Asynchronous Federated Learning Using Outdated Local Updates Over TDMA Channel

Jaeyoung Song; Jun-pyo Hong

REVIEW 3 major objections 4 minor 24 references

Asynchronous Federated Learning Using Outdated Local Updates Over TDMA Channel

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Asynchronous federated learning over a TDMA channel converges at rate O(G^2/√K), where G is the number of TDMA groups; adding an intentional delay lowers the effective staleness and speeds convergence.

desk verdict A new TDMA group-staleness model and a sensible intentional-delay heuristic, but the main convergence theorem is unproven because the proof uses ||x+y||^2 ≤ ||x||^2 + ||y||^2. read the letter →

arxiv 2411.13861 v1 pith:FWTBCI6M submitted 2024-11-21 cs.DC

Jaeyoung Song , Jun-Pyo Hong This is my paper · ORCID

classification cs.DC

keywords asynchronousfederatedlearningTDMAstalenessconvergencerateintentionaldelaycommunication-computationtradeoffdelayedstochasticgradientwireless

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

This paper seeks to establish that asynchronous federated learning over a time-division multiple access (TDMA) wireless channel converges even when every aggregated local update is computed from an outdated global model. The staleness is shown to be fixed by the size of the TDMA groups: after an initial transient, each update is exactly G−1 training rounds old, where G=N/S is the number of groups. The main theorem bounds the average squared gradient norm by O($G^{2}$/√K), making explicit that stale updates slow convergence but do not stop it, under bounded data heterogeneity. Because that rate quantifies the cost of staleness, the paper proposes an intentional delay strategy that lets devices receive a fresher global model before computing gradients, reducing effective staleness to G−1−α without increasing round time. Experiments on MNIST and CIFAR-10 show this intentional-delay version reaches lower global loss within the same time budget than the undelayed asynchronous scheme.

What carries the argument

The load-bearing object is the staleness identity d_{n,k}=G−1: after the first G training rounds, every TDMA group transmits a gradient computed from the global model G−1 rounds old, independent of device index. This converts asynchrony into a deterministic, constant delay and lets the analysis treat every update as uniformly stale. The proof machinery is the four-lemma descent argument that bounds the mismatch between local and global gradients and the accumulated drift across the stale window, then telescopes the stale sums to produce the O($G^{2}$/√K) rate. The second mechanism is the intentional-delay scheduler in Algorithm 2, which changes when each device receives the global model so that training starts from a fresher model; Proposition 1 picks the largest delay α that still hides computation inside communication, giving effective staleness d^* = G−1−α.

What would settle it

Evaluate the inequality in Lemma 2 with ∇f_n(w)-∇f(w)=1 and ∇f(w)=1; it claims 4≤2, so the proof step is false. A concrete check is to redo equation (26) with the valid bound ||x+y||^2 ≤ 2||x||^2+2||y||^2 and see whether the coefficient of the gradient-norm sum stays positive under the stated step-size condition; if it cannot, the claimed O($G^{2}$/√K) rate is not established by the given argument.

Watch

Extended reading notes

Core claim

The central claim is that asynchronous FL over a TDMA channel converges at rate O($G^{2}$/√K) for the average squared gradient norm, where G=N/S is the number of TDMA groups and K is the number of global rounds. The proof works by first establishing the staleness identity d_{n,k}=G−1 for every device in every round k≥G, so the asynchronous process becomes a constant-delay update. Lemma 1 bounds the one-round loss decrease; Lemma 2 bounds the local-vs-global gradient mismatch; Lemma 3 bounds the accumulated parameter drift over the stale window; Lemma 4 converts the stale-gradient sums back into current gradient norms. Theorem 1 then closes the rate under L-smoothness, a bounded second moment, and a bounded heterogeneity assumption, with step size η≤β/√(K+1). A separate claim is that introducing an intentional delay α, chosen by Proposition 1, reduces the effective staleness to d^* = G−1−α, and the experiments report that this accelerates convergence in the tested settings.

Load-bearing premise

The proof's Lemma 2 rests on the inequality ||x+y||^2 ≤ ||x||^2+||y||^2, which fails for nonzero vectors, and this same step is reused in Lemma 3 and the appendix, so the bounds on the stale-gradient error that Theorem 1 depends on are not derived.

Editorial extensions

If this is right

If Theorem 1 is correct, the convergence rate degrades quadratically in the number of TDMA groups, so the group count is the direct lever for how much staleness can be tolerated before the rate worsens.
The intentional-delay scheme should outperform plain asynchronous FL for the same time budget whenever the computation-to-communication ratio allows a positive α, with the largest gains at small group sizes.
Applying the intentional delay shifts the optimal group size: a smaller S becomes preferable because global updates happen more often and the delay recovers most of the staleness penalty.
The staleness identity implies that steady-state behavior, not the initial transient, governs convergence, so performance can be predicted from N, S, and τ_comp/r alone.
In the experiments, the S=1 configuration with intentional delay beats the synchronized S=N configuration on MNIST, indicating that asynchronous operation can be strictly preferable when the time budget is fixed.

Reading between the lines

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

An editorial check: Lemma 2 uses ||x+y||^2 ≤ ||x||^2+||y||^2, which is false; replacing it with the valid 2||x||^2+2||y||^2 bound changes constants in Lemmas 3 and 4, so Theorem 1's rate would need re-derivation to see whether O(G^2/√K) survives the extra factors.
The intentional-delay rule suggests an online tuning procedure: since α is chosen from τ_comp/r, a system could adapt α as channel conditions or device speeds change during training.
The fixed-delay analysis would break under heterogeneous device speeds, because the staleness identity G−1 becomes device- and time-dependent; a natural extension is a delay-distribution analysis rather than a single scalar.
The quadratic dependence on G implies a design optimum that the paper does not state in closed form: for fixed N and time budget, the best S balances update frequency against staleness, and the reported shift from S=10 to S=5 in CIFAR-10 illustrates that optimum moving with α.

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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

Quick take on 2411.13861. The system model is the genuinely new piece: asynchronous FL over a TDMA channel with a deterministic group-based staleness pattern, and the staleness formula (16) cleanly captures how the group count G drives update delay. The proposed intentional-delay algorithm is a sensible heuristic, and the experiments suggest it helps in practice. Credit where it's due: the paper is not parameter fitting in disguise, and the delay rule is derived from the authors' own convergence expression, which is the right way to motivate a design.

The problem is the proof of Theorem 1. Lemma 2 uses ||x+y||^2 ≤ ||x||^2 + ||y||^2, which is false (take x=y=v). That inequality is load-bearing: it produces the heterogeneity bound in (24), and a multi-term version is used in Lemma 3 and again in (A.5). Without it, the bounds leading to (26)–(27) don't hold, so the claimed O(G^2/√K) is not established. The standard fix—||sum x_i||^2 ≤ d sum||x_i||^2—puts an explicit d=G−1 inside the drift bound; that factor can't be absorbed by the step-size condition and would likely change the rate. This is a real, central flaw, not a typo.

Smaller issues: no baseline comparison against existing asynchronous FL methods, and no code. Those are minor relative to the proof.

Who gets value from this? People working on wireless FL and communication-computation trade-offs. The group-staleness model and the intentional-delay idea are worth discussing even if the theorem is currently unsupported. I'd send it to a serious reviewer—this is exactly the kind of paper where a referee might be able to tell whether the proof can be repaired. But I wouldn't cite the convergence result as it stands, and if the authors can't fix the inequality, the contribution reduces to a system model plus heuristic with empirical support.

Referee Report

3 major / 4 minor

Summary. The paper studies asynchronous federated learning (FL) over a time-division multiple access (TDMA) channel. Devices are partitioned into TDMA groups of size S, each group transmits in turns while other groups compute local updates. The paper derives the staleness of local updates as G-1 (where G=N/S), analyzes convergence under smoothness, bounded second moments, and bounded heterogeneity, and claims the rate O(G^2/√K) for the average squared gradient norm. Based on this analysis, the authors propose 'intentional delay FL' (IDFL), which deliberately delays global-model reception to reduce effective staleness, and provide simulation results on MNIST and CIFAR-10 showing empirical improvements.

Significance. If the convergence theorem were valid, the paper would make a useful contribution by linking the TDMA group size to the staleness of local updates and by proposing a simple, communication-side mechanism (intentional delay) to reduce staleness without extra time slots. The system model in Section III and the IDFL algorithm in Section V are clearly presented, and the experiments are extensive. However, the central theoretical claim, Theorem 1, is not established because the proof relies on a false square-norm inequality. Since the convergence rate is the main contribution and motivates the proposed IDFL strategy, the current manuscript cannot be accepted.

major comments (3)

[Section IV, Lemma 2] The proof of Lemma 2 (Eq. (24)) uses the inequality ||∇f_n(w_hat)-∇f(w_hat)+∇f(w_hat)||^2 ≤ ||∇f_n(w_hat)-∇f(w_hat)||^2 + ||∇f(w_hat)||^2. This is false in general; for example, with x=y=1 it would assert 4 ≤ 2. The step is not a valid application of the triangle inequality, and Assumption 3 does not prevent this failure because both vectors can point in the same direction. Since Lemma 2 is used to bound the terms in (26) and in the proof of Lemma 3, the derivation of Theorem 1 is not sound.
[Appendix A, Eq. (A.5)] The same false inequality is used in Appendix A to write ||∇f(w_k)-∇f_n(w_hat)||^2 ≤ ||∇f(w_k)-∇f(w_hat)||^2 + ||∇f(w_hat)-∇f_n(w_hat)||^2. The correct bound is ||a+b||^2 ≤ 2||a||^2+2||b||^2, which introduces an extra factor of 2 in Lemma 1 and hence in Eq. (23). This affects the constants in the subsequent convergence proof, so Lemma 1 as stated is not justified.
[Appendix B, Eq. (B.1)] In the proof of Lemma 3, the squared norm of the sum over j from k_hat_n to k-1 is bounded by the sum of the individual squared norms without a factor for the number of terms. The standard inequality is ||∑_{i=1}^m x_i||^2 ≤ m ∑_{i=1}^m ||x_i||^2, with m equal to the staleness d_{n,k}. Since d_{n,k}=G-1 for k≥G, this repair puts an explicit factor of G into the drift bound in (25), which propagates into Theorem 1 and changes the step-size condition (30). Consequently, the claimed O(G^2/√K) rate in (31) is not derived.

minor comments (4)

[Section IV, Lemma 2 statement] The statement of Lemma 2 writes ∑ E[||f(w_hat)||^2] where the proof and the surrounding text concern ∇f(w_hat); the notation should be corrected to ||∇f(w_hat)||^2.
[Eq. (27)] In the last term of (27), the summation index is written as ∑_{n∈} without specifying the set; it should be ∑_{n∈S_k}.
[Section V, Proposition 1] The symbol d^* is used in (33) before it is defined in (34); please reorder or add a short explanation around its first occurrence.
[Section VI, Table I] The caption of Table I is referenced in the text as 'Table I' but the table itself appears without a numbered caption; please ensure all tables are numbered and referenced consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the convergence analysis is self-contained and the intentional-delay algorithm is validated empirically, not by construction.

full rationale

The paper's central derivation, Theorem 1, starts from standard Assumptions 1-3 and derives the rate O(G^2/sqrt K) by algebra in Lemmas 1-4 and Appendices A-D. The staleness identity d_{n,k}=G-1 is a direct consequence of the TDMA schedule in Algorithm 1, not an assumption of the theorem's conclusion. Lemma 4's combinatorial bounds and the step-size condition eta <= beta/sqrt(K+1) are derived in-text rather than imported from prior work. The intentional-delay strategy in Algorithm 2 is motivated by the authors' own convergence expression, but Proposition 1 derives alpha from the timing constraints (35)-(37), and the claimed acceleration is then checked against MNIST/CIFAR-10 simulations in Section VI; no fitted parameter is relabeled as a prediction. The paper does contain self-citations ([7], [9], [12], [13]) but these appear only in background statements about dynamic batch sizes, device counts, and over-the-air aggregation; they do not carry the convergence proof or the protocol design. The flagged issue in Lemma 2 - the inequality ||x+y||^2 <= ||x||^2+||y||^2 and its reuse in Appendix B/A.5 - is a mathematical correctness concern if true, not circularity: it does not make any input equal to any output by definition. Because the derivation chain is self-contained and the simulations are external evidence, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard smoothness/variance/heterogeneity assumptions plus a homogeneity assumption on device computation speed. The proof also introduces an incorrect inequality in Lemma 2 that is load-bearing for Theorem 1. No new physical entities are introduced.

free parameters (1)

Step size η = 0.01 (MNIST and CIFAR-10 simulations)
Chosen by hand for the experiments; the convergence theorem only requires an upper bound, so it is not fitted to the target result.

assumptions (5)

standard math Global loss is L-smooth (Assumption 1)
Used for the descent lemma in Lemma 1.
standard math Bounded second moment of stochastic gradients (Assumption 2)
Controls gradient noise; standard in FL convergence analysis.
domain assumption Bounded data heterogeneity (Assumption 3)
Assumes ||∇f(w)-∇f_n(w)||^2 ≤ Γ^2 for all n and w; this is strong for realistic non-IID data.
domain assumption Homogeneous device computation capability q (Section II)
All devices process q data samples per time slot, yielding deterministic τ_comp and the exact staleness d=G-1 in (16).
ad hoc to paper Lemma 2 inequality ||x+y||^2 ≤ ||x||^2+||y||^2
Used in Lemma 2 and Appendix B; this inequality is false, so the proof of the convergence theorem is invalid as written.

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Cite this review

Pith. "Pith review of Asynchronous Federated Learning Using Outdated Local Updates Over TDMA Channel." pith.science (2026). https://pith.science/paper/FWTBCI6M

@misc{pith2026241113861,
  author       = {Pith},
  title        = {Pith review of: Asynchronous Federated Learning Using Outdated Local Updates Over TDMA Channel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWTBCI6M}},
  note         = {Machine review of arXiv:2411.13861}
}

read the original abstract

In this paper, we consider asynchronous federated learning (FL) over time-division multiple access (TDMA)-based communication networks. Considering TDMA for transmitting local updates can introduce significant delays to conventional synchronous FL, where all devices start local training from a common global model. In the proposed asynchronous FL approach, we partition devices into multiple TDMA groups, enabling simultaneous local computation and communication across different groups. This enhances time efficiency at the expense of staleness of local updates. We derive the relationship between the staleness of local updates and the size of the TDMA group in a training round. Moreover, our convergence analysis shows that although outdated local updates hinder appropriate global model updates, asynchronous FL over the TDMA channel converges even in the presence of data heterogeneity. Notably, the analysis identifies the impact of outdated local updates on convergence rate. Based on observations from our convergence rate, we refine asynchronous FL strategy by introducing an intentional delay in local training. This refinement accelerates the convergence by reducing the staleness of local updates. Our extensive simulation results demonstrate that asynchronous FL with the intentional delay can rapidly reduce global loss by lowering the staleness of local updates in resource-limited wireless communication networks.

Figures

Figures reproduced from arXiv: 2411.13861 by the authors.

**Figure 1.** Federated learning system with 𝑁 IoT devices and a server convergence analysis, we propose the asynchronous FL using intentional delay in Section V. Moreover, we present results of experiment using real dataset in Section VI. Finally, Section VII concludes the paper. II. SYSTEM MODEL We consider FL over a wireless network which consists of a server and 𝑁 IoT devices indexed by 𝑛 ∈ N = {1, 2, . . . , 𝑁} as shown in t… view at source ↗

**Figure 2.** Example timeline of asynchronous FL over TDMA channel for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Performance of asynchronous FL over TDMA channel [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Performance of IDFL and iterations, respectively, training CNN for MNIST dataset. Consequently, for 𝑆 = 1, when 𝜏 comp 𝑟 = 50, the maximum intentional delay is obtained as 𝛼 = 74, which leads to 𝑑 ∗ = 25. Similarly, for 𝜏 comp 𝑟 = 10 and 𝜏 comp 𝑟 = 2, 𝑑 ∗ = 5 and 𝑑 ∗ =…

**Figure 7.** Figure 7: Comparison of global loss between Asynchronous FL with delayed [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 5.** Figure 5: Comparison of global loss between Asynchronous FL with delayed [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Comparison of global loss between Asynchronous FL with delayed [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 12, 2026 · model on record in the stance chip above.

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