Paper: Investigating the effective dimensionality of a model using a thermodynamic learning capacity (ICLR-2024)

Reviewer comment (Evaluation):
Given that the paper attempts to provide a general-purpose measure of effective parameters with little to no limitations—a very significant claim—the arguments and evidence at this stage are insufficient to recommend acceptance.

Author response:
Thank you for your time and detailed review of our paper. We appreciate your attention to detail and the desire to improve the quality of our paper. A lot of the questions that you had asked were minor technical details which we have clarified in the responses below. We have elaborately addressed your major concerns regarding the learning capacity. If these responses are satisfactory, we would encourage you to increase your original score, which we felt was not commensurate with your concerns.

This is not at all a speculative paper, it has a lot of theoretical and empirical depth; and it does make meaningful process upon the problem. Papers such as this are bring in fresh ideas to work on long-standing problems in deep learning and we need all these diverse perspectives if we are to make progress upon the problem. When someone defines a new quantity, there is of course going to be a lot of criticism and questioning, we completely appreciate that. The merit of a quantity like learning capacity is that it is so general and broadly applicable...from parametric models such as deep networks (MLPs, CNNs, LSTMs etc.) to even non-parametric models such RFs and k-NNs; it is well defined even for a Python function that makes predictions! As other researchers in the field study these ideas further, we will develop a deeper understanding of such definitions. This paper provides sufficient evidence that the learning capacity is a well-defined and meaningful quantity that can help us understand generalization and can also be estimated accurately.

Reviewer comment (Writing):
In Section 2 - Methods, while the authors were inspired by thermodynamic notions, presenting the method in these terms made the reasoning unnecessarily challenging to follow due to switching back and forth between domains. It would be clearer to stick to ML notation (e.g., negative log-likelihood) and insert thermodynamic results when needed without changing notation.

Author response:
**While I understand the authors were inspired by notions from thermodynamics and are using results from that field I don't understand the benefit of presenting the method in these terms. I found the reasoning of this section more challenging to follow than necessary due to the switching back and forward. It would be much clearer to simply discuss the quantities of interest in the ML setting (e.g. negative log-likelihood) and derive an expression of effective parameters from there (using results from thermodynamics when required but in the same notation to avoid switching back and forward).**

This is a reasonable comment and we will consider modifying the narrative to make it more friendly to the machine learning audience. The thermodynamics-specific comments in the paper are actually quite few and one can simply think of these quantities as definitions without worrying about their meanings in thermodynamics.

The original purpose of this writing is trying to provide gradual understanding of the context. For the first paragraph, we give simple example to help people understand or have a grasp of the importance of learning capacity. For the second paragraph, we give a much formal formulation and framework about how calculations are done through the process and how it is related to different generalization calculations. There are also some compromises between going to the detail without resorting to appendix since the context have connection to different areas. Thank you for your advice about arrangement of the context and we will change it and explore different ways to make it clear.

Reviewer comment (Theory):
The choice of setting the Boltzmann constant to 1 is questionable, as it does not reflect its true value and therefore $N$ is not exactly $T^{-1}$. This affects downstream interpretations, including the statement that "the learning capacity is a tiny fraction (~2%) of the total number of weights."

Author response:
**Why is setting the Boltzmann constant to 1 a reasonable choice? This does not seem to be its true known value and therefore is not exactly. This has downstream effects by my understanding since the authors explicitly interpret the effective parameters relative to the number of parameters stating "the learning capacity is a tiny fraction (∼ 2%) of the total number of weights".**

Boltzmann constant is often set to 1 in theoretical physics, there it simply means that we are measuring temperature in the units of energy. It is important for our purposes here, we can simply think of the partition function in Equation 2 as the point of beginning and derive everything from it. Please see Appendix F where we provide more details now. Just to clarify for the Reviewer, the fact that learning capacity is small (which is what we are sating when we say that the learning capacity is a tiny fraction of the number of weights) is not something that depends upon the units in which temperature is measured. Things in physics are quite similar, heat capacity of water is large compared to steel---a statement that can be made independently of the units of temperature.

Reviewer comment (Theory):
The four bullet points arguing that learning capacity is a good measure of effective parameters are unconvincing. The reasoning is sometimes unclear, such as the claim "[...] the learning capacity C equals half the effective number of parameters K/2." There is no clear definition of the true effective parameters here, and the definitions and justifications for H are not convincing.

Author response:
**The four bullet points on Page 2 argue that learning capacity is a good measure of effective parameters (I would suggest convincing the reader on this point is fundamental to the remainder of the paper). However, this is not a convincing argument in my opinion and I struggled to even understand the reasoning in some cases. For example "[...] the learning capacity C equals half the effective number of parameters K/2." How do we know the true effective parameters here, this is not defined clearly? Why is H defined as it is here (this convex expression) and why should this be convincing?**

We believe there is a gap in the understanding here. There are many quantities to estimate the complexity of a model, e.g., VC-dimension, Rademacher complexity, PAC-Bayes bounds etc. These quantities are all characterizing the effective dimensionality of a model in different ways, e.g., generalization gap <= sqrt (VC-dim/n). In our case, we adopt the definition of “true effective parameters” proposed by Watanabe (refer to Algebraic Geometry and Statistical Learning Theory). If a linear model has p parameters then the VC-dimension is p; therefore the effective dimensionality of such a model is p. For some models, e.g., singular models discussed in Appendix C, the effective dimensionality may be smaller. In information theory, there are more quantities such as minimum description length, Bayesian information criterion etc. See the first paragraph of the Discussion for references. 

The biggest issue in deep learning is that these quantities are all very large, they scale with the number of weights in the network. So the existing theories of generalization are rendered vacuous. Effectively, in deep learning these quantities, even if they do tell us some general principles, do not seem to correlate well with the test loss. The learning capacity does correlate with the test loss, so it mitigates this issue. The second issue is that some of the standard learning theoretic quantities are difficult to estimate for deep networks, e.g., PAC-Bayes bounds work well but it is very expensive to calculate them. The learning capacity is an approximation of PAC-Bayes-based effective dimensionality (Equation 13) and easier to calculate.

H is not defined in the four bullets, it is defined in Equation 1. In the bullets, we have simply given examples of how the learning capacity looks like for different values of H and the prior. This helps the reader appreciate why we call C the effective dimensionality of the model.

Reviewer comment (Theory):
In equation (6), it is unclear whether the two log terms are a good approximation for -U, as the justification is merely asserted. This approximation is assumed valid for every data point in equation (7), and the subtle change to N-1 in equation (8) is unexplained.

Author response:
**In eqn (6) it's not clear to me that the two log terms are a good approximation for -U. The justification is just a statement that it is. Especially since in eqn (7) this approximation is assumed to be good for every data point in the distribution. Why the subtle change to N-1 in eqn (8)?**
** **

The definition of U is given in Equation (4). The derivation in Equation (6) is a finite-difference approximation of U. Equation 7 follows from 6 directly due to linearity of expectation. There is no reason for changing to N-1 in Equation (8), it just makes it easy to write the expression of Ubar as the standard leave-one-out estimator (LOOCV) for a dataset of N samples. See Appendix F for more details.

Reviewer comment (Theory):
The rightmost equality in equation (8) is unclear. It appears that the LOOCV estimator should be approximating something, but it is presented as an equality. This step should be explained in detail.

Author response:
**The rightmost equality is eqn (8) is unclear to me. Why is this an equality, I understood the LOOCV estimator was approximating something here. Could this step be spelled out further please?**

The right most equality of Equation (8) follows directly from the definition of the partition function. We now provide more details in Appendix F.

Reviewer comment (Experiments):
In practice, LOOCV is not actually used; k-fold cross-validation is used instead. There is no analysis of how increasing the value of k affects the results, introducing another unexamined approximation.

Author response:
** Then in practice LOOCV is not actually used, k-fold is used instead. There is no analysis of the effect of increasing the value of k. This is another approximation being introduced that requires further consideration.**

We can see the k-fold experiment as an approximation of LOOCV. We train on the k-1 folds of data, and calculate the average test loss on the remaining 1 fold to approximate the result of LOOCV.  It is very expensive to estimate LOOCV in practice. As k gets closer to N, the results from k-fold will be get closer to LOOCV. If k equals N, the N fold is exactly LOOCV. The analysis of how the Monte Carlo error behaves as a function of k is quite standard, see for example https://theory.stanford.edu/~sergei/papers/ics11-cv.pdf. The variance of the estimator using k-fold goes down as ~1/k compared to LOOCV.

Reviewer comment (Reproducibility):
The process described at the start of Section 2.2 (sampling datasets, cross-validation, etc.) should be presented formally as an algorithm.

