The Impact of Geometric Complexity on Neural Collapse in Transfer Learning

Thank you for your comments. We are very happy that you found that the “unified analysis of neural collapse and geometric complexity presents an interesting line of research for the community [...] in terms of studying seemingly disjoint phenomena under a common lens.” We have addressed your comments to the best of our ability below.

Please consider raising your score if your concerns have been resolved.

We'd also like to respectfully note that your concerns seem to focus around only minor aspects of our work that are easily fixable (i.e., clarification, notation, typos, addition of plots). Could you please help us understand which of your concerns prompted you to a "confident reject” so that we can address it?

"There seems to be a mismatch in the log⁡(2/𝛿) term in Proposition 4.2 vs log⁡(1/𝛿) term in Proposition 4.3, which makes the main results inconsistent (yet fixable). Additionally, why do we have an "expectation" over the "error" term in the LHS of Eq (10)? Isn't the "error" itself an expectation?"

Thank you for noticing the typo: there should be $\log(2/\delta)$ instead of $\log(1/\delta)$ in Prop. 4.3. As for the second expectation in the LHS of Eq. (10) it is taken over the sample S used to produce the classifier $h_{f,S}$ on line 230 (similarly to [22, Proposition 5]). We will correct the typo and make the second point clearer in the paper.

"Although the paper presents generalization bounds, a discussion on the generalization performance of the classifier (VGG-13 for Figure 1) seems to be missing. Only the neural collapse and geometric complexity plots are illustrated but a deeper relationship between the trends and the test performance is lacking."

We included plots for the test accuracy in the rebuttal document (see Figure 1, 3rd column). As referenced in our paper, it has already been observed in [22] that lower levels of NC are indicative of higher test accuracy. Similarly, in [17] it has been shown that lower levels of GC also are characteristic of higher test accuracy. We will add a remark to this effect in the paper and include similar plots for our experiments in the appendix.

"Is there an upper limit of "good" learning rates, and batch sizes above which the geometric complexity can reduce but the test performance can get worse?"

Yes, you are correct. As for any form of regularization (implicit or explicit), test performance degrades beyond a certain level of the regularization rate, while the regularized quantity keeps decreasing at the expanse of the original objective.

"Similar to the above point, currently it seems like increasing the learning rate, increasing batch size, and increasing the regularization parameter value results in lower neural collapse values. Ideally, neural collapse is applicable during the terminal phase of training (TPT) where the training accuracy is 1. Does this mean that all the experiments (at least for Figure 1) lead to TPT during training?"

Yes, that is correct and a good clarification. All our experiments lead to training accuracy 1. We included these plots for our main experiments in the attached rebuttal document (see Figure 1, 1st and 2nd columns) and will add similar plots for our additional experiments to the appendix.

"Please fix the notation for the norms as it is unclear if $||\cdot||$ represents $||\cdot||_2$ or $||\cdot||_𝐹$."

Thank you. We will clarify the norm notation in the paper as you suggest.

"Since the geometric complexity-based generalization/transfer learning bounds are the upper bounds of existing neural collapse-based bounds, what can be a potentially new implication of your results? since it is common practice to do a hyper-parameter search over learning rates, batch sizes, and regularization parameter values."

To our knowledge, there are no known theoretical mechanisms which control neural collapse. Our bound in Prop. 4.1 shows that the same mechanisms that control GC also control NC. This is significant because (see [17] from the paper references) there are a number of experimental mechanisms which control the GC and furthermore a strong theoretical justification as to why [17, Thm 5.1]. The current work provides a novel justification that these same mechanisms also control NC. This implication is theoretically motivated via our Prop 4.1 and experimentally validated.

A second new implication that can be derived from our bound in Prop 5.1 is that pretrained networks with lower GC have better transfer learning performance on new tasks. These 2 new implications are indeed the key theoretical findings in our paper. Devising practical schemes based on these theoretical findings is beyond the scope of the current paper, yet opens a new avenue for future work.

"The paper does not discuss any explicit regularization mechanism that aims to reduce the geometric complexity. What are the potential challenges in designing such a regularization mechanism."

This is a good point. While direct regularization by GC computed at the logit layer has been previously studied (for instance in [17]) and found beneficial, direct regularization by GC computed at the embedding layer becomes impractical because of the much higher dimension of the embedding layer, resulting in a very large computational cost.

That being said, explicit regularization by GC computed at the logit layer provides a direct way to control GC computed at the embedding layer. We added this experiment (see rebuttal document Fig. 3) and observed the predicted effect on the embedding GC (rebuttal Fig 3. left and middle) and on neural collapse (rebuttal Fig 3. right). We will add a paragraph to our paper about the challenge of directly regularizing for embedding GC.

Thanks for your thoughtful review. We are very happy that you found our approach to be a “novel perspective” as well as an “original contribution” and that it “could open up new avenues for improving pre-training techniques” and “provide valuable insights into the mechanisms behind transfer learning and self-supervised learning”. We are also glad that you found the “empirical evaluation” to be “thoroughly” done and that the paper was “well-structured” with “ideas communicated clearly”.

Below are our answers to your concerns. They have been very helpful in further strengthening the paper.

We ask that you please consider raising your score if your concerns have been resolved.

"The connection between the linear models and DNNs is not justified"

There is no mention of linear models in our paper. Can you please reference the paragraph creating confusion so that we can clarify?

"Limited theoretical foundation: [..] the assumptions required (e.g., Poincaré inequality) are quite strong [..] the authors could strengthen this by providing evidence that these assumptions hold in realistic settings"

This is a good point but we respectfully disagree with the idea that the Poincare Inequality (PI) is an overly strong or unrealistic assumption. We agree our paper would be strengthened by clarifying this point. Here, we include a short summary of key facts and can provide a more detailed exposition in the Appendix.

Note that the Gaussian distribution; mixtures of Gaussians with equal (or different) variances; mixtures of Gaussians and sub-Gaussians; mixtures of uniform and Gaussian measures; and any log-concave probability measure all satisfy a PI ([1, 2] below). The same is true for most distributions of importance in probability and statistics ([3] below).

The PI has also been assumed to hold for real life image datasets, where it has been used to help improve GAN convergence ([4] below). It is also a key assumption to understand the role of over-parameterization in generalization as happens for large NNs ([5] below).

On the contrary, non-PI distributions are considered pathological; they can be constructed for instance by artificially concatenating distributions with fully disjoint support. The only restrictive assumption for the model is that it's differentiable wrt the input, which is a widely assumed property in the literature.

Furthermore, we examine various real-life datasets like CIFAR10, CIFAR100, and ImageNet as well as common architectures like VGG and ResNet.

[1] Bakry et al. A simple proof of the Poincaré inequality for a large class of probability measures, '08

[2] Schlichting, Poincaré and log-Sobolev inequalities for mixtures, '19

[3] Pillaud-Vivien et al, Statistical Estimation of the Poincaré constant and Applications..., AISTATS '20

[4] Khrulkov et al, Functional Space Analysis of Local GAN Convergence, ICML '21

[5] Bubeck et al, A universal law of robustness via isoperimetry, NeurIPS '21

"Experimental limitations: [..] The generalizability of the results to other domains (eg, LLMs) is unclear [..]"

This is good point. Thanks for raising it. We will definitely clarify this aspect in our paper. There is a limitation of our work when it comes to extending to LLMs. However, this limitation is not specific to our work per-se but it is a general limitation of the application of NC to LLMs (see Wu & Papyan, Linguistic Collapse: Neural Collapse in (Large) Language Models, ‘24 which outlines limitations).

The main problem is that for LLMs the embedding dimension is lower in general than the number of classes (vocabulary size), making the NC simplex impossible to exist. Extending the notion of NC to LLMs is an open question beyond our scope.

"How does GC compare to other measures [..]"

This is indeed interesting. However, this comparison has been done already; e.g. [17] introduces GC and thoroughly relates it to the gradient norm & [41] compares with other complexity measures. The relation with the Fisher information and the gradient norm has also been studied (Jastrzebski et al., Catastrophic Fisher Explosion, ICML '21). These works indicate that the GC as a measure in function space is a similar notion of complexity to the gradient norm or the Fisher information in parameter space.

We are happy to add an Appendix section on this.

"The paper shows correlations with GC [..] but not [..] causation"

We added an experiment (rebuttal:Fig. 3) that directly controls GC by explicitly regularizing it, mitigating confounding factors. We observe the same predictions: lower GC produces lower NC. Note, here we regularized w.r.t. the GC computed at the logit layer rather than the embedding GC. This is because the high dimensionality of the embedding layer makes taking the gradient of the embedding GC very prohibitively expensive.

"How computationally expensive is it to measure GC [..]? Is it feasible to use as a regularization term?"

We have not yet tried to use GC as an early stopping criterion but this could be a good idea. Measuring GC during training is relatively cheap, easy, and robust, especially if one samples it as explained in Section 4.2, see also Fig. 2. We have explored using GC as a regularization term (see above & rebuttal:Fig. 3) as have others, eg [17].

"The generalization bound in terms of GC is interesting, but how tight is it empirically?"

We added an experiment (rebuttal:Fig. 2) where we plotted the LHS and RHS (excepting the Lipschitz term) of this bound Eq. 10 in Prop 4.3, which we will add to our paper. On that plot the bound is not vacuous and quite tight! A full comparison with other generalization bounds is beyond our scope. Our main goals are showing that 1) the mechanisms that control GC also control NC and 2) solutions with lower GC are more performant for transfer learning.

"Have you explored using GC insights to guide models or algorithms?"

This is a very interesting but beyond the scope of the current paper.

Thanks for your review. We are very happy that you found the work “well-written” and “easy to follow”, making the interpretation of the theoretical results “clear” and “intuitive”. We are also glad that you found the relation between “GC” and “NC” to be “quite interesting” and that the “empirical results align well with the theoretical part”.

Below you’ll find below our best effort to answer your questions.

Please consider raising your score if you feel your concerns have been resolved.

"In proposition 4.1, NC is upper bounded by geometric collapse up to some constant scaling. What's the range of this constant c, does it purely depend on Q. If yes, it would be better to directly plot NC and RHS in (6) in the same plot to demonstrate the validity of the bound; and if no, it would be interesting to show how different settings examined in Figure 1 affect c."

Yes, the Poincare constant depends exclusively on the distribution Q. Stacking the plots in Figure 1 as you are suggesting is a great idea, which we will add in our paper. On these plots, by comparing the magnitude RHS and LHS of (6), we can roughly see that the c ~ 1000 (lower bound) for CIFAR-10 in our setting. The Poincare constant can take very high values depending on the spread of the distribution. For instance, for a standard normal distribution the Poincare constant is 1 while for a multivariate normal distribution the Poincare constant equals the largest eigenvalue of its covariance matrix which can be any positive real number. Intuitively speaking, the Poincaré constant will increase if the space is stretched in any direction and decrease if it is compressed in any direction.

"The empirical evidence presented in Figure 2 clearly demonstrates the robustness of geometric complexity. However, I have some concerns regarding the Lipschitz notion. Neural networks are known to be not so Lipschitz, and making them more Lipschitz is an active area of research aimed primarily at improving robustness. Is Proposition 4.2 numerically verifiable by calculating the Lipschitz constant of the tested neural networks?"

The numerical experiment in Figure 2 (plot 1) shows that in practice the empirical GC estimate may be approaching the true value much faster than the bound specifies. Namely after 10 samples only, we already obtain a value very close to what we obtain with 50 times more samples, which is much better than a 1/sqrt(#sample) dependence, even ignoring the Lipschitz constant. This makes us believe that a much tighter bound here should be possible. We agree with you that this bound may be a looser bound than reality.

"In equation (7), does the distance d represent the norm distance? This should be stated clearly."

Thank you for pointing this out. We will clarify that point in the paper. “d_ij” represents the euclidean distance between the mean for class i and the mean for class j.

"Both the paper and the prior work (Galanti et al, 2022) rely on the data assumption where the source and transfer datasets come from the same distribution Q. However, in practical settings, source and transfer datasets typically have domain gaps which can potentially violate the data assumption. Would the results in the paper still hold in this case? For example, pre-training on ImageNet/Cifar-100 while fine-tuning on DTD?"

This is a very good point. In theory any source and target distribution can always be concatenated together mathematically into a single larger distribution. However, as you are pointing out, issues may arise when “gaps” are created by this process. Namely, when the support of the source and target distribution do not overlap, this process results in an overall distribution with non-connected support. This particular configuration is a setting ripe for violations of the Poincare inequality even if the source and target distribution both satisfy it.

If this happens, then the range of validity of our bounds as well as our predictions concerning the impact of the learning rate, batchsize, regularization on the neural collapse and transfer performance may not hold.

However, instead of seeing this as a limitation, we interpret it as a feature that may help us understand what compatibility conditions are necessary for a model pre-trained on a source distribution to transfer well to a target distribution. Namely, in line 286, we give compatibility conditions for the transfer bound in Prop'n 5.1 to be meaningful. This condition in particular implies that no such gaps are created by concatenating the source and target distributions.