length of the biases in the first layer is both lower and upper bounded by terms that depend on the average Lipschitz constants close to the data points. Next, in Theorem 2 it is proven that the Lipschitz constant close to points outside of the training set is upper bounded by the trajectory length of the biases in the first layer. In Theorem 3 the authors give a generalization bound (when training with dropout) which depends on the data manifold dimensionality, instead of growing exponentially with the depth. Finally, several experiments are made to validate the theoretical claims.
  I think this is a very interesting paper, giving several good results about the behavior of neural networks during training, and their generalization capabilities. First, bounding the trajectory length of the parameters (in this case, the bias of the first layer) by the Lipschitz constant of the network close to the dataset is an interesting result by itself. Next, the result in Theorem 2 is interesting for extrapolation analysis, where we want to study the behavior of the network outside the training set. Finally, the generalization bound which depends on the data manifold dimensionality and the steadiness of the function is very interesting in my opinion, as it does not grow exponentially with the depth, like previously known generalization bounds.

The experimental part is satisfactory in my opinion, testing on a synthetic dataset, and a subset of CIFAR-10. To this end, the correlation between the trajectory length and the generalization capabilities is very interesting.

A couple of minor issues:
1) Theorem 1 requires a lower bound on the number of iteration T. Is there some guarantee that this bound can always be achieved? e.g. what if \sigma_n(W_1) goes to zero as fast as 1/T, this would require that T=Omega(T^2), no?

2) I think the authors should explain in more detail the reasoning behind the focus on the bias in the first layer. This is a crucial part of the paper, and the explanation is a bit shallow. A derivation of how in general the Lipschitz constant of the network without the first layer depends on the bias and the singular values of the matrix in the first layer.

3) I think it would be more natural to phrase Theorem 3 where the bound depends on the local Lipschitz constant of the data instead of the \varphi parameter connected to the steadiness of the network. Maybe this can be done by using a similar setting to Theorem 1 and deriving \varphi in terms of the local Lipschitz constants.

4) The choice in CIFAR-10 of separating 'dog' and 'airplane' is a bit weird. Is there a reason these two specific classes were chosen?  It would be more convincing if more classes are considered - i.e. keeping it a binary classification task, but having several experiments for several other classes. Otherwise, it makes one wonder if there is something different for the two chosen classes.

5) Just a suggestion, there are several experiments made in the appendix which are not even mentioned in the main text (MNIST, total trajectory length, effect of batch size and architecture on the trajectory). I suggest mentioning all the experiments made in the appendix, even in a single sentence, and referencing to them in the main text. This would give a better view of the empirical contribution of the paper.

To summarize, I think this is a good paper with interesting results, and it should be accepted to NeurIPS.

----- Post rebuttal update-----

I am satisfied with the author's response and retain my score. The authors adequately addressed the limitations and potential negative societal impact of their work.

 Thank you for your reply.

*The reviewer argued that using biases is not practically relevant ... Thus, not using biases with ReLU networks limits their expressive power.*

A fully-connected layer with biases is equivalent to a layer without biases but having an extra neuron with the constant output 1 in the previous layer. In particular, if the input data happen to have a constant nonzero component, then the network without biases in the first layer is exactly as expressive as with biases. In my experience, removing biases from VGG16 does not degrade its performance on CIFAR. At the same time, I don't claim that biases are not practically relevant in general (that would be wrong). 

*"The experiments ... expected based on general grounds"*

I don't see how experiments performed in section 6 demonstrate the predictive power of the developed theory. 

I agree that the experiments are consistent with the theoretical results, but the authors only discuss this consistency on a relatively weak qualitative level. I don't find this very interesting, because the demonstrated effects (namely, that higher-frequency target functions are harder to fit, are fitted with a worse test loss, the respective optimization is less robust, and the respective optimized weights are further away from initialization) are already heuristically expected. Section 6 does not show that the developed theory can be used to supplement these heuristics with some concrete knowledge (e.g., describe the specific shapes of the trajectories in figure 1, or explain why the variance values in figure 2 apperently exponentially depend on the frequency, etc.). 

At the same time, I acknowledge the general novelty of the work in analyzing connections between bias trajectories and complexity of the target function.

The paper deals with the following question: does the behavior of the NN during training indicate anything about the complexity of the learning function? The authors tackle this question in the case of a deep ReLU network in the case of regression/classification and using SGD. They show that the (average over the training points) Lipschitz constant of the learned function lower and upper bounds the bias trajectory. As a consequence, they deduce that i) large complexity NN need to better fit the training data to have similar variance as the one of low complexity NN ii) the bias will remain close for NNs that have smaller Lipschitz constant. Then, the authors derive a bound on the Lipschitz constant in a region that is not in the training set and finally, derive a generalization bound when dropout is used. They add some numerical experiments to verify the theoretical claims.  Originality: This paper tackles a very exciting question in deep learning theory that is about the relationship between training and the complexity of the learned predictor. I am not aware of a lot of works tackling this question (apart the ones in the NTK regime).  Th tools that are used are from statistical learning theory but i think that they are used in a novel way.  I am not an expert in the field, so I cannot really assess whether the related work has been well cited.  

Quality: The paper seems technically sound. I think it is a high quality paper and even for a non-expert, it is easy to understand. The claims are well-interpreted. The numerical experiments well support the theory and the authors are honest about the limitations of their work. I am not an expert in the intersection between statistical learning and deep learning so I am unable to compare this paper with existing papers in the field. However, I would like to raise some points: 

1) What I always found puzzling in the "statistical learning" papers for deep learning is the fact that they rely on generalization bounds that depend for instance on the Lipschitz constant of the learned function. This is why the authors seem to desire low complexity function obtained through training. However, it is not obvious for me that networks getting the best generalization are ones with low Lipschitz constants. For instance, it has been numerically observed that some networks that generalize well (CNN) have large Lipschitz constant [1] and therefore are complex. I have some doubts about the fact that the Lipschitz constant is a valid indicator on the generalization ability of the learned predictor.


2) I did not manage to fully understand why the authors focus on the 1st layer bias (and not on the 1st weight W_1) in line 150. I think that this paragraph should be clearer.

3) I have some trouble to understand the impact of the optimization algorithm here. How does the algorithm influence the result in Theorem 1? Is it an algorithm specific result? 

4) Does your Theorem 1 cover the case of NTK? If we initialize the parameters such as the lazy training paper [2], I expect that my neural network behaves as the NTK. The bias trajectory is going to be short (since my parameters don't move too much from the initialization) and yet, to my knowledge, the NTK can have arbitrary large Lipschitz constant with respect to the input [3]. Therefore, Theorem 1 seems to not hold in this case.

5) In NN, we know that the eigenvalues of a neural network are extremely spread [4]. Does it mean that the bound can be vacuous in some cases? 

6) In theorem 2, it is not clear: you need the condition that "activation of any point in the space can be written as a linear combination of the training points". Am I correct? Because this is what you say after the regression setting example below the theorem. 

7) The paragraph from Line 231 to 234 is a bit unclear to me. According to what I understood, you need the condition "N \geq \prod_{l=1}^{d-1} n_l" in order to have the following property: the activation of any point in the space can be written as a linear combination of the training points. Is this correct? I think that it should be clearer.

8) Regarding the numerical experiments, do you think that it would be possible to estimate an average Lipschitz constant as done in [1]? It would be interesting to see whether the Lipschitz constant matches with the distance to the bias. 

Clarity: The paper is well-written and well structured. I enjoyed reading it. 

Significance: The paper tackles an important question and I believe that this is a first interesting step to solving it.

Minor comments: 

a) Is there a way to simplify Corollary 2? It is a bit hard to skim it.


[1] Scaman, K., & Virmaux, A. (2018). Lipschitz regularity of deep neural networks: analysis and efficient estimation. arXiv preprint arXiv:1805.10965.

[2] Chizat, L., Oyallon, E., & Bach, F. (2018). On lazy training in differentiable programming. arXiv preprint arXiv:1812.07956.

[3] Bietti, A., & Mairal, J. (2019). On the inductive bias of neural tangent kernels. arXiv preprint arXiv:1905.12173.

[4] Sagun, L., Bottou, L., & LeCun, Y. (2016). Eigenvalues of the hessian in deep learning: Singularity and beyond. arXiv preprint arXiv:1611.07476.

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I would like to thank the authors for their reply. I still believe that this paper is good. If this paper gets accepted, I would like to ask the authors to be transparent on the fact that a small Lipschitz constant is sufficient but not necessarily for generalization. Besides, I think it is important to be more explicit on why the analysis depends on the bias of the first layer. What if the network does not have any bias? Except these two points, I am happy about this paper. The authors mention the limitations of their work in Section 5 which is definitely an advantage of this paper.

 Thank you for your constructive review! We answer your comments below:  

### 1. Is the lower bound on the number of iterations $|T|$ always achievable?

This is an excellent point. You are correct that the lower bound on $|T|$ might not always be achievable. 
The good news is that we can guarantee independently of $|T|$ that the normalized bias update will give upper and lower bounds for the Lipschitz constants at the specific training points sampled by SGD (rather than on the average region). We will update the main text to reflect this.    

### 2. Better explanation of why we focus on $b_1$

We realized that the choice of $b_1$ was a common point of confusion for a number of reviewers. 
We will add a more in-depth explanation of why the gradient of $b_1$ (and not $W_1$) is indicative of the Lipschitz constant, enhancing our current discussion in lines 150-157. (See also section of our reply to reviewer KizT)

### 3. The phrasing of Theorem 3

We initially intended to pursue the suggested route. We had to reconsider because we realized that, when Dropout is employed, the notion of "local Lipschitz constant of the data" partially loses its meaning. Specifically, with Dropout enabled, the bias is updated based on a quantity that relates to a stochastic perturbation of the regions with training points in their interior. The bias trajectory and $\varphi$ are thus affected by the Lipschitz constants of regions that may be far from any training point. 

### 4 \& 5. Experimental results

The choice of the CIFAR-10 classes was arbitrary.  We can repeat the experiment with alternative classes if you believe it is pertinent. We did not do it in the submitted version to avoid over-averaging the trends, choosing instead to focus on a different dataset (MNIST in Section B.5 of the supplementary material).

Following your suggestion, we will add to the main text explicit references to the experiments done in the appendix. 



 
We found your review very helpful---thank you for that. We answer your comments below:

### 1. Lipschitz constant as an indicator of complexity

This is a very good point.  From a theoretical point of view, we care about the global Lipschitz constant because, otherwise, we cannot guarantee generalization. However, this might indeed be an artifact of the proof techniques.  

In practice, it might be more indicative to focus on the average Lipschitz constant (rather than the worst-case one). The latter describes the typical behavior of the function close to the data manifold and, thus, statistically, we would expect it to be more relevant. This is also why Section 4 of the paper considers the Lipschitz constant close to the training data. Deriving relevant generalization bounds with this quantity is an open and exciting direction.          

### 2. Why do we focus on $b_1$ and not $W_1$?

It might be originally surprising that the bias is indicative of NN complexity.  
While the bias by itself is not indicative of much, it turns out that the way the bias changes over successive SGD iterations reflects the operation of the entire NN.    
Indeed, as shown in the proof of Lemma 1, the expression of the gradient of $b_1$ with input $x$ is very close to the expression of the NN's Lipschitz constant at $x$. 
On the other hand,  the gradient of $W_1$ features additional terms that confound its meaning. 

Following the reviewer's recommendation, we will explain this in detail in the relevant paragraph of Section 4. 

### 3. What is the impact of the optimization algorithm? 

Our theoretical results are only applicable to SGD. In the supplementary material, we also tested empirically the effect of batch size, and we found similar trends. 

We suspect that adding historical information in the gradient update (e.g., momentum) will render the change of the bias more reflective of the average Lipschitz constant close to training points.  However, we cannot say for certain as the analysis becomes more convoluted.  We will clarify this limitation in the main text. 

### 4. Does Theorem 1 cover the case of NTK? 

Being non-experts on NTK, we are not sure if the assumptions render the analysis of [3] applicable here. Nevertheless, we suspect that in the example you mentioned, even if the bias update is small, the loss derivative will also be comparably very small, and thus their ratio (bounding the Lipschitz constant in Theorem 1) might not necessarily be a small number.  

### 5. Effect of the singular values of the NN on the bound

Ideally, we would hope to avoid any dependence of our bounds on singular values.  
However, thankfully our bounds depend on the largest and smallest singular values of only the first layer -- which renders it significantly tighter than a product-of-norms upper bound. 

We think that the results of [4, from review] are not applicable in our case since they focus on the eigenvalues of the Hessian, whereas we care about the singular values of $W_1$. 
Based on our experiments the largest singular value is typically a moderate constant and does not pose a big issue, whereas the requirement on the smallest singular value renders our proposed lower bounds on the Lipschitz constants less tight.      

### 6 \& 7. Clarifying the condition of Theorem 2

Thank you for raising this point. You are indeed correct: Theorem 2 has predictive power only for those points $x$ whose activation can be written as a linear combination of activations of the training points. Thus, the global Lipschitz constant can be bounded if the above condition holds for every $x$. The condition $N \geq \prod_{l=1}^{d-1} n_l$ is necessary for the matrix $S_T$ to be full rank (which would also imply that Theorem 2 gives a bound on the global Lipschitz constant). 

We will make this clearer in the camera-ready version.  

### 8. Estimate an average Lipschitz constant to see whether it matches the distance to the bias

We were also thinking about this. We tested it when performing experiments (for the case of the MLP) and found that the curve was almost identical to the one presented in Fig 1. We did not include it since we thought that it was not offering additional information.  

### Minor comment. Simplifying Corollary 2

We agree that the corollary might be hard to parse for the casual reader. We will try to simplify the statement in the camera-ready version.  

 
Thank you for your comments. We answer them one by one below:

### 1. Why is the Lipschitz constant a good indicator of complexity

This is an excellent question. The Lipschitz constant is a very natural complexity measure for ReLU networks: ReLU NNs implement piece-wise linear functions, and the Lipschitz constant corresponds to the maximum norm of the gradient at any piece-wise region.  Moreover, a key benefit of the Lipschitz constant is that it can be used to predict whether a model will generalize (which we exploit in this paper).  Note also that there is a large literature that focuses on the importance of the Lipschitz constant as well as on finding ways to estimate it (see the relevant paragraph in the related work section).  

There are two additional comments we would like to make here: 

* One could argue that, instead of the Lipschitz constant, we should be focusing on the number of piece-wise linear regions. However, this choice comes with a big drawback: even if the number of regions is exponentially large, this does not mean that the NN implements a different function on each region (making it complex). Indeed, it is also commonly observed empirically that most nearby regions in the input space implement approximately the same function. (see also discussion in lines 39-41 about a separate issue).  

* One could argue that the Lipschitz constant is too pessimistic since it depends on the largest gradient of any linear region. This, however, is not an issue for the results of Section 4 that focus on the Lipschitz constant close to the training data (rather than on the largest gradient). 

### 2. Why it's practically relevant to focus on biases

The reviewer argued that using biases is not practically relevant. 
We respectfully disagree. From a practical point of view, using biases is recommended when employing ReLU activations. 

To understand why, consider the first layer $f_1(x) = \rho(W_1 x)$ of a NN $f$ without a bias. Then, for every $W_1$, $f_1(x)$ will have the same output for all $x$ for which $W_1 x \leq 0$ (with the inequality applied element-wise), meaning that also the NN $f(x)$ will necessarily assign the exact output for all such $x$. Thus, not using biases with ReLU networks limits their expressive power.

### 3. Singular values of weight matrices

We agree that, ideally, we would hope to avoid any dependence of our bounds on singular values. Thankfully, our bounds depend on the largest and smallest singular values of *only* the first layer and thus are independent of depth. 

This comes in contrast with previous analysis of generalization through Lipschitz constants that depend multiplicatively on the largest singular values of all layers 
(the product-of-norms upper bound) and can thus be *exponentially* larger than the bounds provided herein. From that perspective, our analysis presents a significant improvement over the state-of-the-art.

### 4. The reviewer mentions: "*The experiments presented in the end of the paper (...)  could be anyway expected even without the developed theory, just on general grounds.*"

We are not sure what the reviewer means by "*expected based on general grounds*":  
To the extent of our knowledge, no previous work has proved comparable theoretical claims to our work. Further, proving the dependence between complexity and training time is by no means solved but remains an open question. 

The experimental findings that (a) more complex functions take longer to learn, and (b) have higher test losses have been shown empirically before (and, indeed, are cited in the intro as a motivation of this study). 

On the other hand, the explored connections between the complexity of the learned function, the rate of bias update, the distance to initialization, and the variance of bias are unique to this paper.  

 Thank you for your comments.  Below we discuss your three main questions: 

### 1. Why do we focus on bias rather than only on the Lipschitz constant and what happens when there is no bias?

The reviewer expressed doubts about why it's meaningful to analyse the bias and asked why not "*simply give your assumptions on the Lipschitz constant?*"  (rather than going through the bias).

To start, we clarify that the paper's primary concern is not to connect Lipschitz continuity with generalization (this is already known). Rather, the paper asks the question of whether the complexity (and generalization) of a NN can be deduced from its training behavior (i.e., how the parameters are updated during training). This is why focusing on bias is key.  As shown, a bias that is steadily updated implies a small Lipschitz constant close to the training data, and (under assumptions) also globally.

We should also mention that it's problematic to pose a direct assumption on the Lipschitz constant (i.e., beyond that there would not be any connection with training, which is the main point of the paper). Unfortunately, assuming a small Lipschitz constant is something that (a) is not always true (it depends on the task one is solving) and (b) cannot be easily verified (computing the Lipschitz constant is NP-hard). This is the reason why previous works on NN generalization through robustness employ the, easier to compute, "product-of-norms" bound on the NN Lipschitz constant (rather than pose an assumption as suggested). However, as explained in the paper, such approaches can yield loose bounds as the product-of-norms upper bound is often very far from the actual NN Lipschitz constant.    

The reviewer correctly points out that the arguments provided do not work when the first layer has no bias. We would like to make some remarks here: 

* From a practical point of view, biases are very useful when using ReLU activations. Consider the first layer $\rho(W_1 x)$ of a NN $f$ without a bias. For every $W_1$, $f(x)$ will have the same output for all $x$ for which $W_1 x \leq 0$ (with the inequality applied element-wise). Thus, not using biases with ReLU networks limits their expressive power.  

* The analysis employed can be easily extended to the biases of higher layers (see proof of Lemma 1). However, doing this yields looser bounds. As an example, if we are focusing on $b_2$, then the resulting bound on the Lipschitz constant will depend on $||W_2|| ||W_1||$, which can be arbitrarily larger than the real Lipschitz constant (this is a product-of-norms bound for the subnetwork consisting of the first two layers).
        
### 2. Dropout

This is a great question. Let us start with the reason why the argument proposed by the reviewer cannot be applied. In short, by looking at what happens on training data one cannot guarantee that the Lipschitz constant in different (though close) regions will not be large. Recall that the bounds derived in Section 4 only concern the regions of the ReLU NN that contain at least one training point. However, due to the geometry of these regions, it is possible (and likely) that small empty regions will be located near those that have training data inside them. Deriving a generalization bound would require upper bounding the Lipschitz constant of the NN on these empty regions. However, to our knowledge, the latter is not possible without resorting to loose product-of-norms bounds or imposing strong additional assumptions that assert that the training regions are sufficiently diverse (as Theorem 2 does).  
  
Here is where Dropout comes in. Dropout introduces randomness in the training procedure that provides information of the NN function in the gaps between training data. This allows us to infer the Lipschitz constant of the function in larger portions of the space from the training behavior --- and thus to relax the assumptions of Theorem 2.    

### 3. Is a NN trained on a target function with large Lipschitz constant non-Lipschitz?

The reviewer pointed out that "*the theoretical analysis focuses on the complexity (i.e., Lipschitz constant) of the function being learned, while the experiments focus on the complexity of the target function.*" and asked if we "*assume that a network trained on a target function with large Lipschitz constant will also be non-Lipschitz?*".

There is a direct relation between the two Lipschitz constants: To make things concrete, say that the target function and NN have Lipschitz constants $\mu$ and $\lambda$, respectively. If the NN fits the target function well enough (e.g., the loss is small as in our experiments) then $\mu \approx \lambda$. So, by varying the complexity of the target function in our experiments, we are effectively controlling the Lipschitz constant of our NN. 

Controlling the NN Lipschitz constant more directly would require changing either the architecture (artificially restricting all $W_l$ so that the Lipschitz constant is easily computable/controlable) or the training procedure (imposing some regularization). We believe that such experiments deviate from the main question of the paper, which is to understand how the *normal* training dynamics (i.e., how the bias is updated), relate to the complexity of standard NNs. 

### Minor comments 

Good catches. We will incorporate your suggestions in the camera-ready version.

The paper studies the connection between the complexity of a function fitted by a neural network and the complexity of the training trajectory. The paper is mostly theoretical and contains several main results. The first main theorem gives bound on the length of the weight trajectory in terms of the Lipschitz constant of the network. The second theorem gives a bound for the Lipschitz constant of the network at a point away from the training set. The third result gives a bound on generalization error. Some experiments are provided to illustrate the conclusions of the paper.  I'm not sure that I understand this paper. My general impression is that it poses reasonable questions and makes some interesting analysis, but somehow I don't see a clear picture resulting from it. For example, already after Theorem 1 the authors state that this theorem  "shows that lower complexity learners will have a shorter bias trajectory". This is debatable for many reasons. First, the choice of the Lipschitz constant as an indicator of complexity is questionable (is a linear function with a large Lipschitz constant more complex than a nonlinear function with a small Lipschitz constant but many different local variations?). Second, the established inequalities only concern the biases, which are often not even used in practical networks. Third, the inequalities involve singular values of the weight matrices, which may vary in a very wide range and be quite different for different learners. So Theorem 1 doesn't really clear things up for me. I also have similar impressions from the other results of the paper. The experiments presented in the end of the paper demonstrate some natural effects of complexity (more complex functions take longer to learn, have higher test losses, correspond to a higher distance from initialization, etc.) that agree with the results of the paper but could be anyway expected even without the developed theory, just on general grounds.

 -

Meta-review: The majority of reviewers were positive about this paper, commending the connection it makes between generalization and training dynamics.  Nonetheless, several technical concerns were raised, in particular:

* The paper equates "complexity" with "small Lipschitz constant".  Obviously small Lipschitz constant is sufficient to ensure generalization, but as far as the committee is aware of it is not necessary.  Identifying the two will potentially mislead the uninformed readership.

* The learned parameter to which the analysis mostly applies is the bias in the first layer.  It is perhaps a bit trivial to associate Lipschitz constant with the first layer bias, as the gradient with respect to this bias and with respect to the input are equal.  Reviewers felt that this might be an indication of the analysis being somewhat fragile and not representing overarching phenomena.

Despite the above, the paper was deemed interesting and significant enough to warrant acceptance.  I encourage the authors to be more transparent with regards to the above points and clearly highlight the limitations of their framework in the text.

Decision: accept