Author response:
**The process described at the start of Sec 2.2 (sampling datasets, cross-validation, etc.) should be described as a formal algorithm somewhere.**

The narrative in Section 2.2 is the algorithm right? It describes each step of the calculation in complete details.

Reviewer comment (Evaluation):
The authors report taking 4 bootstrap samples of the data, but it is unclear what this is intended to approximate and whether it is meant to account for the $\mathbb{E}_{x,y \sim P}$.

Author response:
**The authors report taking 4 bootstrap samples of the data. What exactly is this intended to approximate? Is this supposed to account for the E_{x,y ~P}?**

No E_{x,y ~P} is estimated directly when we calculate the average test loss over the validation fold. The bootstraps are done for a different reason. The quantity Z(N) is a random variable because it depends upon the samples in the training dataset; for different draws of the training dataset with the same N, we will have different values of Z(N). In order to improve our estimate of the learning capacity, i.e., to ensure that the estimate is not a function of the specific samples in the training dataset but only of the number of samples N, we do a bootstrap. This is a very common application when we are estimating quantities that depend on the realization of the particular training set.

Reviewer comment (Theory):
It is unclear what the calculation $\bar{C} = - N^2 \nabla_N \bar{U}$ means exactly, and whether the intent is to take the gradient of equation (9) with respect to the number of samples N.

Author response:
**At this point it's not entirely clear to me what calculating C = -N^2 grad Ubar means exactly.**

To calculate C, we are taking the gradient of Ubar with respect to the number of samples N. This gradient is calculated using finite differences as shown in Equation 8.

Reviewer comment (Writing):
The phrase "we used a polynomial mode" is initially misleading, as it could suggest that the underlying model is polynomial. The authors should clarify that they are referring to fitting a model to U and C values.

Author response:
**"We used a polynomial model" - Initially thought the authors meant that the underlying model was polynomial (which it technically could be since its a single hidden layer MLP). They should specify that they are referring to a model being fit to U and C values.**

Easy to fix, we will make it clear.

Reviewer comment (Theory):
The claim that "this procedure also gives an estimate of the uncertainty" is questionable, as the estimates are likely invalid given that the observations are not independent.

Author response:
**"this procedure also gives an estimate of the uncertainty" - These estimates are unlikely to be valid I would imagine since the observations are not independent.**

This is not true. The observations that are used to calculate the estimate of uncertainty are obtained from bootstrap, random weight initializations and multiple folds. They are correct numerical estimates of the uncertainty when the assumptions of bootstrap are valid, which are rather broad.

Reviewer comment (Theory):
The fitting of a sigmoid seems to undermine the paper's fourth contribution that "The learning capacity saturates for very small [...] and very large [...] N," since the saturation is effectively enforced by modeling it as a sigmoid.

Author response:
“We therefore fit a sigmoid [...]" - Does this not undermine the fourth contribution of the paper that "The learning capacity saturates for very small (equivalent to high temperature) and very large (equivalent to low temperature) sample sizes N ." Since you are effectively enforcing this by modeling it as a sigmoid?
Responses: The model is fit on U values with explicit form described in eqn(10). We experimentally found that the taking derivatives of polynomial fitting of the U curves show large and small sample size saturation in general. The drawback of polynomial fitting is that we need to specify the degree of polynomials which can be subjective, and the damping of polynomials fitting is inevitable. However, based on the observation we have, if we fit the U curve with the eqn(10), we can prevent the overfitting or underfitting due to subjective choice of degree of the polynomial and we also capture the characteristic of most of the characteristics showed in the experiment data if we assume smooth change in the U and C curves.

Reviewer comment (Theory):
In Section 3 - Connection to PAC-Bayesian framework, while the idea of making this connection is interesting, the execution lacks clarity and could have benefited from a formal presentation of the equivalence (e.g., a theorem). The discussion is notation-heavy and insufficiently explicit, making it difficult to assess the strength of the connection.

Author response:
**While the idea of making this connection to the PAC-Bayesian framework is interesting, I found the execution could have been made more clear. I think this section could have benefitted from a more formal presentation of the equivalence (e.g. a theorem). As with the previous section, including all theory as part of a long discussion section with extensive notation and several steps being insufficiently explicit harms the paper as a whole. I found it difficult to judge just how strong the connection claimed in this section really is.**

We are a bit puzzled at the point that the Reviewer is making here. The equivalence between the learning capacity cannot be a theorem! We are simply saying that the expression of the learning capacity can be simplified to be the expression of the effective dimensionality given by PAC-Bayes calculation. This is indeed a “strong connection” that the Reviewer is looking for. Because it is not derived from ad hoc assumptions or imprecise derivations, it is a direct calculation. This connection says that even if the expression for the learning capacity was motivated from thermodynamics and, a priori, doesn't have anything to do with learning theory, this expression can be connected directly established statistical learning theory. It is therefore a meaningful connection.

Reviewer comment (Evaluation):
In Section 4 - Experiments, even setting aside the earlier concerns about the proposed measure, it is questionable whether the actual raw value (rather than relative values) should be considered accurate. Comparing this to the actual number of parameters is problematic given the approximations.

Author response:
**Even setting aside my aforementioned concerns with a proposed measure of effective dimensionality, I am skeptical that the actual raw value rather than the relative values it outputs should be considered accurate. Comparing this value to the actual number of parameters seems problematic given the number of approximations taken in its estimation.**

This is not true. The learning capacity is a number that we have proposed as an estimate of the effective dimensionality. This number correlates with the test loss, is much smaller than the number of weights (as we would expect any estimate of the capacity of deep networks to be) and has many other interesting properties. The readers are free to use this number, use relative values if they want to model selection. We do not understand how the Reviewer can conclude that the actual values of the effective dimensionality are not accurate. We defined it, they are accurate by definition! The merit of a definition is whether it is useful, which we have shown that it is.

Reviewer comment (Experiments):
The claim that low effective parameter count is consistent with literature on distillation could be tested experimentally by training a large model and distilling it into a much smaller model to see if effective parameters remain constant or decrease slightly.

Author response:
**The authors mention that this low effective parameter count is consistent with other aspects of the literature e.g. distillation. This would seem to be a testable hypothesis. If we trained a large model with a given effective parameter count and nearly perfectly distilled it into a much smaller model, the number of effective parameters should remain constant or decrease very slightly. Without being convinced by the underlying measures U and C, the results in Table 1 are insufficiently granular to convince the reader of the effectiveness of these measures. I also find it surprising that regularization wasn't a primary consideration in attempting to validate the metrics.**

This is an interesting idea. We believe it will also work. A partial evidence for whether it will work is found in Fig. 3. Each of the colored lines are networks with many different parameters trained on the same number of samples. For the brown line in both Fig 3a and 3b, which are networks trained on all the training data in MNSIT and CIFAR-10, the learning capacity is essentially constant as a function of the number of weights in the model. 

Regularization is a tricky thing to do while working with even standard PAC-Bayes bounds. The effect of regularization interacts in complicated ways with the PAC-Bayes optimization. The paper by Jiang et al. (2019) also found that regularization does not affect the test error much in many cases. For example, one natural choice of regularization is the L2 norm. The L2 norm can only restrict the parameters from being too large. It did not prevent the learning capability from being larger or smaller so it is hard to have a dicisive conclusion from this experiment. L2 regularization is not actually that effective for deep learning, see https://arxiv.org/abs/1905.13277. More formally, L2 norm is meaningful for models where the classical bias-variance tradeoff holds. For neural networks with symmetries, the origin is not any more special than some other point in the weight space. For example, there is a scale symmetry in networks with batch-normalization and therefore any two weight configurations along a ray that intersects the origin express the exact same function.

We did perform some experiments using different learning rates, see the last question in the FAQ in Appendix G.

Now that we have defined this quantity and provided an easy way to estimate for deep networks, other researchers and evaluate whether it is useful to understand regularization.

Reviewer comment (Evaluation):
Without convincing evidence for the underlying measures U and C, the results in Table 1 lack sufficient granularity to demonstrate their effectiveness. It is also surprising that regularization was not a primary consideration in validating the metrics.

Author response:
**The authors mention that this low effective parameter count is consistent with other aspects of the literature e.g. distillation. This would seem to be a testable hypothesis. If we trained a large model with a given effective parameter count and nearly perfectly distilled it into a much smaller model, the number of effective parameters should remain constant or decrease very slightly. Without being convinced by the underlying measures U and C, the results in Table 1 are insufficiently granular to convince the reader of the effectiveness of these measures. I also find it surprising that regularization wasn't a primary consideration in attempting to validate the metrics.**

This is an interesting idea. We believe it will also work. A partial evidence for whether it will work is found in Fig. 3. Each of the colored lines are networks with many different parameters trained on the same number of samples. For the brown line in both Fig 3a and 3b, which are networks trained on all the training data in MNSIT and CIFAR-10, the learning capacity is essentially constant as a function of the number of weights in the model. 

Regularization is a tricky thing to do while working with even standard PAC-Bayes bounds. The effect of regularization interacts in complicated ways with the PAC-Bayes optimization. The paper by Jiang et al. (2019) also found that regularization does not affect the test error much in many cases. For example, one natural choice of regularization is the L2 norm. The L2 norm can only restrict the parameters from being too large. It did not prevent the learning capability from being larger or smaller so it is hard to have a dicisive conclusion from this experiment. L2 regularization is not actually that effective for deep learning, see https://arxiv.org/abs/1905.13277. More formally, L2 norm is meaningful for models where the classical bias-variance tradeoff holds. For neural networks with symmetries, the origin is not any more special than some other point in the weight space. For example, there is a scale symmetry in networks with batch-normalization and therefore any two weight configurations along a ray that intersects the origin express the exact same function.

We did perform some experiments using different learning rates, see the last question in the FAQ in Appendix G.

Now that we have defined this quantity and provided an easy way to estimate for deep networks, other researchers and evaluate whether it is useful to understand regularization.

Reviewer comment (Evaluation):
The experiments seem intended to investigate effective parameters assuming a valid measure, but the overarching issue may be that the proposed measure is not a good choice.

Author response:
**It seems that the general intention of these experiments is to investigate the effective parameters of models, datasets, etc. assuming we have a valid measure of effective parameters. I think the overarching problem is that the proposed measure may not be a good choice.**

This is an appropriate characterization of the paper. We have defined the learning capacity and studied it under various settings to see if it works well, e.g., in terms of its correlation to the test loss, its utility to see/unsee double descent, for different non-parametric models etc. We are not assuming in these investigations that our estimate of the effective dimensionality is correct. That part comes from the theory, e.g., the PAC-Bayes calculations, and experiment where we directly checked the learning capacity against PAC-Bayes effective dimensionality (Fig 4).

So far, in literature, the definitions of effective dimension are based on the analysis of the generalization performance of the models. For example, VC dimension and effective dimension mentioned in Yang et al.’s work. They first provide generalization upper bound and collect part of the terms and define them as effective dimension. However, the upper bound can be loose or vacuous due to many assumptions or approximations and this can lead to an overestimation of the effective dimension. In our work, the goal is a direct estimation of the effective dimension without layers of assumptions. The definition is sound and we also provide calculation and experimental results to link to PAC-Bays theorem, which shows similar result with less assumption or approximation and is more general. If you find the effective dimension under PAC-Bays framework meaningful, then our work is just a cheaper way to calculate it and it has the exact same place in the literature.

Reviewer comment (Evaluation):
The double descent results in Section 3.2 suggest problems with the proposed measure. The LHS plot does not show a double descent shape, and the RHS plot is (a) extremely noisy and (b) does not show the expected U-shape, indicating the measure may not be working effectively.

Author response:
**I found the double descent results in Sec 3.2 suggested issues with the proposed measure. Firstly, I don't see a double descent shape in the LHS plot. Then the right-hand plot seems (a) extremely noisy and (b) does not capture the correct U-shape that we would expect if it was working as we would hope. This seems to me to provide evidence that the measure is not working effectively.**

The left hand side panel does show double descent. Usually in many papers, researchers do not use bootstrap to calculate the test error and that is why they see a distinct shape of the double descent. When you do bootstrap, there will be error bars on the double descent curve at all points and the curve will be more diffuse. The curve for MNSIT in Fig 3a uses bootstrap and has error bars, the curve for CIFAR-10 in Fig. 3b does not use bootstrap and you will see the familiar double descent plot.

We have implemented these experiments carefully over almost two years, we do not believe there are mistakes here. For example, the training loss of models in our work is close to zero (0.01 - 0.1) in every case for deep learning models, so there are no under-fitting issues and that fits real practice well. The main spirit of the U-shape for deep learning is to reconcile the over-parameterization and generalization performance. Under our experimental condition that deep learning rarely underfit, we do show that higher effective dimension results in higher test loss.

It is true that the right hand side plot does not show a U shape, we have discussed this in the paper. It is not noisy actually if you see it, the p-value for the slope being non-zero is significant in all cases except the one with N=50,000. This is very surprising in our opinion and should be investigated further. This does not mean that it is wrong. If the Reviewer can make a concrete argument otherwise, we will gladly consider it. The U shaped curve of MSE is a very formal argument and the details of this differ in different models.

The brown line on the right panel which has a small slope corresponds to N=50,000. But it is actually very interesting that it does not show a linear trend. This shows that the learning capacity is almost constant as a function of the number of weights in the network (different brown points are different networks). This is in stark constant to the left panel where the test loss is non-monotonic in the number of weights of the network. In essence, when the number of samples available for training is very large, networks of the same architecture but different sizes have the same number of effective number of dimensions; this is presumably because there is not enough information in the dataset to constrain the large number of weights. A complement of this result is seen Fig 2 b (right) where networks of different architectures have different learning capacity at large sample sizes.

Reviewer comment (Evaluation):
In Section 3.3 and Figure 4, the evidence provided does not strongly support the claimed connection, and the reported Pearson correlation of 0.99 between the solid and dashed green lines is visually implausible.

Author response:
**In Sec 3.3 and Fig 4, in contrast to the authors, I would find it hard to conclude that there is a particularly strong connection based on the results provided in the plot. I also find it hard to believe that the Pearson Correlation between the solid and dashed green lines is 0.99 based on visual inspection.**

Sorry it should be Kendall-tau correlation, not Pearson. We have fixed this in the PDF now. The correlation is large because of the large error bars, and this why calculating the error bars is important.

Reviewer comment (Theory):
It is argued that the estimator of learning capacity can be used to calculate a PAC-Bayes bound. ("Our techniques to estimate the learning capacity are computationally inexpensive ways to calculate PAC-Bayes bounds.") This is not true. Such a bound must account for the uncertainty in the estimate of learning capacity, however, this work does not present bounds on the quality of the estimator.

Author response:
**It is argued that the estimator of learning capacity can be used to calculate a PAC-Bayes bound. ("Our techniques to estimate the learning capacity are computationally inexpensive ways to calculate PAC-Bayes bounds.") This is not true. Such a bound must account for the uncertainty in the estimate of learning capacity, however, this work does not present bounds on the quality of the estimator.**

Sorry for the confusion. In Section 2.3, the PAC-Bayes bound is presented, encompassing a term that approximates the learning capacity. In our analysis, minimizing the expression introduced in Section 2.3 can be interpreted as a trade-off between the distance separating the posterior and prior distributions, model complexity, and the training loss.

Reviewer comment (Writing):
The arguments are difficult to follow. Among others, I found the presentation of the connection between learning capacity and PAC-Bayes theory in section 2.3 rather confusing.

Author response:
**The arguments are difficult to follow. Among others, I found the presentation of the connection between learning capacity and PAC-Bayes theory in section 2.3 rather confusing.**

The presentation in Section 2.3 is a bit difficult to follow because it is a compilation of many results in the PAC-Bayes literature. Specifically, we are discussing the techniques used by the paper by Yang et al. (2022b) to define a kind of effective dimensionality (Equation 13). That paper gives rigorous and elaborate derivations of these formulae; it also showed that PAC-Bayes bounds can give a non-vacuous analytical expression for the upper bound on the generalization error. In Section 2.3, we are making the argument that their expression for the effective dimensionality is close to our definition for the learning capacity (Equation 15). This connection with PAC-Bayes also gives learning capacity a firm theoretical foundation.

Reviewer comment (Presentation):
The Monte-Carlo based estimator for learning capacity, which is listed as a main contribution, is only presented in the supplementary material.

Author response:
**The Monte-Carlo based estimator for learning capacity, which is listed as a main contribution, is only presented in the supplementary material.**
 

We had to make some editorial choices to fit the material within the 9 pages.

Reviewer comment (Theory):
Given these estimates of the average energy, we could now calculate \( C = -N^2 \partial_N U \) but it is difficult to do so because the noise in the estimation of U gets amplified when we take the finite-difference derivative. Could you clarify, whether this is an observation or a mathematical fact?

Author response:
**Given these estimates of the average energy, we could now calculate C = − N 2∂N U but it is difficult to do so because the noise in the estimation of U gets amplified when we take the finite-difference derivative." Could you clarify, whether this is an observation or a mathematical fact?**

This is an empirical observation. But you can appreciate why it could be true by looking at the expression in Equation (6,7,8).

