Paper: Towards Explaining Deep Neural Network Compression Through a Probabilistic Latent Space (ICLR-2024)

Reviewer comment (Writing):
The English of this paper needs to be improved significantly.

Author response:
Thank you for your point,  throughout the paper, we have improved the language and provided further clarifications.

Reviewer comment (Writing):
Many statements and definitions are unnecessarily wordy and long, making reading and following the paper difficult.

Author response:
Thank you for your feedback. We have made revisions throughout the paper.

Reviewer comment (Writing):
What is the meaning of $\mathcal{F}$ in the empirical risk minimization? I don't follow the notation used in equation (1). If $w_F$ denotes the whole set of weights, what is the meaning of the product between $Y$ and this term (loss is supposed to be a scalar value)? What kind of product is considered in the expression of $l(w)$? The same question applies in Lemma 2 for the function $g$.

Author response:
Thank you for your question. 
$\mathcal{F}$ represents a set of networks, each characterized by a set of parameters denoted as 
 $w_{F}$. The objective of minimizing the loss function is equivalent to finding a network, or more precisely, a set of parameters, that minimizes the value of the formula given specific inputs and labels. In addition, one way to think of loss function in deep neural networks is a correlation between the class label and the last layer of the network. More specifically, training a classifier is defined by minimizing a loss function that decreases with the correlation between the weighted combination of the layers and the class label.
 In our paper, for simplicity, this correlation is defined as an inner product between $Y$ and the last layer features, but it is straightforward to generalize this to more practical loss functions such as logistic or hinge loss. Note that the same definition is used in the following papers to name a few.

 Ilyas A, Santurkar S, Tsipras D, Engstrom L, Tran B, Madry A. Adversarial examples are not bugs, they are features. Advances in neural information processing systems, 2019.

Craig J, Andle J, Nowak TS, Sekeh SY. A Theoretical Perspective on Subnetwork Contributions to Adversarial Robustness. arXiv preprint arXiv:2307.03803, 2023.

Reviewer comment (Theory):
In the example on page 3, the authors have said that "without loss of generality, assume that ...". Why should the covariance of the projected pruned weights be positive definite (all eigenvalues are strictly positive)? This is actually used in the experiment of section 5.2.

Author response:
We thank the reviewer for the constructive feedback. In the revised version, we have made the sentence more explicit to clarify. In this paper, we exclusively utilize positive definite covariance matrices for the latent spaces, and our proofs are based on this assumption. This assumption is now explicitly stated in the introduction for transparency. Additionally, in the example provided, we assume, without loss of generality, that the two covariance matrices are identical. Given the assumption maintained throughout the paper, we can confirm those covariance matrices are positive definit.

Reviewer comment (Experiments):
I understand this is a theoretical paper, but choosing only CIFAR10/100 datasets, experimenting in a controlled setup like a diagonal covariance matrix, and on the last layer, using only three architectures for a simple classification problem (no other tasks have been considered), and no experiments for other architectures (RNNs, Transformers, etc.) is not a completely convincing approach.

Author response:
Regarding the setup and the use of non-diagonal covariance matrices, we conducted experiments as detailed in the Appendix, specifically in the "MULTIVARIATE GAUSSIAN DISTRIBUTION WITH NON-DIAGONAL COVARIANCE MATRICES" section. While our primary goal was to establish theoretical explanations through our theorems, we acknowledge the limited inclusion of experiments with networks beyond the three utilized in this study.

Reviewer comment (Experiments):
The focus on specific architectures (ResNet50, AlexNet, VGG16) may limit the general applicability of the findings. The behavior of the pruning methods on newer or different architectures remains unexplored.

Author response:
We'd like to point out that in the Appendix, we have provided an extensive range of experiments including various architectures. In this paper, we intend to provide an in-depth theoretical analysis of our pruning explanations along with evaluating them on common datasets CIFAR 10 and 100 and VGG, ResNet, and AlexNet networks. To our knowledge, we are the first to explain DNN pruning analytically from a novel information theoretical angle.

Reviewer comment (Experiments):
The paper could benefit from a direct comparison with state-of-the-art pruning techniques in terms of performance, computational efficiency, and model robustness. This would give readers a clear benchmark on where the introduced methods stand relative to existing methods.

Author response:
Note that, the AP2 and AP3 metrics are not presented in this paper as newly proposed pruning approaches. While AP2 may resemble weight magnitude pruning, it is not employed here as a method to prune the network. Instead, these metrics are utilized to elucidate why the accuracy of the network does not significantly decrease when optimal pruning is applied.

It's noteworthy that, in future research, based on our in-depth analysis in this paper we plan to introduce a novel pruning approach utilizing the AP3 metric. This new approach will be compared with existing techniques to assess its efficacy in network compression.

Reviewer comment (Experiments):
There is no mention of the robustness of the pruning methods against adversarial attacks or their performance under distribution shifts. Given the increasing importance of model robustness, an assessment in these areas could significantly strengthen the paper.

Author response:
Thank you for your inquiry. It's important to clarify that AP3 is not designed as a pruning approach but rather serves as an explanatory metric for network pruning in this study. However, in future research, as we introduce a novel pruning approach based on AP3, we will thoroughly investigate its robustness.

Reviewer comment (Novelty):
The paper fails to make a distinguishable contribution to understanding why and when compression works. It does not live up to the term "explaining" in its title since it contains no arguments why or when compression should be applicable without harming performance. In my estimation the paper ends up correlating some probabilistic divergences with performance difference between original and pruned network, but even then 1- using the Frobenius norm of the parameters' difference seems to work better, 2- the mapping of parameters to probabilistic measures are not motivated in any systematic way.

Author response:
Our paper primarily offers a theoretical explanation of deep neural network pruning. Our objective is not to advocate for a specific pruning approach or demonstrate the superiority of our method over others. Rather, we provide an explanation of the pruning and performance relationship using the Frobenius norm. It's important to note that our experiments serve as evidence supporting our theorems, illustrating why optimal pruning leads to nearly identical accuracy. However, in future works, we plan to propose a novel pruning approach that leverages the AP3 metric and our studies in this paper. This involves fine-tuning hyperparameters and investigating various non-Gaussian distributions to achieve improved accuracy for the pruned network, compared to utilizing the Frobenius norm.

Reviewer comment (Writing):
Pg. 1 onwards: I find the usage of "probability space" and related terms confusing throughout the paper. Here $\mathcal{Z}$ is introduced as a "probability space" but I strongly suspect the authors mean the domain (state space) of the random variable $Z$. Similar for $\tilde{Z}$.

Author response:
We recognize potential confusion and agree with your suggestion to clarify. Regarding $\mathcal {Z}$, our intent is to represent a probability space, denoted as $(\Omega, \mathcal{F}, P)$.

Reviewer comment (Writing):
Pg. 1: Neither in the beginning, nor anywhere the paper makes an explicit, mathematical definition of compression. Given the centrality of the notion for the paper, and the various ways in which it can be defined, I find this a very important shortcoming.

Author response:
As stated in the abstract, our focus in this paper is on pruning as a compression method. We provide the mathematical definition of pruning on page 2, immediately following Equation 1. "Note ${\widetilde{\omega}}^*=m^*\odot \omega^*$ is optimal sparse weight where $m\in[0,1]^d$ is the binary mask matrix ($d$ is dimension). The optimal sparse weight is achieved by  $(m^*, \omega^*):= \underset{\omega,m}{\arg \min}  \mathbb{E}_{(\mathbf{X},Y)\sim D} \\{ Y \cdot \ell(\omega) \\}$".
This definition aligns with standard approaches, and we refer to a specific paper as an example.

 Blalock, Davis and Gonzalez Ortiz, Jose Javier and Frankle, Jonathan, and Guttag, John, "What is the State of Neural Network Pruning?," Proceedings of Machine Learning and Systems, 2020 (2), 129--146.

Reviewer comment (Writing):
Pg. 2: What is a "probabilistic space"? If the intended meaning is probability space, more care should be taken for using the established terminology. If not, explicit definition is required.

Author response:
By probabilistic space, we meant the same as probability space and we fixed this in the revised version.

Reviewer comment (Writing):
Pg. 2: I find it extremely confusing that the authors use the same terminology for the parameter matrix and its vectorized version. This confusion goes beyond this point as well. E.g. although the authors state that they will proceed with the vectorized parameters, Definition 1 clearly refers to parameter matrix.

Author response:
We maintain consistent notation for both weight matrices and vectors for simplicity, explicitly noting in the Notations section that weight tensors can be viewed as high-dimensional vectors. We have made adjustments in the introduction for enhanced clarity and acknowledge and correct the noted typo in Definition 1.

Reviewer comment (Writing):
Pg. 2: In (1), why do the authors refer to a weighted combination of models in the model space? As far as I can see this construction never referred to again.

Author response:
Indeed, it is a common definition of loss function. 
 Moreover, we defined $\ell(\omega)$ as the weighted combination of the networks and we used this definition in our proofs.
For reference, Please see the loss function definitions in equations 3 and 2 in the following papers, respectively.

1. Ilyas A, Santurkar S, Tsipras D, Engstrom L, Tran B, Madry A. Adversarial examples are not bugs, they are features. Advances in neural information processing systems (NeurIPs), 2019.

2. Craig J, Andle J, Nowak TS, Sekeh SY. A Theoretical Perspective on Subnetwork Contributions to Adversarial Robustness. arXiv preprint arXiv:2307.03803, 2023.

Reviewer comment (Theory):
Pg. 2: I think the paper needs to justify that $\tilde{w}^* = m^* \odot w^*$, that the ideal compressed parameters are a masked version of the ideal uncompressed parameters. But given no explicit definition of compression/pruning has been presented, this is difficult to do here.

Author response:
As previously mentioned, employing  ${\widetilde{\omega}}^*=m^*\odot \omega^*$ serves as one approach to define pruning.

Reviewer comment (Writing):
Pg. 2 Definition 2: What is a filter? Above $f^{(l)}$ has been explicitly introduced as a network layer. But in this definition $f^{(l)}$ is called a "filter" and the layers are referred to through scalar $l$. I think that for a paper with theoretical aspirations the notations and terminology are too chaotic.

Author response:
In general, the notation $f^{(l)}$ (referring to the l-th layer of the network) denotes the filters of that particular layer. We have clarified this in the revised paper.

Reviewer comment (Writing):
Pg. 2: The paper refers to $\omega^{(l)}$ as both vectors and "parameter set"s. What is a parameter set? Why is the naming for this term changing?

Author response:
The term "parameter set" refers to the parameters associated with a distribution. The name of the term $\omega^{(l)}$ remains unchanged but now serves as a parameter of the distribution. The revised sentence in the updated version aims to provide clarity.

Reviewer comment (Theory):
Pg. 2: "$\mathcal{Z}$ and $\tilde{\mathcal{Z}}$ are latent spaces that could overlap." Should these two spaces not be the same in order for KL divergence to be computable? What are we integrating over in Eq. 2, if not?

Author response:
To have a compatible kl divergence between two distributions, they should have the same domain. So either two probability spaces should have some common elements or be aligned. Although, the distributions can be different or have different parameters. The integration is on the same domain of distributions.

Reviewer comment (Theory):
Pg. 4 Definition 5: Why is the probability space in the first sentence defined? Is it used in any way? Is $\mathcal{F}$ the event space here, was this notation not reserved for the model space? I strongly suspect that the authors intended mapping is to a space of probability spaces defined on the same measurable space, with the state space being $\mathcal{Z} = \tilde{\mathcal{Z}}$. If this is true, the definition strays far away from this purpose, and the use of terms such as "probabilistic space" adds to the confusion.

Author response:
Thank you for pointing that out. We acknowledge the error, and it has been appropriately addressed in the revised version.

Reviewer comment (Writing):
Pg. 5 Theorem 1: The theorem starts with a compressed training scheme. This is not introduced beforehand or afterwards.

Author response:
Thank you for the feedback. We have rephrased the expression to convey the concept in an alternative manner.

Reviewer comment (Reproducibility):
Pg. 6: How are these models pre-trained?

Author response:
As indicated in the "EXPERIMENTS" section, the models utilized in our study have been pre-trained on the Imagenet dataset. We employed pre-trained models available in the PyTorch framework.

Reviewer comment (Novelty):
Pg. 8: Given these results, do not the papers' findings boil down to suggesting using Frobenius norm of the original and pruned parameters difference for examining compression and performance? If so, is this not a very novel finding, and should be considered in light of previous results such as [1].

Author response:
Our paper primarily offers a theoretical explanation of deep neural network pruning. Our objective is not to advocate for a specific pruning approach or demonstrate the superiority of our method over others. Rather, we provide an explanation of the pruning and performance relationship using the Frobenius norm. It's important to note that our experiments serve as evidence supporting our theorems, illustrating why optimal pruning leads to nearly identical accuracy. However, in future works, we plan to propose a novel pruning approach that leverages the AP3 metric and our studies in this paper. This involves fine-tuning hyperparameters and investigating various non-Gaussian distributions to achieve improved accuracy for the pruned network, compared to utilizing the Frobenius norm.