__ Summary and Contributions__: This paper study the convergence properties of Sign and Median SGD algorithm federated learning in heterogeneous data distribution regime. The paper makes several interesting contributions:
1) Making connection between Sign and Median SGD by showing that SignSGD updates along the direction of signed median of gradients.
2) Propose a simple but efficient method for reducing the gap between mean and median statistics.
3) Showing the convergence of Sign and Median SGD in terms of the expected median and mean of gradients at different workers.
4) Using 2 and 3 to propose convergent variant of Sign and Median SGD.

__ Strengths__: - The paper is theoretically sound and very well written. Providing intuitions about each theoretical contribution helps me understand the paper and design choices very well.
- The paper is novel and the contributions are significance. Note that heterogeneous data distribution is typically the case in federated learning since each user creates and stores their own data.
- I admire the authors for pointing out the superiority of their algorithm in iid setting. It helps the future research understand the problem in more depth.
- While the idea of adding noise to reduce the gap between mean and median statistics is intuitive and straightforward to prove it has been not discovered in any other work. I agree with the authors that this could be of independent interest.

__ Weaknesses__: Sign and Median SGD both shows some robustness against Byzantine workers in iid setting. The paper could discuss if similar properties are held in the proposed method when the model is trained on heterogeneous and non-heterogeneous data.
After Rebuttal: The author discussed my question regarding the robustness of their method for iid and non-iid data. I hope they add a more detailed discussion in the paper. My overall recommendation is still a clear accept.

__ Correctness__: The method and the experimental protocol seem correct and well designed.

__ Clarity__: The paper is very well written in much details. I appreciate the effort for explaining all the details in a simple language.

__ Relation to Prior Work__: The paper provides a well description of the prior works and their relation to the proposed method.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: In this paper, the authors focus on the problem of median-based distributed training algorithm. The main result is that when the data is draw in an non-IID setting, there is a gap between median-based and mean-based algorithms. And more fascinatingly, this gap can be closed by adding noises to the gradient.

__ Strengths__: - Relevant and important problem.
- Interesting result and solid theoretical analysis.

__ Weaknesses__: ~

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__: This is an interesting, solid work and I do not have any major concerns.
In my opinion, the problem is relevant and important, and the results are
insightful and interesting.
One thing that i found interesting is that in Algorithm 3 and 4, the noises
added to the gradient is similar to how people achieve differential privacy
during training --- since the paper talked about robustness, I wonder what
claims the authors can make in terms of privacy --- it seems to me that
preserving certain level of privacy is actually helping the convergence for
a class of algorithms, improving the utility instead of decreasing it. Maybe
this is something to discuss in the paper of having a simple lemma about.
### After rebuttal
I liked this paper before rebuttal, and continue to believe that it should be accepted after seeing the author feedback.

__ Summary and Contributions__: This paper studies the convergence of sign-SGD and median-SGD under heterogeneous setting. The main conclusion this paper draws is the gap between median and mean will leads to non-convergence. To deal with this issue, a treatment by adding noise is proposed and the convergence has been theoretically justified. Numerical experiments are provided to support the conclusion.

__ Strengths__: Federated learning is really popular recently and the theoretical guarantee is still far from enough. This paper tries to bridge the gap and provides useful intuition.

__ Weaknesses__:
My major concern is on the treatment part. Adding noise does make the median closer to mean. However, it also increase the variance.Though the author has carefully studied this in Theorem 6 and shows some trade-off. I still feel this is the most important question to answer here for this treatment.

__ Correctness__: Correct from my side.

__ Clarity__: The paper is overall well written and very easy to read. It has a good introduction part and provide enough intuition to help the readers understand the work.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: