From: ICLR.cc/2023/Conference/Program_Chairs
title: Paper Decision
decision: Reject
metareview:_summary,_strengths_and_weaknesses: The paper provides rigorous theoretical guarantees for a federated sampling algorithm using using Langevin dynamics. The theoretical results are interesting and novel: they include fixed and varying learning rate, correlated and un-correlated noise, and full and partial device participation. Moreover, the results also consider the (differential) privacy aspects of SGLD. 

The reviews had some major concerns about the proofs of the paper (since the paper is mainly a theoretical paper) as well as the quality of the bounds, and the presentation of the results -- please see the detailed review (and the discussions). While some of the concerns were addressed, some others still remained. In particular, the reviewers are unable to certify the correctness of some of the main results of the paper. I appreciate the authors responses and revisions. However, I recommend that the authors revise the proofs and make sure that everything is clearly explained with proper details. I am confident that, after all the reviewers comments are addressed in a revised version, the paper would be a very interesting contribution to the literature of Federated Learning and Langevin dynamics.  

justification_for_why_not_higher_score: Two of the reviewers are in favor of rejecting the paper. 
justification_for_why_not_lower_score: -

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_Nk8K
title: Thanks
comment: Thanks for your responses, I have updated my score accordingly.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Updated version to improve the presentation on the continuous time diffusion
comment: We appreciate your thoughtful comments and have made the second minor revision to improve the presentation of the continuous diffusion. Please the see details in the public thread. The main results are not affected.

We hope this revision could address your concerns and we are happy to hear any insightful feedback.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: second minor revision
comment: We appreciate reviewer PJdt's constructive suggestions on the derivation of the continuous diffusion. We further made the following changes in the second minor revision  (marked with red font) and the main results are not affected:

4) we included section A.1.1 in the appendix to detail the derivation of the continuous diffusion.

5) we fixed some typos in section D.2 and fully adopted the continuous diffusion based on Eq.(20) without using local diffusion.

6) we simplified the derivation of Eq.(26) since the property of the continuous-time diffusion is standard based on frequent enough synchronization (almost at any time).

7) we included some discussions on a concurrent work of federated averaging Langevin diffusion based on probabilistic averaging schemes [1] in the related work section H.

[1] Federated Averaging Langevin Dynamics: Toward a unified theory and new algorithms. 2022.

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_PJdt
title: Please only post results that you think are correct
comment: Otherwise erroneous results will confuse other researchers.

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_PJdt
title: The third equation in your comment is evidently wrong
comment: If we accept first two equations in your comment, your third equation should be
$$d \bar\theta_{t}^c =-\sum_{c=1}^N p_c \nabla f^c(\bar\theta_t^c) \cdot d t+\sqrt{2\tau} \cdot d W_t \text{ for  any } c$$
but not 
$$d \bar\theta_{t}^c =-\nabla f^c(\bar\theta_t^c) \cdot d t+\sqrt{2\tau} \cdot d W_t \text{ for  any } c$$
due to your definition
$$\bar\theta_{t}^c=\sum_{c=1}^N p_c \bar\beta_{t}^c.$$

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Could you be more specific?
comment: 
**A different SDE is needed**

We did have written an informal SDE for $\theta_t^c$ after Eq.(20) such that
$$d \bar\beta_{t}^c =-\nabla f^c(\bar\theta_t^c) \cdot d t+\sqrt{2\tau/p_c} \cdot d \overline{W}^c_t$$
where the synchronization follows that $$\quad\bar\theta_{t}^c=\sum_{c=1}^N p_c \bar\beta_{t}^c.$$ for every time interval $\nabla t$. Taking the limit $\nabla t\rightarrow 0$ allows us to adopt the desired Langevin SDE in Lemma B.2:
$$d \bar\theta_{t}^c =-\nabla f^c(\bar\theta_t^c) \cdot d t+\sqrt{2\tau} \cdot d W_t \text{ for  any } c$$


**Lemma B.2 uses the SDE without synchronization**

Similar to the argument above, taking $\nabla t\rightarrow 0$, the synchronization is implicitly adopted and it is sufficient enough for us to upper bound the only required term $\int_{k\eta}^{(k+1)\eta} \sum_{c=1}^N p_c E[|| \bar\theta_s^c -\bar\theta_{k\eta}^{c}||] ds$ in Eq.(26).

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_PJdt
title: If infinite times of synchronization is needed, then you need to write a different SDE.
comment: Moreover, your proof (e.g. Lemma B.2) uses the SDE without synchronization.
Therefore your proof is wrong.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: what is the frequency of synchronization?
comment: Thanks for the timely feedback.

At any time $t\geq 0$. It is essentially the standard Langevin diffusion. We write it in federated settings to facilitate the analysis of the important dominated contraction property in Lemma B.1.

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_PJdt
title: For SDE, what is the frequency of synchronization?
comment: > Note that this is fundamentally different than no-synchronization during training

If authors claim synchronization is needed for SDE so that $\bar\theta_t^c$ converges to $e^{-\frac{f(\theta)}{\tau}}$, then what is the frequency of synchronization?

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Summary on Minor Revision
comment: We appreciate the reviewers carefully checking the details. We have made the following minor changes (using red font in the revision) to improve the presentation of the paper.

1) In Lemma B.2, we *removed $/p_c$* and confirm the continuous-time diffusion of $\bar\theta_t^c$ is the same as $\bar\theta_t$ due to the synchronization at any time $t\geq 0$; The factor $N$ in Eq.(26) is also removed. Although the main convergence rates are not affected at all, this minor typo may mislead the reviewer PJdt to speculate that $\bar \theta_t^c$ does not converge to $\pi(\cdot) \propto e^{-\frac{f(\cdot)}{\tau}}$.

2) We have changed the notation $\gamma^2$ to $\Gamma$ in the definition of $W_2$ and changed $\psi$ to $\phi$ in Burkholder  inequality.

3) We have included a short discussion on the optimality of our convergence rate w.r.t. dimension and compared it with [1,2] in section 4.3.1.

[1] High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm.

[2] Sqrt(d) Dimension Dependence of Langevin Monte Carlo.

We hope the revision has addressed your concerns and thank you again for your insightful comments, which have further improved the clarity of this paper.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Thanks for the comments.
comment: **The algorithm is not novel**

Our novelty is the first convergence guarantee of the promising FA-LD in Bayesian federated learning instead of the methodology, as we stated in the introduction.

**The upper bound does not go to zero**

Such a result is standard in the literature. SGLD with a constant learning rate and infinite time cannot obtain a zero upper bound either [1].

[1] User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient.

**Why FA-LD might outperform SGLD**

As shown in **optimal choice of $K$** in section 4.3.1, the 
number of communication round to achieve accuracy $\epsilon$ is $\frac{T_{\epsilon}}{K}=\Omega(K+\frac{\kappa}{K})$, clearly $K=1$ requires too many communications compared to $K=\Omega(\sqrt{\kappa})$.

**The continuous-time diffusion does converge to the correct distribution**

Yes, that is right.

**$\theta_t^c$ seems to be using only the local dynamics**

We found a typo in $d\bar \theta_t^c=- \nabla f^c(\bar \theta_t^c) dt + \sqrt{2\tau /p_c} d W^c_t$ in Lemma B.2 and have removed $/p_c$ to be consistent with Eq.(20). The main results are not affected.

We hope our answer has addressed your concerns. Thanks again for your suggestions to improve the presentation of the paper.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: minor issues
comment: **relation between $k$ and $K$**

Every server takes $k=1, 2, ..., K$ local step updates and then averages the parameters.

**$\gamma^2$ as a measure is unusual**

We have defined a more appropriate notation $\Gamma$ in the revision. 

**$\phi=\psi$ and reference?**

Yes, we have fixed the typo in the revision. Please see the reference in Lemma A.32 [1].

[1] Yin, Zhu. Hybrid Switching Diffusions. Springer.


**Is Bukholder inequality a general version of Doob's martingale inequality or Ito's isometry?**

Yes, exactly.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Thanks for the valuable suggestions
comment: **technical challenges**

The biggest challenge is that the parameter at step $k$, where $k \text{ mod } K \neq 0$, is not accessible. Such a complication requires a delicate proof framework to construct the dominated contraction property detailed in Lemma 4.4 (B.1). It is challenging to analyze the specially discretized Langevin diffusion with periodically inaccessible parameters while maintaining the same optimal order of $\Omega(d/\epsilon^2)$ iterates as in [1] to achieve $\epsilon$ error in $W_2$ distance.

[1] High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm. Bernoulli. 2019.

**why can we remove $l_2$ assumption?**

Because the convexity assumption essentially leads to a bounded second moment.

**Is the result optimal with respect to the dimension?**

Yes, our algorithm in section 4.3.1 shows that $\epsilon$ error ($W_2$ distance) requires $\Omega(d/\epsilon^2)$ steps given Lipschitz smooth gradient, which is consistent with [1, 2]. Notably, such a result holds even with multiple local steps $K>1$. Moreover, given more smoothness, we believe the dimension dependence can be improved following [2,3]. We have included such a discussion in the revision.

[1] User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient. 2019.

[2] High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm. Bernoulli. 2019.

[3] Sqrt(d) Dimension Dependence of Langevin Monte Carlo. 2021.

**challenges in non-convex cases**

Presenting some upper bound in non-convex cases is not difficult, however, proving a sharp non-asymptotic convergence, where the rate in federated settings is the same as standard cases, is highly non-trivial and may require significant delicate computations.

From: ICLR.cc/2023/Conference/Paper3225/Authors
title: Thanks for carefully checking the details.
comment: **SDE is not well defined and $\bar\theta$ fails to converge to $e^{-\frac{f(\theta)}{\tau}}$**

We acknowledge that there is a typo in $d\bar\theta_t^c=- \nabla f^c(\bar\theta_t^c) dt + \sqrt{2\tau /p_c} d W^c_t$ in Lemma B.2, which may mislead the reader to conclude $\bar \theta_t^c$ converges to $\pi^c(\cdot) \propto e^{-\frac{p_c f^c(\cdot)}{\tau}}$.

We have removed $/p_c$ so that $\bar\theta_t^c$ converges to $e^{-\frac{f(\theta)}{\tau}}$ due to the synchronization at any time $t\geq 0$. Note that this is **fundamentally different** than no-synchronization during training, where $\bar \theta_t^c$ converges to $\pi^c\propto e^{-\frac{p_c f^c(\theta)}{\tau}}$.

We appreciate the reviewer checking the details and hope this adjustment has addressed your concerns.



**Contradiction between assumption 4.2 and assumption G.1**

Assuming assumption G.1 and assumption 4.2 at the same time does limit the space of applicable functions, however, it is not a conflict but rather a compromise to show differential privacy guarantees. For example, Langevin dynamics can be studied in bounded support [1] where both assumption G.1 and assumption 4.2 hold. We will leave the study of more consistent assumptions on differential privacy guarantees for future works.

[1] Convergence of score-based generative modeling for general data distributions.

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_PJdt
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: This paper extend Langevin Monte Carlo into federated learning setting with non-i.i.d. data and partial participation. Authors tried to develop convergence guarantee for this algorithm under assumption that the target distribution is strongly log-concave. According to their results, the convergence of their algorithm depends on data heterogeneity, learning rate, gradient noise and correlation between Gaussian vectors among clients. A differential privacy guarantee is also provided.
strength_and_weaknesses: **Strength**:

Some claims in this paper appear to be new, including the convergence of FA-LD with correlated Gaussian noise and partial participation.

**Main Weakness**:

The soundness of the theoretical analysis is questionable.

Authors define the auxiliary continuous-time processes
$$
\mathrm{d} \bar{\theta}_t=-\nabla f\left(\bar{\theta}_t\right) \mathrm{d} t+\sqrt{2 \tau} \mathrm{d} \bar{W}_t
$$
and authors claim that "continuous-time algorithm is known to converge to the stationary distribution $\pi(\theta) \propto e^{-\frac{f(\theta)}{\tau}}$".

When the term $\nabla f\left(\bar{\theta}_t\right)$ means the gradient of function $f$ at point $\bar{\theta}_t$, I agree the above SDE converge to the Gibbs distribution.

However, authors interpret that term as 
$
\nabla f\left(\bar{\theta}_t\right)=\sum\_{c=1}^N p_c \nabla f^c\left(\bar{\theta}_t^c\right)
$.

The first problem is that, author left out the dynamics for $\bar{\theta}_t^c$ on each client, so the above SDE is not well-defined.
In Section D.1, authors use $\bar{\theta}\_s^c=-\int\_{k \eta}^s \nabla f^c\left(\bar{\theta}\_t^c\right) d t+\sqrt{2 \tau / p_c} \int\_{k \eta}^s \mathrm{~d} \bar{W}\_t$, and claim this equation is a consequence of eq. (20) (which is the SDE above for $\bar{\theta}_t$). But that is not true. Eq. (20) only defines dynamics for $\bar{\theta}_t$ but not for $\bar{\theta}_t^c$.

The second problem is that, even if we append the missing definition, the convergence claim is still wrong.
Let's choose following SDE for $\bar{\theta}_t^c$ (which matches the integral authors used in Section D.1):
$$
\mathrm{d} \bar{\theta}_t^c=-\nabla f^c\left(\bar{\theta}_t^c\right) \mathrm{d} t+\sqrt{2 \tau/p_c} \mathrm{d} \bar{W}_t
$$
It is clear that the distribution of $\bar{\theta}_t^c$ converge to $\pi^c(\theta) \propto e^{-\frac{p_c f^c(\theta)}{\tau}}$.

$\bar{\theta}_t$ is weighted average of $\bar{\theta}_t^c$, thus the distribution of $\bar{\theta}_t$ will converge to some kind of mixture of $\pi^c$, but not $\pi(\theta) \propto e^{-\frac{f(\theta)}{\tau}}$.

In summary, the auxiliary continuous-time processes defined in this paper doesn't converge to desired distribution $\pi$, and all convergence theorems w.r.t. $\pi$ are wrong.

**Other questions**:

The assumption G.1 is clearly in conflict with assumption 4.2, thus Theorem 4.7 and Theorem 5.7 cannot hold at same time, or more specifically, we can guarantee either convergence or privacy, but not both. Is there any possible mitigation to that dilemma?

Lemma 4.4 is called "informal version of Lemma B.1". Which part of its statement is informal?
clarity,_quality,_novelty_and_reproducibility: The writing is clear and the contribution of the paper is presented with clarity.
summary_of_the_review: My main concern is the soundness of the theoretical analysis.
correctness: 1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.
technical_novelty_and_significance: 4: The contributions are significant, and do not exist in prior works.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_mK8G
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: The paper studies non-asymptotic convergence of federated averaging Langevin dynamics in 2-Wasserstein distance, where the target is log strongly concave. The authors consider stochastic gradient, constant and varying learning rates, privacy-accuracy trade-off, partial device participation. They also discuss differential privacy guarantees. 
strength_and_weaknesses: This is a solid theory paper, and the analysis seems to be rigorous. 

However, just in my opinion, I feel that the authors should highlight a bit more about their technical contributions. For example, under the log-strongly-concave setup for overdamped Langevin, the method to obtain convergence is well known, see e.g. Arnak S Dalalyan and Avetik Karagulyan (2019) "User-friendly Guarantees for the Langevin Monte Carlo with Inaccurate Gradient." The authors perhaps should comment on what is the technical difficulty under the federated averaging setting and how they overcome it. Right now, the impression I have is that even though the authors discusses many model setups, for non federated averaging case, they are known in the literature, so the current paper is just a collection of many results, where the only novelty is the presence of the federated averaging. 

Also, the authors mentioned that The frequently used bounded gradient assumption of $\ell_{2}$ norm in FedAvg optimization is not required. It would be nice if the authors can elaborate more on this point and why they can remove this assumption. Also, if they can remove this assumption for Langevin dynamics for sampling, does that mean their techniques can be used to remove this assumption for optimization as well?

Moreover, I am just wondering whether the non-asymptotic convergence is sharp in the current paper. For example, there have been many works in recent years, and one of them is Li et al. (2021) "Sqrt(d) Dimension Dependence of Langevin Monte Carlo", where they used mean-square analysis to obtain better dependence on the dimension d. I am wondering whether that is possible in the federated averaging setting. At least, the authors should make some comments on whether the convergence rates they obtained are optimal or not depending on the model parameters. For example, if you reduce your results to the special case for standard SGLD, are your results sharp according to the literature? This will help the readers understand how good the results are presented in the current paper. 

I am also wondering what is the technical difficulty if you consider non-convex $f$, as in Raginsky et al. (2017). It is possible that non-convex setting is very challenging for the federated averaging setting, but that can also help the readers appreciate the log-strongly-concave assumption made in the paper and make it more clear about the technical challenge in the federated averaging setting. 

clarity,_quality,_novelty_and_reproducibility: Overall, the paper is well written and the presentations are clear. I have some minor suggestions.

(1) on page 2, when you introduced local updates, "then conduct $K\geq 1$ local steps", please make it more clear whether you take $k=0,1,\ldots,K-1$ or not? That is, what is the relation between $k$ and $K$?

(2) In section B.1., it is a bit unusual to use $\gamma^2$ to denote a measure.

(3) In the paragraph about Bukholder-Davis-Gundy inequality in section F, is it true that $\phi=\psi$? Please provide a reference. It seems to me that you are simply stating a very special case of Bukholder-Davis-Gundy inequality which is a direct result of Doob's martingale inequality and Ito's isometry. 
summary_of_the_review: The paper is a theory paper that provides non-asymptotic convergence results for federated averaging Langevin dynamics. It would be better if the authors can make the technical contributions more clear and also comment about the quality of their non-asymptotic convergence results as I mentioned previously. 
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 6: marginally above the acceptance threshold

From: ICLR.cc/2023/Conference/Paper3225/Reviewer_Nk8K
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper studies a federated version of stochastic gradient Langevin dynamics (SGLD) that the authors dub FA-LD. Motivated by the shortcoming of existing federated optimization research which do not allow for uncertainty quantification, they consider a Bayesian version of the federated optimization problem where one wishes to sample from a posterior, and the negative log density of the posterior is distributed across multiple devices. They study the analog of federated averaging in this setting, where each device does a few steps of SGLD locally before the weight vectors are re-aggregated across devices. Their main findings are convergence bounds in $W_2$ for the law of the iterates, including for fixed learning rate, varying learning rate, correlated noise, and partial device participation. The authors also provide a result giving some differential privacy guarantees for their algorithm, as well as some empirical exploration of their method, including comparisons to vanilla SGLD.
strength_and_weaknesses: Strengths:

- This work provides rigorous theoretical guarantees for a federated sampling algorithm using SGLD.

- The theoretical guarantees are extensive: they include fixed and varying learning rate, correlated and un-correlated noise, and full and partial device participation.

- It includes a result addressing differential privacy of their algorithm

- The authors mention that they do not require $\ell_2$ bounds on the gradients as in the federated averaging optimization framework. I am not familiar enough with the federated averaging literature, nor their proof, to judge if this is significant or not

Weaknesses:

- Their algorithm and setting are not novel, having been already considered in the work Al-Shevidat et al 2020 (the authors address this in their paper; I am by no means saying they missed this reference)

- Their upper bounds (except for the upper-bound for varying learning rate) involve a term that does not go to zero as iterations go to infinity, apparently due to the independent SGLD steps

- The paper is written in a confusing fashion, it is difficult to understand what the definitions of the continuous time dynamics are. Also there is significant repetition between the appendix and main text, with the appendix basically being an extended version of the main text. The related work is in the appendix.

- While there are many results, it is not clear which is most important, what the key difficulties in establishing those results are, and none explain why FA-LD might outperform SGLD (the bounds as-is suggest that with more independent SGLD steps the algorithm performs worse)

- The experiments are not in a setting where the data is non-i.i.d. (in this context meaning the devices see fundamentally different portions of the data distribution). In fact, the data is split randomly between devices, undermining the need for federation

- Issues with correctness (added after reading reviewer PJdt's review): I agree that there seem to be major issues with correctness, greatly compounded by confusing definitions of the continuous time dynamics. I interpret them to be using continuous time dynamics where at all times t, the gradients are shared, which seems to be what they are indicating on the footnote on page 4 as well as in section A.1. As in the footnote on page 4, this means that
$$
\bar \theta_t^c = \bar \theta_t
$$ for all $t \in \mathbb{R}$ and $c \in [N]$. It follows, therefore, that
$$
\sum_{c = 1}^n p_c \nabla f^c (\bar \theta^c_t) = \sum_{c = 1}^n p_c \nabla f^c (\bar \theta_t) = \nabla f(\bar \theta_t),
$$ where the last gradient is in the usual sense. Hence the continuous time dynamics do indeed converge to the correct distribution. However, as reviewer PJdt points out, the integral formula for $\bar \theta_t^c$ seems to be using only the local dynamics
 $$
d \bar \theta^c_t = - \nabla f^c(\bar \theta^c_t) + \sqrt{2\tau/p_c} dW_t.
$$ These are obviously at odds with one another and seem to be a major problem with the proof of the discretization error bound in Lemma B.2. That said, the authors also have some continuous time variables $\beta$ which only follow the local gradients defined in subsection A.1, and it may be that the authors mean to prove Lemma B.2 with these alternate variables (which may not resolve their issues). Clarification on this point is seriously needed.


Al-Shedivat, Maruan, et al. "Federated learning via posterior averaging: A new perspective and practical algorithms." arXiv preprint arXiv:2010.05273 (2020).
clarity,_quality,_novelty_and_reproducibility: See strengths and weaknesses above.

Suggestions for improvement:

- The definition of the auxiliary continuous-time process $(\bar \theta_t)_{t \geqslant 0}$ could be made more clear, the usage of "synchronizes" is confusing and I needed to read the footnote to understand its meaning

- In my opinion, the paper would be improved if the discussion of related work was moved to the main text, and some of the variants on your main results be relegated to the appendix

- The sentence "Notably, since the Brownian motion..." on page 5 is confusing but seems like it is talking about novel aspects of their proof. Perhaps this could be expanded?

- More indication and discussion of the main result (which of the theorems is the most important and why?) would greatly help in comprehension. There was no heading that said "main results", nor links in the contributions section to these theorems. And more discussion of the significance of these results would helpful, in my opinion.

- The experiments don't seem to incorporate any use of uncertainty quantification. Perhaps a set of experiments that use the additional information you gain from a Bayesian approach over the vanilla FedAvg approach would be helpful to indicate the importance of your setting.

- Please clarify the nature of the continuous time dynamics.
summary_of_the_review: This work provides novel theoretical guarantees for a federated version of SGLD. While they do give several theoretical bounds, they generally suffer from an additive term that does not go to zero and it is difficult to judge their significance.  Moreover, their experiments are not extensive and don't incorporate a use of uncertainy quantification fitting their motivation of studying a Bayesian setting. Also, there are definitely issues with clarity of presentation and likely issues with correctness. Overall, I think that the work is not suitable for acceptance.
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold