__ Summary and Contributions__: Provides a new analysis with a clearer dependence on key problem constants then existing approaches to the analysis of clipped gradient descent with and without momentum.

__ Strengths__: The analysis performed in this paper seems like a good contribution. I've not seen the Lyapunov function they use before, the use of a non-squared norm is unusual and interesting. The interplay between the constants also provides insight into what's going on, the fact that the convergence rate is dominated by the L_0 constant matches intuition, as well as the pragmatic fact that gradient clipping is most useful at the earliest iterations.
Analyzing methods with step-dependence scaling in the stochastic case is challenging, and so the contribution in this respect is strong.
The experiments in the paper are good, they perform a hyper parameter search and the plots are clear and easy to read.

__ Weaknesses__: A more in-depth discussion of the mixed method which appears to work better empirically may improve the paper. I would move the imagenet experiments into the main body of the paper as well, as having three datasets in the main body of the paper is the informal standard.
The final test errors should be reported in each case, with the test error shown. Ideally 10 trials instead of 5 should be used.
Is there a reason to use "energy function" instead of "Lyapunov function"? The later is more standard.

__ Correctness__: I have not checked the proofs in the appendix.

__ Clarity__: This paper is very clear and concise.

__ Relation to Prior Work__: There appears to be a reasonable set of references to existing research.

__ Reproducibility__: Yes

__ Additional Feedback__: UPDATE: I have discussed the paper with other reviewers.

__ Summary and Contributions__: The authors improve on an analyis by Zhang et al. 2020 which introduced the notion of (L0,l1) smooth functions and derived bounds on clipped GD and SGD.
The improvements contributed by this work are:
- a more general framework of analysis which covers the work of Zhang et al, but also allows for the consideration of clipped/normalized momentum as well as "mixed clipping", a less commonly used variant which yielded good empirical results
- obtaining tight bounds for this general framework, yielding much sharper rates than the original work
- experimentally verifying the theoretical results

__ Strengths__: The paper gets nice empirical results with a relatively simple analytical framework which seems to yield good bounds.
The fact that the proof is based on an energy function means that it can handle momentum and the temporary increases in function value as well and so could possibly be used as a stepping stone to also analyse the behaviour of adaptive algorithms under clipping (although I have to couch this assessment since I am not very familiar with the theoretical works in this area).

__ Weaknesses__: - small and easily fixable: I think the experiment section would gain additional context if there was a comparison with AdaM with default parameters included, since it's the current "default" optimiser and could give context for the performance reached by SGD. In particular, since the work [https://arxiv.org/abs/1705.08292](https://arxiv.org/abs/1705.08292) showed the tradeoffs involved between training speed of of adaptive methods and performance at test time, seeing whether the proposed method retains the improved generalization while benefiting from a speedup using momentum would be quite interesting
Edit: authors addresed this in rebuttal
- in order to compare the results of this work with those of Zhang et al. 2020, using the same resnet architecture+hyperparameters for them for the CIFAR10 evaluation and reporting the validation loss on top of perplexity (while also matching hyperparameters) would be nice
Edit: fixed in rebuttal
- also, adding a run of the algorithms analysed in Zhang et al. for direct comparison of empirical behaviour to see the impact of momentum and mixed clipping
- reporting standard deviation as shaded area in the plots would be appreciated
- some discussion on what the theory implies for the values of the clipping parameter gamma, the step size and the momentum parameters given specific L0,L1 of the functions (+ the sigma for the stochastic gradient oracle) and some constructed toy examples to verify predictive power of the framework would be a nice
Edit:
the rebuttal did not comment on the request for standard deviations across multiple training runs, which I would stress on adding, otherwise I feel most of these were adressed

__ Correctness__: I have not been able to go through the proofs in depth, but the general arguments seem to make sense to me.
The empirical methodology seems sound, although reporting the standard deviation of the runs would be appreciated

__ Clarity__:
The paper is well written and clear to me, despite not being an expert I felt I could follow the general shape of the proofs.

__ Relation to Prior Work__: The paper builds heavily on Zhang et al 2020 and does adequate comparisons with it and other relevant work as best as I can judge, but the experimental section could use some more direct comparisons.

__ Reproducibility__: Yes

__ Additional Feedback__: The broader impact discussion is well done, although the privacy comment seems a bit thrown in. Another (half) sentence how momentum/other gradient information can leak training data information and in what settings would help clear this up

__ Summary and Contributions__: This paper present a new analysis of (momentum) SGD with gradient clipping. Its contributions are two fold: (1) improve the rate compared with a recent work; (2) include more variants in the framework.

__ Strengths__: The strengthen of this paper lies at the theoretical improvement in terms of the problem parameter L1. It improves our understanding of SGD with gradient clipping.

__ Weaknesses__: Although the improvement is interesting, it has some drawbacks in analysis:
1. It is unfair to compare with Ghadimi and Lan 2013, because they assume weaker assumptions about stochastic gradient, which is very important in the context of gradient clipping. The reason that gradient clipping is useful is that stochastic gradient might have large error, it could be heavy-tail or unbounded. However, this work simply assume that the difference between stochastic gradient and true gradient is bounded, which implies that stochastic gradient is bounded if we assume the true gradient is bounded. This is very problematic. Even one can relax the assumption to sub-Gaussian light tail assumption, it is still not valid in practice.
2. The theoretical version of the proposed algorithm is very impractical. In particular, the clipping parameter is very small in the level of \epsilon^2, and the step size \eta is also very small. This makes a huge gap between theory and experiments, where \gamma is set to be a constant.
3. The theoretical results seems indicate the setting \beta=0 gives the fastest convergence. This seems problematic. There should be some tradeoff in the iteration complexity involving \beta.

__ Correctness__: The small value of \gamma seems problematic and its gap with the empirical study exists.

__ Clarity__: Yes

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__: