__ Summary and Contributions__: This paper provides the first convergence analysis of data echoing-based extensions of SGD, proximal SGD, and Nesterov's accelerated SGD, in which the corresponding algorithm takes additional gradient steps with the current minibatch before a new minibatch is arriving; hence the term "echoing". It focuses on the convergence and generalization for the described echoed gradient methods in convex optimization, and provides an understanding of bias-variance trade-off as batch sizes grows in the data echoing regimes.

__ Strengths__: - Data echoing is a simple strategy that increases hardware utilization when the training pipeline has a bottleneck of data loading and processing. Previous empirical experiments already demonstrate the superiority of data echoing in large-scale machine learning problems but lacks theoretical support for this strategy. This work takes the first step to provide a theoretical understanding of convergence guarantees for several common data-echoed algorithms.
- The data echoing analysis in this work characterizes the inherent bias-variance tension well, i.e., the optimization gains from repeating gradient updates versus the degrade of generalization from the overfitting to the persistent batches. It is interesting to see that the bias term is related to the echoing steps K while the variance term is related to the minibatch size B. As a result, the convergence results for several different SGD variants nicely reveals this property in the data-echoing setting.
- The analytical framework in this work is very generic, and is extensible to other gradient based methods in the data-echoing regime. It provides a main data echoing theorem, which can be applied to certain specific algorithm by considering its stability and potential bounded regret properties.

__ Weaknesses__: As is mentioned in the paper, one of the major applications of the data echoing algorithms is the massively parallel training of deep neural networks, which is non-convex. The current theoretical analysis and empirical study are only restricted to convex case. Although it is challenging to deal with the general non-convex case, I wonder if it is possible to derive any result with respect to some simple shallow networks, or some certain class of non-convex functions with bounded degree of nonconvexity. This would make the work even more interesting and contribute more to the community.

__ Correctness__: The results make sense and seem to be correct, but I did not go through the proof details in the supplemental.

__ Clarity__: The paper writing is very good, but I find several small problems related to notations, which could make confusion:
- Between line 108-109, the authors use both the \bf\xi with a supscript "t" and the \bf\xi without a supscript "t", I guess for the latter the authors mean a general batch of samples does not depend on "t", but it is not explained clearly. Also, sometimes it has "i" in the supscript while othertimes it has "i" in the subscript.
- In the Definition 1, line 120-121, I understand that \bf\xi and \bf\xi' are two batches of data differing in exactly one example, and the superemum is taken over all \xi, which is independent of \bf\xi or \bf\xi'. However, the reuse of the same notation really makes me confused for a while since it looks like \xi is some element belong to \bf\xi or \bf\xi'.
- In Algorithm 1, I wonder why the output is an arithmetic average of all the w_t's? Is this a proof artifact? It makes more sense that if we want to do an averaging here, the w_t's should better have different weights such that the recent updates get higher score.
- In Algorithm 1 line 4, we update w_{t} to w_{t+1} by executing algorithm A for K steps. However, in the equation below line 131, the subscript for w changes for each of those K steps. All these inconsistencies downgrade the readability of the paper.
- Line 140 is a duplicate of line 132, which should be a typo here.

__ Relation to Prior Work__: It clearly discussed the difference of this work from the previous one whose focus is empirical study only while this paper provides the theoretical convergence analysis.

__ Reproducibility__: Yes

__ Additional Feedback__: ********************* After rebuttal ***************************
The authors' response has addressed my concerns, and I prefer to keep my original score.
******************************************************************

__ Summary and Contributions__: This papaer provides a convergence and generalization analysis of a model problem of data echoing-based gradient extensions of convex optimization methods.
========================
I have read the author's rebuttals, and the authors did not really address my concerns during rebuttal. I slightly raise my rating.

__ Strengths__: 1. The topic under study, i.e., theoretical analysis of optimization with echoed gradient, is of general interesting.
2. The paper is overall well-written.

__ Weaknesses__: 1. If the stochastic gradient in the echoed gradient-based gradient is computed with the same batch of training data? If so, this raises the efficiency issue of the use of training data; in particular, when K is large. I would expect that in each gradient computation step, randomly sample a large portion of these K examples would give better performance. This issue should be addressed in the paper.
2. Does Stochastic AGD converge (Table 2)? This result seems wrong. It is a well-known result in the first-order optimization community that any first-order optimization algorithm will accumulate error, in stochastic gradient, in SGD.
3. The experiments are too simple to verify the advantage of the echoed gradient-based SGD. The comparison of state-of-the-art algorithms, e.g. ADAM, SGD with momentum, etc., should be provided. Moreover, the experiments should be extended to large-scale experiments.
4. Can the analysis be generalized to nonconvex optimization? The most used machine learning algorithms are based on deep neural networks, which corresponds to highly nonconvex optimization problems. Convergence of echoed gradient based SGD for nonconvex optimization should be included in this paper.

__ Correctness__: The convergence of stochastic AGD (in Table 2) seems incorrect, and the authors should verify this.

__ Clarity__: The paper is well-written and easy to follow.

__ Relation to Prior Work__: The paper does a good job in the literature review.

__ Reproducibility__: Yes

__ Additional Feedback__: Please see the weaknesses section.

__ Summary and Contributions__: This paper investigates data echoing on some simple gradient-based methods; namely, SGD, SGD + Nesterov acceleration, and prox SGD. In particular it investigates the bias/variance tradeoffs of extra passes through data for assisting with communication lag.

__ Strengths__: The idea is well-motivated, and the structure / type of results are clean and useful. I can't speak too much on previous work; I have seen data echoing in numerical papers before, and believe there has been some theoretical work, but I haven't seen any on these specific methods.

__ Weaknesses__: There is a lot of carelessness in the proofs. I think the claims are not outlandish, and the general proof structure seems reasonable, but the paper and the math seem unconnected at times. If not for that I would raise my score.

__ Correctness__: - The citation [17] for the accelerated method is not correct. It refers to Lan's method, not Nesterov's acceleration, which are very different methods.
Appendix:
- line 322 What is equation 4, 6? same with line 323
Lemma 8 proof: Lambda, gamma, and eta included. Which is step size used? Also for prox, it should be the gradient at w_{t+1}.
Lemma 11: indexing seems off, since d_t involves w_{t-1}, not w_{t+1}
- line 351: do you mean gradient?
What are the proofs in appendix B for? They seem somewhat unrelated and not used anywhere.

__ Clarity__: Yes, for the most part the paper is well-written
Figures 2,3, its a bit hard to figure out what I should be looking at. I assume that red line going down means data echoing has less overhead when you do it less?

__ Relation to Prior Work__: I have seen previous works on data echoing, but those have mostly been numerical. There is most likely overlap between this work and stochastic optimization in general, e.g. the lan paper actually cited, where controlling the variance term is done by sampling more gradients. This should be discussed more.

__ Reproducibility__: No

__ Additional Feedback__: After rebuttal: I have read the author's rebuttal. While the contributions are interesting (resubmission encouraged), the presentation of the proofs were such that it's not easy to be 100% certain they are error-resistant. Additionally, the comment about the superfluous proofs (lemmas 14,15) were not addressed. Overall, I will keep my borderline score.