Title: Tight analyses of first-order methods with error feedback
Venue: NeurIPS 2025
Accepted: yes
Keywords: distributed optimization

distributed learning

error feedback

EF

EF21

tight analysis

performance estimation

convex optimization

large-scale machine learning
Abstract: Communication between agents often constitutes a major computational bottleneck in distributed learning. One of the most common mitigation strategies is to compress the information exchanged, thereby reducing communication overhead. To counteract the degradation in convergence associated with compressed communication, error feedback schemes---most notably EF and EF21---were introduced. In this work, we provide a tight analysis of both of these methods. Specifically, we find the Lyapunov function that yields the best possible convergence rate for each method---with matching lower bounds.  This principled approach yields sharp performance guarantees and enables a rigorous, apples-to-apples comparison between EF, EF21, and compressed gradient descent. Our analysis is carried out in the simplified single-agent setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper provides a tight analysis of CGD, EF, and EF21 through the lens of the meta-optimization problem defined in equation (6). The authors detail the formulation of this meta-optimization problem, specify the set from which the Lyapunov function is selected, and explain the procedure used to construct the optimal Lyapunov function (i.e., how the meta-optimization problem is solved). The formal proofs of the convergences have been presented.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper addresses a compelling research question and utilizes an interesting tool to tackle it.\n2. The theoretical analysis is rigorous, with carefully derived proofs.\n\nWeaknesses:\n1. The analysis is restricted to the single-client setting. This significantly limits the impact of the results, as some of the known challenges of CGD in the distributed setting, when $n > 1$, are not present in the $n=1$ case. In particular, as noted by the authors and shown in prior work \\[1, 2], CGD fails to converge in the distributed setting. Therefore, the phrasing in line 239  “These results challenge the prevailing intuition that error feedback ensures convergence comparable to that of uncompressed methods” is misleading: the lack of convergence is not a matter of intuition but a rigorously established fact. On the other hand, CGD converges without issue in the single-node case. Additionally, in the single-node case, error feedback appears unnecessary and overly complex.\n2. The paper lacks a clear rate comparison with classical results for EF [3] and EF21 [4]. In particular, Theorems 1 and 2 do not explicitly state the iteration complexity rates, and require additional manipulation to extract them.\n\nReferences:\n[1] Beznosikov et al, \"On Biased Compression for Distributed Learning\"\n\n[2] Karimireddy et al, \"Error Feedback Fixes SignSGD and other Gradient Compression Schemes\"\n\n[3] Stich et al, \"SGD with memory\"\n\n[4] Richtarik et al, \"EF21: Error Feedback 2021\"\n\nQUESTIONS:\n1. How do the convergence rates obtained in this work compare to the classical results for EF and EF21?\n2. What's the primary challenge of analyzing the distributed setting?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper provides tight bounds for Compressed GD, EF, and EF21. It employs the performance estimation framework to find the optimal Lyapunov functions for EF and EF21. The paper demonstrates that CSG actually converges for the larger set of problem parameters compared with EF and EF21.

STRENGTHS AND WEAKNESSES:
The paper is very well-written and the main technique for finding optimal Lyapunov functions was very interesting.

Concerns:

The paper only considers the exact-gradient case. Note that EF was largely motivated by the need of compressed GD to handle the stochastic noise (see [“Error Feedback Fixes SignSGD and other Gradient Compression Schemes”](https://arxiv.org/pdf/1901.09847), Section 3). Since the practical scenarios have stochastic noise, I question the significance of this work, in particular the conclusion “We also observed that their performance is strictly worse than that of compressed gradient descent – an outcome that, we believe, challenges the intuition of many in the field.”

Section 3: what is the optimal convergence rate for CGD?

In Section 2.1, you don’t talk about EF. Also, please mention EF21 when using reference [14].

QUESTIONS:
Minor concerns:

Line 188: redundant “and”

Line 220: better to move the reference to the end of the phrase

Line 237: naemly

Line 565 is strange

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors provide a tight analysis of both EF and EF21 methods. They identify the Lyapunov function that gives the best possible convergence rate for each method and, using computer-assisted techniques, establish matching lower bounds that confirm the optimality of these rates. This careful approach provides sharp performance guarantees and allows for a clear, direct comparison between EF, EF21, and Compressed Gradient Descent (CGD). The analysis is conducted in a simplified setting, focusing on the single node and deterministic case, which highlights the core algorithmic ideas and enables clean theoretical insights without the added complexity of other settings.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper is well written, carefully structured, and easy to follow in most parts. The authors present their ideas clearly and maintain a logical flow throughout the paper, which makes it accessible to the reader.\n\nAll theorems, assumptions, and lemmas are clearly stated and supported by intuitive explanations. The theoretical sections are presented in a way that is both rigorous and understandable, which I find valuable.\n\nThe background section and the discussion of related work are thorough. The authors provide a solid overview of the existing literature and clearly explain how their work connects to and differs from previous studies.\n\nThe pseudocode is well presented and easy to interpret. It offers a clear description of the algorithms, making the proposed methods easier to understand and potentially easier to implement.\n\nFigures 1 and 2, as well as Tables 1 and 2, are well designed, clearly labeled, and well explained in the text. They effectively support the theoretical findings of the paper. I appreciate the clarity and care in the visual presentation.\n\nI briefly checked the theoretical results. They appear to be correct and rigorous, although I might have missed some minor details.\n\nOverall, the results seem to be sound and are well explained. The paper demonstrates careful work and contributes insights to the topic.\n\n\nWeaknesses:\n\nThe abstract is confusing and potentially misleading, as it does not mention that the results are obtained in the single-node setting.\n\nAssumption 1 is made for deterministic compressors, while in the EF21 paper by Richtárik, the contractive compression operator is defined for stochastic compressors in expectation. Is it possible to extend the current results to the class of stochastic contractive compressors?\n\nIn line 132, there might be a typo in the formula (though I am not certain). I believe it should be $\n\\Vert g - \\nabla f(x) \\Vert^2 \\leq \\epsilon \\Vert \\nabla f(x) \\Vert^2.\n$\n\nPlease clarify.\n\nThe main issue with the paper is its focus on the single-node regime. The core motivation for using gradient-type methods with compressed updates, such as EF, EF21, and CGD (Compressed Gradient Descent) is typically in the multi-node setting. It is unclear why it would be meaningful to consider these methods in the single-node setting, where compression is not useful. Could the authors clarify any potential benefits of using such methods in the single-node case?\n\nThe authors claim that studying the single-node setting can provide intuition. However, I find this potentially misleading. In the single-node setting, CGD can converge with an appropriate stepsize without any issues. In this scenario, EF and EF21 mechanisms seem redundant, since the standard CGD method can work without additional memory or algorithmic adjustments. On the other hand, in the multi-node setting, as the authors themselves mention, CGD may diverge regardless of the stepsize. This indicates that the two settings are fundamentally different. It is unclear how insights from the single-node case can reliably transfer to the multi-node case.\n\nAnother important point is that EF21 is typically analyzed in non-convex settings and also under the Polyak-Łojasiewicz (PL) condition. This gives EF21 a theoretical advantage in non-convex problems. The results provided in this paper focus on the strongly convex case and do not address the more interesting non-convex scenario. Showing that EF and EF21 perform identically in the strongly convex setting does not necessarily imply that the same will hold in the non-convex or multi-node case. In fact, theory currently favors EF21 in the non-convex regime.\n\nFinally, another limitation is that the analysis assumes full-gradient updates. Stochasticity is not considered, even though stochastic gradients are essential in practical applications. Therefore, it would be valuable to compare these methods in the stochastic case.\n\nQUESTIONS:\nPlease check the weaknesses section.\n\nI think the paper is rigorous and clear, but the issues raised in the weaknesses section are important.\n\nLIMITATIONS:\nThe results are obtained only in the single-node case and the deterministic setting.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work applies performance estimation framework (PEP) via SDP reformulation to convergence analysis of popular federated learning algorithms (CGD, EF and EF21) using contractive compression. While the analysis is restricted to smooth, strongly convex, single node (agent) setup where EF and EF21 are not helpful (meaning that CGD already converges and achieves tight rate), the use of PEP framework delivers some interesting insights. First, two new Lyapunov functions are found for EF and EF21, and contraction rates is shown to be equivalent despite different Lyapunov functions. \n\nWhile I believe this work bring some useful insights for the study of compressed gradient methods, there are some formal inaccuracies in the Appendix and lack of explanation of some key steps. These issues summarized below are critical and have to be addressed in detail in the next revision before the paper is ready for publication. I am happy to increase my numerical rating if these concerns are addressed in the rebuttal.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1. The paper applies a general methodology of performance estimation problem to a relevant and modern machine learning problem. The analysis yields interesting conclusions about two most popular error feedback schemes reinforcing their equivalence. \n\n2. New tight Lyapunov functions are identified, which can potentially help to further improve the analysis of these algorithms in distributed settings and design even better algorithms. \n\n**Weaknesses:**\n\n1. Why interpolation conditions for the compression look as in equations (35,36) in supplementary material? This requires more detailed explanation as it is the key non-trivial extension of the prior work on PEP.\n\n2. Since I don't fully understand the point (could be due to poor explanation), it seems to me that the analysis does not actually cover the entire class of contractive compressors. I am afraid that it is restricted to linear contractive compressors such as the one shown in tightness analysis in equation (50). If that is the case, then such analysis is very limited and not particularly insightful. Specifically, EF21 is equivalent to heavy ball momentum (if restricted to linear compressors), which has already been analysed using such PEP framework, and for such linear compressors EF = EF21 as shown in (Richtarik et al 2021), thus most claims in the paper follow immediately. In fact, from the practical viewpoint, the most interesting contractive compressor is Top-$K$, but I am not sure this is covered in this framework. \n\n3. It is unclear why the resulting Lyapunov function for EF21 does not include the function value? I would expect it should since analysis in (Richtarik et al 2021) is conducted using the function value as the potential and is tight if I understand it correctly. \n\n\nTypos and Mistakes:\n\nIn equation (21), $\\nabla f(x_0)$ should be $f_0$.\n\nIn equation (22), it is unclear how to pick elements with $i = 1$ and $i = *$, because neither the gram matrix $G$ nor matrix $B$ contain elements with $i = 1$. \n\nIf I understand it correctly, the definition of large $C_{i,j}$ coefficients in line 591 can be simplified using a norm squared instead of inner products. \n\nOn line 595 there is likely a wrong pointer to equation (27) which wasn't discussed at this point yet. \n \nI assume it is just a typo, but coefficients $C_i$ in Lemma 1 should have a superscript EF. Same issue for analogous lemma for EF21. Moreover there is an issue with superscript in Lemma 2 in the SDP.\n\nQUESTIONS:\nFrom analysis, CGD allows larger step-size of order $1/L$ than EF and EF21 which require $(1-\\sqrt{\\epsilon})/L$. This can be also seen from upper bounds and convergence analysis. Is there an intuitive explanation why this happens?\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}