Reviewer comment (Experiments):
Which value did you set \(\epsilon\) parameter for computing the effective dimensionality from Yang et al. in section 3.3 to (Yang et al. state that the parameter has to be user chosen)? How did you select it?

Author response:
**Which value did you set ε parameter for computing the effective dimensionality from Yang et al. in section 3.3 to (Yang et al. state that the parameter has to be user chosen)? How did you select it?**

We optimized the PAC-Bayes bound using the methods/code of Yang et al. to select the values of epsilon. We will make a table in the Appendix that details these values. Optimizing PAC-Bayes bounds correctly is tricky but we have a lot of experience in doing so. And as the numerical estimates suggest, our results are also consistent with existing results in the literature.

Reviewer comment (Theory):
Derivation of the PAC-Bayes bound could be much more rigorous than presented. The assumptions should be made clear, and the full bound provided as a theorem.

Author response:
Thank you. We have expanded the Appendix to include the derivations of the different expressions of the learning capacity, the log partition function and the PAC-Bayes bound.

Reviewer comment (Theory):
Theoretical arguments all rely on a non-singular Hessian, which is never the case for practical neural networks. See, for example [1].

Author response:
Thank you for pointing us to this paper. We will add a discussion to the updated manuscript. There are indeed direct connections between our calculations and singular learning theory. We provided a majority of our derivations using non-singular Hessians because this is easy. But we also discuss singular Hessians, e.g., in the narrative on Page 3 before Equation (5). Appendix C also details these connections precisely; in an initial submission of our paper we found that the readers were confused with the exposition of singular learning theory (which is rather uncommon in mainstream machine learning literature...this is the reason we moved that content to the Appendix.

As the Reviewer might realize, the merit of our calculations is that the learning capacity gives the correct expressions for the effective dimensionality even for singular models. And this is also evident in our experiments with deep networks, which are singular almost everywhere.

Reviewer comment (Experiments):
Models considered are relatively small, and not nearly state-of-the-art. This can be forgiven, due to the complexities associated with estimating Bayesian quantities, but is nonetheless an issue that would need to be overcome eventually for the work to be practical.

Author response:
Fair point. As we discuss in the Introduction, our perspective on this is as follows. PAC-Bayes bounds correlate very well with the test error of deep networks. But beyond small multi-layer preceptors on MNIST of CIFAR-10, it is very difficult to calculate PAC-Bayes bounds without using compression. We see our work as approximating the PAC-Bayes calculations using Monte Carlo and this allows us to scale to much larger networks.

To wit, the networks considered here may be small by the standards of foundation models, but they are not really all that small. The LSTM on WikiText-2 has ~14M weights, and the WideResnet on CIFAR-10 achieves a test accurate of 93.7%  which is very good. These techniques can be implemented as is for large networks. The MCMC procedure in Appendix B gives an even more efficient mechanism to estimate the learning capacity. Altogether, we believe that these techniques provide evidence that deep networks have much fewer effective degrees of freedom than what the number of weights indicate and that when we consider the test loss as a function of these effective degrees of freedom, we do not see double decent. These are the main points of our paper.

Indeed, research on Bayesian statistic for large models will require high computational complexity but we regard our work as a start. The calculations and experimental results give intuitive understanding and effectiveness of the quantity. By proving that this quantity can be a correct measure, the result can be used as a foundation to justify the soundness of a theory or can be used as a quantity to check whether a model saturate or not in terms of performance. We leave the further improvement as future work.

Reviewer comment (Experiments):
While the approximation methods for the learning capacity appear to be effective, they also appear to be very expensive, requiring multiple trained models over bootstrapped versions of the datasets, or a lengthy MCMC chain.

Author response:
Yes, quantity involving Bayesian statistic often require boostrapped or sampling from a posterior given an algorithm. For this paper, since our goal is to make a convincing argument for the utility of the learning capacity, we have focused on accurate estimation using a lot of Monte Carlo. For practical applications, one could implement fewer bootstraps, or using the MCMC method described in the Appendix. This may not be accurate but it will likely be good enough for decision making, e.g., when a model is data starved, vs. when there is sufficient amount of data and we should focus on the architecture; this is an important practical implication of the freezing phenomenon discussed in our paper.