__ Summary and Contributions__: Focus on minimization of a convex combination of losses over N parties holding data, with uncertainty over what convex combination is desired, formulated as minimax optimization problem.
This formulation was studied previously and this contribution is quite clear in what sense it extends the prior work.
Appears to be valuable theoretical contributions (low confidence) but experimental evaluation is weak.

__ Strengths__: Well grounded in prior work and clear in what is the novelty.

__ Weaknesses__: Section 5

__ Correctness__: Probably.
I am not familiar with the theoretical results built upon so I don't have a good prior on what results to expect.

__ Clarity__: Yes, with the exception of Sec 3 which reads harder.

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__: I have ready the response and other reviews.
For concenrn about experimental evaluation, the response is mostly "[27] did it too", which is the concern I flag at the end of review. Not meeting a simple baseline is ignored. Moreover, [27] also tries to do experiment in a more realistic setup, where the simple baseline would not hold, which this work does not reproduce response does not mention.
So I see this concern as not addressed.
Based on opinions of other reviewers, I think the theoretical contribution is valuable but I can't comment on details. I feel the way this fits into the FL setup overall should be improved, or perhaps deemphasized - in which case my concerns would not be as valid. My overall rating stays the same, including the (low) confidence.
Initial content:
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Minor personal opinion - It would be more beneficial if the title was a little more concrete
What is in the introduction is very clear and well presented, but I would recommend adding details to it.
In particular, the proposed algorithm would not be feasible for all scenarios where federated learning is relevant. For instance, this would not work for the system described by Bonawitz et al. "Towards Federated Learning at Scale: System Design" but would likely work for the system described in Ludwig et al. "IBM Federated Learning: An Enterprise Framework". See also Kairouz et al. "Advances and Open Problems in Federated Learning" for discussion on cross silo and cross device federated learning.
I do not think this would make this work worse, quite the opposite. Scoping the work more clearly makes it easier to ask better questions of relevance to that setup, as well as highlights open problems for future work - I believe this notion of distribution robustness is equally important for both scenarios.
L81: I do not think this can be called "real federated data".
L86+ paragraph on convergence rates.
I recommend reading Woodworth et al. "Is Local SGD Better than Minibatch SGD?" which just appeared in ICML and discusses how and when the existing comparisons are insufficient, especially when (not) compared with minibatch SGD. I think the discussion therein could also help better present the theoretical results in this work.
L112 shwoing typo
Section 3.1
I found the text is relatively hard to follow, but the Alg 1 is presented clearly. I suggest rethinking how exactly it is presented. Perhaps one can assume familiarity with the related work and ground the presentation in differences from such baseline.
Sec 4
I cannot provide really confident assessment of the contribution here, as I am not very familiar with minimax optimization theory, and thus do not have a good idea what kind of rates should be achievable, and thus whether this contribution is significant or not. I hope other reviewers will comment on this. I did not look at the proofs.
On the surface, it looks interesting to me, and difference from relevant [27] is quite clearly called out. Rather than being uniformly better, the contribution is presented as having worse convergence rate, but lower number of communication rounds to get there. As such, it is a different point on a possible pareto-frontier.
Question: How do the rates compare with a basic algorithm such as gradient descent?
Sec 5, Experiments.
A major weakness of the paper.
Only showing experiments on an artificially partitioned dataset is not meeting a bar for a persuasive evaluation today. For the specific setup presented here, comparison with a trivial baseline would be missing - predict most common label locally - It does not require any communication and achieves 100% accuracy. So I interpret the experimental result as showing poor performance on a very simple problem.
The work [27] most closely related presents more complex experimental setup which is not even reproduced here. Other previous works such as [16] and also Caldas et al., "LEAF: A Benchmark for Federated Settings." or Reddi et al., "Adaptive Federated Optimization". present baselines based on dataset with more realistic partitioning. I recommend using those as starting point for design of a better experimental setup.
If the other reviewers can confidently claim the theoretical results are interesting and bring significant novelty, I would be more willing to accept the work without experiments, compared to the current state.
I am afraid that accepting the work in its current form would encourage others to submit works without proper evaluation and might thus be detrimental to the broader clarity in the field.
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Potntial inspiration re Sec 4. "Technical challenge"
Perhaps see recent Charles and Konecny, "On the Outsized Importance of Learning Rates in Local Update Methods" for inspiration on how the "changing minimizer" can be handled.

__ Summary and Contributions__: The paper studies distributionally robust (agnostic) federated learning as a minimax optimization problem for both strongly-convex-strongly-concave and nonconvex-strongly concave cases. The propose an algorithm with theoretical guarantees that is also communication efficient since it has local updates and partial participation of devices. Experimental results are also presented.

__ Strengths__: Overall, I enjoyed reading this paper. It tackles an important problem in FL that is data heterogeneity and tries to approach it using distributionally robust or agnostic FL formulation. The proposed algorithm is clever. The theoretical guarantees are quite strong though they come with a weakness about gradient bounded assumption that I will mention.

__ Weaknesses__: - For the theoretical results, the authors assume that the gradients are bounded. Many works in FL literature (See E.g. "FedPAQ: ..." by Reisizadeh et al. and "First Analysis of Local GD on Heterogeneous Data" by Khaled et al.) have tried to relax this assumption and it is well-known that assuming it makes the theory much simpler. Why do the authors need bounded gradients? Should be explained clearly.
- A major promise of FL is preserving privacy. Local updates indeed increase privacy guarantees but the proposed algorithm requires sending a random w in the previous epoch to track a history of local gradients. This indeed compromises privacy and is in contrast with the key promise of FL. (From previous model, new model and an estimate of history of local gradients, information about data points will be leaked, right?) Can you comment on this?
- As the authors mention, the communication cost will also be doubled. I believe that for theoretical purposes, this is fine but it would be good to add experiments as follows:
(i) Add some random noise to model updates to increase differential privacy. How will the algorithm work then?
(ii) Compress the models (quantize) to reduce communication load. This will add quantization noise. How will the algorithm work?

__ Correctness__: I didn't check every detail but they look correct.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes, but it is good to add some references about quantization/compression and preserving privacy in federated learning.

__ Reproducibility__: Yes

__ Additional Feedback__: I have read the response and other reviews. I agree the numerical results can be improved, but I still find the theoretical contribution strong, so I would like to keep my score.

__ Summary and Contributions__: The paper generalizes the agnostic federated learning method of Mohri et al. [27] to allow communication between the nodes and central server to occur at given time intervals.

__ Strengths__: The paper is clearly written, the proofs are detailed and the proposed method is a simple extension of previous algorithms.
Convergence is proved in various settings, including convex, and under a PL condition.

__ Weaknesses__: Motivation:
One of the challenges mentioned in the introduction is low participation. This is not addressed, since the central server chooses which nodes participate at each round. This needs further discussion.
Results:
The convergence bounds assume a particular choice of the synchronization interval \tau. Since there is inherently a trade-off between communication cost and convergence rate, why was this particular trade-off chosen? The result would be more compelling if the trade-off could be adjusted, for example by giving the convergence rate (along with the optimal choice of parameters) when \tau is a given polynomial function of T.
Another weakness is that these results assume a fixed horizon T.

__ Correctness__: I only partially reviewed the appendix. The parts that I reviewed are technically correct.

__ Clarity__: The paper is well-written overall, but suffers from frequent typos, undefined quantities, and a general lack of polish.
Some examples below (apologies if I missed something):
- The quantity \kappa (appears on line 78 and later in Section 4.3) is never defined.
- The quantity f_i(x, \xi) was never formally defined, beyond saying (line 132) that \xi is randomly sampled from the i-th dataset. It is also not stated that f_i(x, \xi) is an unbiased estimate of f_i(x). This may be a common assumption, but it needs to be stated.
- In Algorithm 1, there is confusion between \bar w and w. Should line 4, 12, 13 be \bar w instead of w?
- Algorithm 2, line 2, should \xi be \xi_i?
- line 163: "between local and global (*) at each iteration" missing noun at (*).
- line 166: what is meant by a minimizer of a vector-valued function?
- line 187: The bound involves a random variable \xi. Is it meant that the inequality holds almost surely? Is there a missing ^2 on the norm?
- Missing ^2 in definition 3.
- line 218: "centralized setting". Do the authors mean non-distributed? In distributed optimization one distinguishes between centralized (all nodes communicate with a central server) and decentralized (general communication graph). The setting of this paper is distributed centralized.
- line 230: what is meant by "the choice of $\psi$ includes optimal transport"? Do you mean a Wasserstein distance? Please be more precise.
- The bibliography needs to be more carefully reviewed. Refs [4] and [5] are the same. [9] should reference the original paper from the 40s.
- Many typos in the appendix.

__ Relation to Prior Work__: The literature review covers related work in federated learning.
The problem studied here is a robust optimization problem, and there is a rich literature on the topic that should be reviewed, see for example Ben-Tal, El Ghaoui and Nemirovski, Robust Optimization.

__ Reproducibility__: Yes

__ Additional Feedback__: - Line 38: distribution drift is mentioned, which implies non-stationary distributions, but this is not the setting of the paper. Please rephrase.
- If the experiments are simulated on a single machine, the communication cost should be inexistent or negligible. So how is the wall-clock time for DFRA worse than AFL?
- In Figures 1 and 3, why was training stopped when accuracy reached 50%? It would be better to train the model until convergence, to verify experimentally whether the methods will indeed converge to a similar loss as \tau grows.
- The conclusion can be improved, it currently has no meaningful discussion.
- The broader impact section mentions that the method is designed to preserve the privacy of users, but there are no privacy guarantees provided here. The fact that the model is sent to the central server makes the method potentially vulnerable to privacy attacks, and this should be mentioned. Another issue, alluded to in the text but omitted from the broader impact section, is that when optimizing the worst-case distribution, the model could be more vulnerable to adversarial attacks from a single malicious node, and this limits the applicability of the method.
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Thank you for the response.
One of my questions was why the authors made such specific choices of \tau, for example in Theorem 1. The response claims that this choice optimizes the rate (I am assuming this means the rate in T). I fail to see why this is the case.
For example for Theorem 1, looking at the final bound in Section C.4 in the appendix (lines 540 and 542), wouldn't taking \tau = 1 (synchronize at each step), \gamma = \eta = 1/\sqrt T give an overall better rate of 1/\sqrt T?
I think the result should be presented as a trade-off between communication cost and convergence rate: given a communication constraint \tau (for example a monomial function of T), what is the best choice of parameters \gamma, \eta that gives the best rate.

__ Summary and Contributions__: In this paper, the authors propose a new algorithm for robust FL with efficient communication.

__ Strengths__: The strengths of the paper are as follows.
1. A nice robust FL with less cost communication compared to conventional FL is developed.
2. Nice theory is provided.
3. Experiments have been done to support the theory.

__ Weaknesses__: The weaknesses of the paper are as follows.
1. To gain some advantage in communication, the convergence of algorithm is slower.
2. From the experiments, it seems that the proposed algorithm and conventional FL are not much different in terms of efficiency. (FIG2)
3. It seems that this framework may not be as efficient as asynchronous FL. In my opinion, it would be much better and convincing if the authors can compare their algorithm with asynchronous FL.
4. The same constant \tau should be used for all clients. It is not nice because some clients have many samples while other clients may have a small number of samples.
5. The assumption of bounded gradients and the assumption of strongly convex functions are used in the proof. However, these two assumptions cannot be used together as explained in https://arxiv.org/pdf/1802.03801.pdf. Therefore, the results for strongly convex/ convex cases seem pretty weak.
6. It would be very good for the reader if the authors can explain why the result in Theorem 1, i.e., max_\lambda E[...] - min_w E[...] <= O(1/sqrt(T)), tells us about the convergence rate of Algorithm 1. Actually, i do not see why w^ and lamda^ outputted by Algorithm 1 should be good solutions for the problem described in Equation 3?

__ Correctness__: In my opinion, the claims are correct.

__ Clarity__: Yes, it is.

__ Relation to Prior Work__: Yes, it is.

__ Reproducibility__: Yes

__ Additional Feedback__: