Rethinking the initialization of Momentum in Federated Learning with Heterogeneous Data

Chenguang Xiao; Shuo Wang

REVIEW 4 major objections 4 minor 13 references

Rethinking the initialization of Momentum in Federated Learning with Heterogeneous Data

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that reversing the weight order in momentum accumulation—favoring early local gradients over recent ones—consistently improves federated learning accuracy and F1-score under heterogeneous data.

desk verdict The method's defining equation contradicts its own motivation, and the empirical claims are unverifiable without code or a clear update rule. read the letter →

arxiv 2411.19798 v1 pith:CW4ISMSN submitted 2024-11-29 cs.LG

Chenguang Xiao , Shuo Wang This is my paper · ORCID

classification cs.LG

keywords federatedlearningmomentumdataheterogeneitynon-IIDreversedinitializationlocalepochssensitivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the standard way of accumulating momentum in federated learning is suboptimal when client data are heterogeneous: the usual exponential moving average gives the most weight to the most recent local gradients, but those are also the most biased by client divergence. It proposes Reversed Momentum Federated Learning (RMFL), which assigns exponentially decaying weights as training proceeds so early local gradients dominate the momentum sent to the server. Across MNIST, CIFAR10, and CIFAR100 under three heterogeneity levels, RMFL reports higher accuracy and F1-score than the momentum baseline in every setting tested, with the largest gains under the most skewed data. The result suggests that a simple reweighting of the momentum initialization can offset a major source of bias in federated learning.

What carries the argument

The central object is the reverse-exponential weighted average of Eq. (3), $\hat v_t = (1-\beta)v_0 + (1-\beta)\sum_{i=0}^{t-1}\beta^i \nabla f(x_i) + \beta^t \nabla f(x_t)$. It assigns the largest weight to the first local gradient and exponentially smaller weights to later gradients, the opposite of the standard momentum update in Eq. (1), so that the later, more biased local gradients contribute least to the momentum used to initialize the next local training round.

What would settle it

Measure the average cosine similarity (or projection) between client gradients and the mean gradient at each local step in a heterogeneous federated task. If the similarity does not decrease monotonically with the local step — for instance if it dips and then recovers — the premise of RMFL fails. A second check: run RMFL on a task where later gradients are known to be more aligned with the global objective (e.g., synthetic noise added only to early steps); if RMFL still wins, its benefit does not come from the claimed mechanism.

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Extended reading notes

Core claim

The central claim is that the bias introduced by heterogeneous client data accumulates during local training, so a momentum buffer that emphasises recent gradients carries more bias than one that emphasises early gradients. The paper therefore replaces the standard exponential moving average of local gradients with the reverse-exponential weighted average defined in Eq. (3), and shows empirically that this estimated momentum outperforms the standard momentum baseline across three datasets, three heterogeneity levels, and local epochs of 2, 5, and 10. The authors interpret the gains as evidence that later local gradients are less useful for the global objective, supporting the reverse weighting.

Load-bearing premise

The method assumes that the bias of local gradients grows monotonically with the number of local training steps, so that early gradients are always less biased than recent ones; if that curve is not monotonic, the reverse weighting has no principled reason to help.

Editorial extensions

If this is right

RMFL improves accuracy and F1-score over the MFL baseline on all three benchmark datasets and all heterogeneity levels tested, with the largest gains at the most skewed data (CIFAR10 with $\alpha=0.01$: accuracy 0.472 vs 0.247).
The advantage of RMFL grows with the number of local epochs, so it makes longer local training less harmful than standard momentum in heterogeneous settings.
RMFL shows smaller standard deviation than MFL in most settings, indicating more stable convergence across random seeds.
The reweighting idea could be applied to other momentum-based federated optimization methods that use exponential moving averages, as the paper itself notes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct testable extension is to measure the actual gradient-bias-versus-local-step curve on a real heterogeneous task; the claimed mechanism predicts a monotonically increasing bias curve, and the benefit of RMFL should correlate with the slope of that curve.
The case-study figures in Section III appear to conflate cumulative momentum with gradient cosine similarity, so the direct empirical support for the monotonic-bias premise is thinner than the main accuracy comparison; readers should verify that premise independently.
If early local gradients were not more aligned with the global objective, RMFL could underperform standard momentum, so the method's success hinges on the direction of the bias, not merely its existence.
The same reweighting idea could generalize to adaptive optimizers such as Adam-style second-moment estimates in federated learning, where the bias pattern is likely similar.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: the paper doesn't do what its headline says. Eq. (3), which is the entire proposal, is not a reversed exponential weighting. For β=0.9, the most recent local gradient gets 9× the weight of the one before it and about 9× the weight of the oldest, so RMFL as defined still emphasizes exactly the late, biased gradients the authors claim to suppress. The stress-test note is right; I checked the arithmetic.

What's genuinely worthwhile: the underlying hypothesis — that in heterogeneous FL the later local steps carry more bias and standard momentum overweights them — is plausible and clearly stated. The experiments cover three datasets and three Dirichlet α levels, plus sensitivity to local epochs, and the reported gains for RMFL over MFL are consistent. If a corrected reversed-weighting scheme outperforms MFL that would be a simple, useful contribution.

But the soft spots are serious. Beyond the Eq. (3) mismatch, the paper never states how v̂_t enters the parameter update (the usual x ← x − η v̂_t is only implied). The comparison is against MFL only, not any of the other momentum/adaptive methods cited. The learning rate is tuned per method for 'fastest convergence', which can conceal optimizer-specific sensitivity. Table I says 5 runs while Section IV.A says 10. Figures 4 and 5 have captions that don't match what the text says they show. No code or data are provided. Any one of these would be minor; together they make the empirical claim unverifiable.

The math flaw is load-bearing. If the authors actually implemented Eq. (3), then the method is just a variant of MFL with a peculiar weighting, and the narrative is wrong. If they implemented the prose description, then Eq. (3) is a typo and we need a corrected paper with the true formula and re-run experiments. Either way, the current version cannot be used as is.

Bottom line: this is a paper with an interesting idea but a critical internal contradiction. I wouldn't cite it in its current form. A serious editor could send it to peer review in the hope that the authors fix the equation and provide code, but the review would be a heavy revision request. I'd rather treat it as a cautionary example in reading group than as a building block.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes Reversed Momentum Federated Learning (RMFL), a modification of momentum in federated learning in which the standard exponential moving-average weighting of gradients is replaced by a weighting intended to give larger weight to early local gradients and smaller weight to later, putatively more biased gradients. The method is defined in Eq. (3). The authors report experiments on MNIST, CIFAR10, and CIFAR100 under Dirichlet heterogeneity levels alpha = 1, 0.1, 0.01 and local epochs 2, 5, and 10, claiming that RMFL consistently outperforms MFL in accuracy and macro F1-score.

Significance. The motivating observation that late local gradients are more biased in heterogeneous FL is worth studying, and the experimental matrix is broad. If the proposed weighting were correctly specified and robustly validated, the method would be a simple, practical contribution. However, the central definition in Eq. (3) does not realize the claimed reverse weighting: for beta = 0.9 the latest gradient receives nine times the weight of the immediately preceding one, so the experiments evaluate a method that is inconsistent with the paper's own description. In addition, the local update rule is under-specified and the experimental protocol has reproducibility and tuning issues. The contribution is therefore not established as written.

major comments (4)

[§III-B, Eq. (3)] The proposed momentum estimate is not reverse-exponentially weighted. Expanding Eq. (3), the weights on ∇f(x_0), ..., ∇f(x_t) are (1−β), (1−β)β, ..., (1−β)β^{t−1}, β^{t}. For β = 0.9 and t = 10, ∇f(x_10) has weight 0.3487 while ∇f(x_9) has weight 0.0387, and the ratio of the most recent gradient to the immediately preceding one is β/(1−β) = 9. Thus the latest gradient receives the largest weight, directly contradicting the Section III claim that the weights are exponentially decayed with time going forward. Since Eq. (3) is the definition of RMFL, the experiments do not evaluate the proposed method as described.
[§III-B and §IV] The local update rule is under-specified. Eq. (3) defines a momentum estimate, but the manuscript never states how this estimate is used to update the model parameters, for example whether x_{t+1} = x_t − η \hat v_t or whether \hat v_t is combined with an additional recursion. Without this equation, the implementation of RMFL is ambiguous and the reported results are not reproducible from the text alone, especially since no code is released.
[§IV-A and Table I] The experimental protocol is reported inconsistently and the comparison is not fully controlled. Section IV.A states that 10 repeated runs are conducted, while the Table I caption reports results averaged over 5 runs. The learning rate is searched per method in [0.3, 0.1, 0.03, 0.01, 0.003, 0.001] for the fastest convergence, but neither the selected learning rates nor the convergence criterion are reported; a per-method search without a fixed budget can favor one method. These issues make it impossible to verify that the reported gains are due to the momentum weighting rather than to tuning or implementation details.
[§III-A] The monotonic-bias premise is not established by the presented evidence. The text claims that gradient bias increases with local steps, but the supporting figures are mislabeled: Fig. 4 is captioned as cumulative momentum while the text describes average gradient cosine similarity, and Fig. 5 is captioned as active clients while the text describes average gradient projection. No numerical summary or statistical test is provided. If the bias does not increase monotonically, the reverse weighting has no principled basis; this premise must be quantified before the method can be assessed.

minor comments (4)

[Throughout] Typos include 'beed proved' in the abstract, 'address more basis' in Section III, 'As the MFL parer suggested' in Section IV.A, and 'the most challenge CIFAR100' in Section IV.A; these should be corrected.
[§IV.B and Table I] The text says Table I shows results on MNIST, CIFAR10, and CIFAR100, but the table caption says the results are on MNIST and CIFAR10 datasets; the caption should be aligned with the reported content.
[Figures 4 and 5] The figure captions do not match the quantities described in the text; Fig. 4 is captioned as cumulative momentum while the text reports average gradient cosine similarity, and Fig. 5's caption says active clients while the text reports average gradient projection.
[§II.B, reference [4]] The cited FedDyn paper is described as a federated distillation approach on recommender systems, but the dynamic-regularization FedDyn commonly referenced in federated learning is a different paper (Acar et al., ICLR 2021); the reference should be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RMFL weighting is a hand-designed heuristic and the reported accuracy comparisons are external, not derived from the proposal's definition.

full rationale

The paper proposes RMFL as a heuristic reweighting of local gradients, motivated by the observed increase in gradient divergence during local training. The central empirical claim is that RMFL outperforms MFL on MNIST, CIFAR10, and CIFAR100 across multiple heterogeneity levels. This comparison is independent of the method's definition: no parameter is fitted to the reported accuracy or F1-scores, and the learning rate is tuned for both MFL and RMFL under the same protocol. The motivating observation of increasing bias is supported by the authors' diagnostic measurements (Figs. 4 and 5), which are separate from the final benchmark comparison, so the empirical success is not used as evidence for the premise in a way that makes the claim self-validating. There is a notable internal inconsistency: Eq. (3), as written, does not actually realize the claimed reverse exponential decay, since for beta = 0.9 the most recent gradient receives beta/(1-beta) = 9 times the weight of the immediately preceding gradient. However, this is a correctness or specification flaw, not circularity: the empirical comparison still tests the formula that is stated, even if that formula conflicts with the prose. No self-citation is load-bearing, no uniqueness theorem is imported, and no known result is merely renamed. Therefore the paper does not exhibit the circularity patterns targeted by this review.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The central claim depends on the monotonic-bias assumption, on the fixed β value, and on the per-method learning rate search. No quantities are derived from first principles; the method is a heuristic evaluated empirically.

free parameters (2)

Learning rate = not disclosed per setting
Searched in [0.3,0.1,0.03,0.01,0.003,0.001] for 'fastest convergence' separately for each method, which can bias the comparison.
Momentum coefficient β = 0.9
Fixed to 0.9 following MFL, but β plays a different role in Eq. (3) (a decay factor for past gradients) than in Eq. (1) (an EMA coefficient), so the optimal value likely differs. This is a chosen constant that the central claim depends on.

assumptions (2)

domain assumption Gradient bias increases monotonically with the number of local training steps
Stated in Section III: 'With longer local training, the gradient bias increases gradually as a result of the heterogeneity.' This is load-bearing for the reversed weighting design.
domain assumption Dirichlet-distributed class splits with α ∈ {1, 0.1, 0.01} adequately simulate realistic non-IID conditions
Section IV-A uses this simulation; the conclusions are claimed to hold across 'different heterogeneity levels' based on these simulated distributions.

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Cite this review

Pith. "Pith review of Rethinking the initialization of Momentum in Federated Learning with Heterogeneous Data." pith.science (2026). https://pith.science/paper/CW4ISMSN

@misc{pith2026241119798,
  author       = {Pith},
  title        = {Pith review of: Rethinking the initialization of Momentum in Federated Learning with Heterogeneous Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CW4ISMSN}},
  note         = {Machine review of arXiv:2411.19798}
}

read the original abstract

Data Heterogeneity is a major challenge of Federated Learning performance. Recently, momentum based optimization techniques have beed proved to be effective in mitigating the heterogeneity issue. Along with the model updates, the momentum updates are transmitted to the server side and aggregated. Therefore, the local training initialized with a global momentum is guided by the global history of the gradients. However, we spot a problem in the traditional cumulation of the momentum which is suboptimal in the Federated Learning systems. The momentum used to weight less on the historical gradients and more on the recent gradients. This however, will engage more biased local gradients in the end of the local training. In this work, we propose a new way to calculate the estimated momentum used in local initialization. The proposed method is named as Reversed Momentum Federated Learning (RMFL). The key idea is to assign exponentially decayed weights to the gradients with the time going forward, which is on the contrary to the traditional momentum cumulation. The effectiveness of RMFL is evaluated on three popular benchmark datasets with different heterogeneity levels.

Figures

Figures reproduced from arXiv: 2411.19798 by the authors.

**Figure 2.** Stochastic Gradient Descent with Momentum [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Illustration of the momentum cumulation in FL systems. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: Box plot of the cumulative momentum in different global iterations. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

**Figure 5.** Figure 5: Box plot of the active clients gradient projection length. [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]

**Figure 6.** Figure 6: Number of samples for each class of the first 30 clients sampled from [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 9.** Figure 9: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]

**Figure 7.** Figure 7: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 10.** Figure 10: Accuracy of FL models with MFL and RMFL on CIFAR10 dataset [PITH_FULL_IMAGE:figures/full_fig_p005_10.png]

**Figure 8.** Figure 8: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 14.** Figure 14: Accuracy of FL models with MFL and RMFL on CIFAR100 dataset [PITH_FULL_IMAGE:figures/full_fig_p006_14.png]

**Figure 15.** Figure 15: Accuracy of FL models with MFL and RMFL on CIFAR100 dataset [PITH_FULL_IMAGE:figures/full_fig_p006_15.png]

**Figure 13.** Figure 13: Accuracy of FL models with MFL and RMFL on CIFAR100 dataset [PITH_FULL_IMAGE:figures/full_fig_p006_13.png]

**Figure 16.** Figure 16: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p007_16.png]

**Figure 17.** Figure 17: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p007_17.png]

**Figure 18.** Figure 18: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p007_18.png]

**Figure 22.** Figure 22: Accuracy of FL models with MFL and RMFL on MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p008_22.png]

**Figure 26.** Figure 26: Accuracy of FL models with MFL and RMFL on CIFAR10 dataset [PITH_FULL_IMAGE:figures/full_fig_p008_26.png]

**Figure 27.** Figure 27: Accuracy of FL models with MFL and RMFL on CIFAR10 dataset [PITH_FULL_IMAGE:figures/full_fig_p008_27.png]

Discussion (0). Continue with ORCID to comment.

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