Thank you very much for your thoughtful review. It is gratifying to know that you found the method theoretically sound and intuitive, and that the experimental results were a strength of the paper.

Whereas all the components in S$\mu$Par seem to be important, the importance of each individual parameterization is not studied. It could be the case that S$\mu$Par for weights while keeping $\mu$P parameterization for gradients and learning rate may suffice and produce the same performance. For the sake of completeness, I would suggest ablating individual terms. Prior [1], [2] work shows that proper weight initialization (scaled according to sparsity) already improves training dynamics.

This is an excellent suggestion! In Section 1 "Individual ablations of S$\mu$Par initialization and learning rate corrections" from our "Global Author Rebuttal" we perform the ablation you suggested and show that both the initialization and learning rate correction are required for S$\mu$Par to achieve transfer of optimal initialization standard deviation or optimal learning rate across sparsity levels. We will include these ablations in the camera version of the paper.

Does the S$\mu$Par transfer to other domains in the same way? For example, one could try pretraining a vision transformer in a setup similar to "Scaling laws for sparsely-connected foundation models".

This is a good question because much of the sparse neural network literature has focused on the vision domain. Since ViT shares most of its architecture with language transformers, we believe the extension of $\mu$P and S$\mu$Par to ViT should also follow Table 1 from our main submission, but we leave testing it as future work.

However, the setup you are referencing also uses gradual magnitude pruning (GMP). In Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal", we show that S$\mu$Par is not correct for GMP and provide an explanation as to why (continuing our explanation from Lines 274-278).

"Scaling laws for sparsely-connected foundation models" showed that given sufficiently long compute, sparse models turn out to be compute-optimal. Given that S$\mu$Par finds better hyperparameters, could one expect that higher sparsity levels may be favored compared to the recipe used in the aforementioned work?

This is a very interesting paper with potentially high impact. On Lines 262-266, we discuss how the setup used in "Scaling laws for sparsely-connected foundation models" adjusts the learning rate proportional to sparsity. This correction likely has a similar effect to the S$\mu$Par learning rate correction however without both an appropriate initialization and learning rate correction, optimal hyperparameter transfer is not guaranteed. Without HP transfer, it is likely that many of the models trained in this paper experience "de-tuned hyperparameters" as width and sparsity are varied. As we discuss in Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal", if one could extend S$\mu$Par for arbitrary sparse algorithms, then that parameterization would likely improve the sparse models in the aforementioned work, especially at high sparsity levels. We leave the development of this method as future work, as stated on Line 278.

It seems like the value of train loss decreases with the increase of sparsity (i.e., sparse models are better given the same number of parameters), whereas in Figure 10, the performance of S$\mu$Par loss seems to be independent of sparsity. I would expect an increase in the quality gap for longer training according to sparsity scaling laws.

Iso-FLOP efficiency isn't the focus of our paper. However in Figure 10 we do include the iso-Parameter scaling tests similar to the scaling strategy in "Scaling laws for sparsely-connected foundation models" as well as several other papers. However "iso-Parameter" doesn't necessarily mean iso-FLOP because increasing $d_\text{model}$ also increases attention dot product FLOPs, which can't be sparsified. This increase in attention FLOPs is so significant that our 87.5% sparse model from Figure 10 has just over double the training FLOPs of the dense baseline, with virtually unchanged loss. Unfortunately, "Scaling laws for sparsely-connected foundation models" did not model the effect of attention FLOPs in their scaling law functional forms, instead assuming that sparsity and parameter count alone are sufficient. We suspect that many of the sparse models in that paper use considerably more total training FLOPs than their dense baselines, making for an unfair comparison. It is unclear whether their results hold under more robust FLOP comparisons. For this submission though, we consider the work of beating dense FLOP efficiency with sparse training as future work. We will update our manuscript to reflect the subtleties of FLOP counting in sparse models.

Would be interesting to see if S$\mu$Par is optimal for DST (eg. RigL, Top-KAST, AC/DC)

This is a good suggestion for an experiment! Unfortunately S$\mu$Par is not optimal for DST methods. We demonstrate this and provide an explanation in Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal". We leave the solution as future work.

Thank you for your detailed and constructive feedback, and your kind words regarding the paper’s writing, experimental results, and core idea.

SuPar is discussed and evaluated only with the random unstructured sparsity pattern, which is hardly used in practice to speed up the DNN training and inference.

It is true that the current adoption rate of unstructured sparsity in training and inference is low, as we acknowledge on Lines 35-36 and 73-75 of our submission. While we currently mostly motivate unstructured sparse training on Lines 27-28 and 279-286, we will note in the paper introduction that unstructured sparsity remains a very important technique and intense focus of research for ourselves and other labs, as accelerators such as CPUs, Cerebras, GraphCore, and SambaNova can take advantage of unstructured sparsity during training and inference. Furthermore, GPU and TPU systems support 2:4 block structured sparsity which is quite similar to 50% unstructured sparsity.

Research has shown that the less structure imposed on sparsity masks, the better the performance, making this setting relevant to study in training and inference. For example, SparseGPT [Frantar and Alistarh(2023)], the SOTA LLM pruning technique creates unstructured sparse masks that require hardware acceleration to realize an inference benefit. They report a CPU inference speedup, demonstrating the potential for real-world impact of unstructured sparsity.

Additionally, in Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal", we test RigL and GMP dynamic sparse training and show that none of SP, $\mu$P, or S$\mu$Par achieve transfer of optimal learning rate across sparsity level. We hope this helps address your concerns around applying S$\mu$Par outside the context of random unstructured sparsity.

The theoretical contribution is somewhat weak. The proposed parameterization method is simple extension of maximal update parameterization to the sparse version, which I don't see significant technical improvement.

It is true that S$\mu$Par is a simple extension of $\mu$P. However, we note that we are the first to acknowledge and propose a solution to the systematic misuse of dense hyperparameters in sparse training. On Lines 90-91 and Footnote 1 (page 3) from our submission, we demonstrate how widespread the issue is by providing numerous examples of sparse training works which use dense hyperparameters for sparse training. So, regardless of the perceived significance of technical improvement, the work is important and the potential for impact is fairly high.

Why does S$\mu$Par stabilize the sparse network training even after multiple training steps? How is S$\mu$Par technically different from the previous sparse weight initialization methods?

We take your point, and indeed we discuss this on Lines 267-273. $\mu$P and S$\mu$Par differ from pure initialization-based methods due to the presence of a learning rate correction. The combination of an initialization and learning rate correction allows S$\mu$Par to stabilize sparse network training even after multiple training steps, as seen in Figure 5 of our main submission. In Section 1 "Individual ablations of S$\mu$Par initialization and learning rate corrections" from our "Global Author Rebuttal", we also show that the S$\mu$Par initialization adjustment alone is not sufficient to achieve transfer of optimal HPs across sparsity levels.

Additionally, please see Section 3 "Downstream Task Evaluations" from our "Global Author Rebuttal" where we provide accuracy comparisons between SP, $\mu$P, and S$\mu$Par.

We also think the comparison across PaI methods is somewhat orthogonal to our work because we are not studying the optimal sparse training recipe (eg. PaI, DST, prune-retrain), but instead how to best control sparse training dynamics.

Is the proposed parameterization method generalizable to any structured sparsity patterns (e.g., channel pruning) for the density-invariant hyperparameters? As mentioned in the paper, the unstructured sparsity is extremely hard to speed up. Discussing S$\mu$Par on more structured sparsity patterns, such as channel pruning, will make the paper stronger.

Great question! On Lines 160-161 we remark that S$\mu$Par should easily extend to random 2:4 structured sparsity as it closely resembles random unstructured sparsity. S$\mu$Par would also work for random structured pruning of entire neurons at initialization because this case simply reduces to training with a narrower dense model. However, S$\mu$Par does not transfer to dynamic sparse training because weights become non-Gaussian. Please see Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal" for a discussion on the limitations of S$\mu$Par in dynamic sparse training. We will update the main body to more thoroughly discuss these possible extensions. Thank you for the feedback.

Can S$\mu$Par improve the final performance of IMP?

This is an interesting question. S$\mu$Par improves random static unstructured sparse training and IMP rewinds weights back to their initial values while maintaining the same mask. If the IMP mask at initialization still allows the non-zero weights to have a Gaussian distribution, then S$\mu$Par would apply to this case. Therefore, S$\mu$Par could prevent ``hyperparameter detuning'' in later IMP iterations, and potentially improve IMP losses. Given the popularity of IMP/LTH, we think this discussion would make an excellent addition to Section 5 of our paper. Thank you for bringing this up!

DeltaX vs DeltaW notation

$\Delta W$ refers to the weight update whereas $\Delta \mathbf{X}$ refers to "the effect of the weight update”. We will update Line 113 accordingly.

[Frantar and Alistarh(2023)] Elias Frantar and Dan Alistarh. 2023. SparseGPT: Massive language models can be accurately pruned in one-shot. In International Conference on Machine Learning.

Thank you very much for the helpful comments, and for your kind words regarding the paper's key idea, motivation, and clear and well-structured presentation.

This paper has many supporting experiments reporting the training loss, validation loss, and transfer loss for SµPar. However, loss values are not always reliable for evaluation. The accuracy data is a more convincing data to demonstrate the performance. I wonder what is the accuracy for the experiments. It would make reader easy to compare against other methods.

Thank you for suggesting this. Please see Section 3 "Downstream Task Evaluations" from our "Global Author Rebuttal". We hope this addresses your concerns.

What is the method that is used to prune the network? There are lots of method such as SNIP and GraSP for static sparse training. However, this part is unclear according to the paper.

Thank you for bringing this lack of clarity to our attention! In this work we solely used random static unstructured pruning at initialization, as stated on Lines 107, 512, 528, 549, and 564. To improve clarity we will update our abstract, intro, and conclusion to further emphasize our choice of pruning method. As we noted on Lines 27-28, recent work has shown random sparsity to be a surprisingly effective strategy for sparse training, so we adopted this in our work.

The author claims SµPar is a holistic solution. However, from what the paper has demonstrate (discussion, experiments), the reviewer cannot be convinced. According to weaknesses 2, we don’t know what is the pruning method used in sparse training, and according to the paper, the author also not performs applying SµPar on different sparse training methods. Therefore, we don’t know if SµPar works for different sparse training algorithms. Furthermore, I assume this paper only discuss static sparse training (such as SNIP), without discussing the performance on dynamic sparse training (DST) which is a more reliable and promising sparse training domain (e.g., better accuracy, flexible). Therefore it is not suitable to claim SµPar is a holistic solution. Even the authors don’t make such claim, without sufficient experiments the reviewer mentioned in this part, this paper is still lacks fundamental evidence to demonstrate effectiveness.

Thank you for raising these points. Please see Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal". This section shows that none of SP, $\mu$P, or S$\mu$Par achieve transfer of optimal learning rate across sparsity level for RigL and GMP. We hope this addresses your concerns around applying S$\mu$Par to dynamic sparse training methods.

Also, we agree that ``holistic'' is a vague term, as it might imply S$\mu$Par solves challenges with other forms of sparse training, such as dynamic sparse training. We should have been more clear that S$\mu$Par holistically controls the scale of the forward, backward, and weight update operations with respect to model width and level of unstructured static random sparsity –- some of these operations have been handled in isolation in prior work, but not all together.

We will also update our introduction to make it clear that S$\mu$Par does not extend to pruning or dynamic sparse training because these methods can cause the unmasked/non-zero weight distribution to become non-Gaussian (as noted currently only in L274-L278).

Thank you for your very useful suggestions, your positive comments regarding the promising/interesting findings, and your note on the potential for impact in the community.

This is an interesting study with a relatively low level of originality (in my opinion). It practically puts together and discuss a bit more various findings from various papers. Also, the paper structure and clarity could have been improved. Currently, it is not easy to understand it. If the paper would be more mature, may have a relatively fair impact in the community.

We take your point regarding the level of originality, however we note that we are the first to acknowledge and propose a solution to the systematic misuse of dense hyperparameters in sparse training. On Lines 90-91 and Footnote 1 (page 3) from our submission, we demonstrate how widespread the issue is by providing numerous examples of sparse training works which use dense hyperparameters for sparse training. So, regardless of the perceived level of innovation, the work is important and the potential for impact is fairly high, as you noted.

I believe that one main weakness is that the networks performance is reported just with loss values. For a more comprehensive view, and deeper understanding other performance metrics (task dependent) shall be also presented for all experiments.

Thank you for suggesting this. Please see Section 3 "Downstream Task Evaluations" from our "Global Author Rebuttal". We hope this addresses your concerns.

With respect to the problem motivation, I believe that the paper exaggerates a bit the difficulty of training static sparse neural networks. Perhaps, for static sparsity, the effect of hyperparameters on network performance is more pronounced, but the alternative dynamic sparsity is quite robust to hyperparameter choice (e.g., sparsity level) and can obtain quite easy performance on par (or even better) with dense neural networks while having much fewer theoretical computational requirements as reported in a number of related works. For a fair overview, the paper shall discuss also comparatively the performance of static sparsity against dynamic sparsity and pruning, or alternatively, to motivate better its settings.

This is very perceptive –- and a point that we did not fully appreciate previously. But first of all, we should have done a better job of motivating static random sparsity – the topic of the paper – to begin with. We will note in the paper that static sparsity remains a very important technique and intense focus of research for ourselves and other labs, not least because it enjoys native hardware support (e.g., 2:4 sparsity in NVIDIA GPUs, full unstructured sparsity in Cerebras chips).

That being said, investigating hyperparameter shift in DST methods is insightful. Please see Section 2 "Dynamic sparsity hyperparameter transfer" from our "Global Author Rebuttal". In this section we show that the optimal learning rate significantly changes across sparsity levels for both RigL and GMP, contradicting your claim that dynamic sparsity is quite robust to hyperparameter choice and further highlighting the importance of developing systematic solutions like S$\mu$Par. As we discussed on Lines 274-278 of our submission, S$\mu$Par is not correct for dynamic sparse methods, leaving an impactful direction for future work.

We really appreciate your suggestion here as these interesting findings definitely help expand the paper’s contribution!

(minor) The source code shall be provided to ensure easy reproducibility.

Unfortunately we do not have an open-source repository for S$\mu$Par. However, we do provide Table 1 in our submission, which outlines the simple implementation changes required to implement $\mu$P and S$\mu$Par. Based on this it should be straightforward to extend existing implementations of $\mu$P to implement S$\mu$Par (e.g. https://github.com/microsoft/mup).