Stackelberg Game Based Performance Optimization in Digital Twin Assisted Federated Learning over NOMA Networks

Bibo Wu; Fang Fang; Xianbin Wang

REVIEW 4 major objections 6 minor 40 references

Stackelberg Game Based Performance Optimization in Digital Twin Assisted Federated Learning over NOMA Networks

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A digital twin at the server, NOMA uplinks, and reputation-based client selection let federated learning cut total latency and energy, with a Stackelberg game yielding closed-form optimal resource allocations.

desk verdict A useful DT-NOMA-FL resource allocation paper whose main convergence proof has a concrete normalization error; fixable with major revision. read the letter →

arxiv 2501.01584 v1 pith:HEZLV5TE submitted 2025-01-03 cs.LG cs.CRcs.GTcs.NI

Bibo Wu , Fang Fang , Xianbin Wang This is my paper · ORCID

classification cs.LGcs.CRcs.GTcs.NI MSC 68T0791A6590C26

keywords federatedlearningdigitaltwinNOMAStackelberggameclientselectionpoisoningattacksresourceallocationlatency-energytradeoff

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 tries to show that a server-side digital twin can absorb most of a client's training load, that NOMA lets many clients upload simultaneously, and that reputation-based selection keeps poisoners out of aggregation. On top of that, it formulates the tension between clients (who want low energy) and the server (who wants low latency) as a Stackelberg game. The claim is that this game has a Stackelberg equilibrium with closed-form strategies: clients map the maximum allowed data to the twin, choose the smallest feasible local frequency, and set transmit power through a Dinkelbach iteration, while the server equalizes its per-client compute times. If true, the scheme gives a low-complexity resource-allocation rule that reduces both energy and latency, even with 30–50% poisoners, and it degrades gracefully as twin-mapping error grows.

What carries the argument

The load-bearing object is the Stackelberg game itself, with the selected clients as leader (minimizing total energy) and the server as follower (minimizing total latency), solved by backward induction: first the follower's optimal $\alpha_n$ is derived, then it is substituted into the leader's problem, which decomposes into independent subproblems for $v_n$, $f_n$, and $p_n$. The mechanism is supported by two structural assumptions: NOMA transmission gives all clients one common transmission time, and the compute-then-transmit split lets the leader stretch local training to the common deadline so that energy is minimized. The transmit-power subproblem is a concave-convex fractional program solved by the Dinkelbach method with KKT-based closed-form iterates. A secondary mechanism is the reputation score $Z_n=\xi_1 AC_n+\xi_2 MS_n+\xi_3 PI_n$, which sorts clients before each round and is what blocks poisoners from aggregation.

What would settle it

Take a quadratic loss such as $L(w)=\tfrac12\|w\|^2$ and run the paper's global update with $\varepsilon>0$ and $N$ clients. Plain gradient descent converges for $\eta<1$, but the scaled update multiplies each step by $\Gamma>1$; if the iterates grow without bound or converge to a point different from the minimizer, the claimed convergence is false. The same check can be run on MNIST or CIFAR-10 by comparing the paper's aggregation rule against the same rule with $\Gamma$ replaced by $1$.

Watch

Extended reading notes

Core claim

In the proposed DT-assisted FL system over NOMA, each client can split its data into a local portion $(1-v_n)D_n$ and a portion $v_nD_n+\varepsilon$ mirrored at the server's digital twin. The paper's central result is that the latency-energy objectives of clients and server can be modeled as a Stackelberg game and solved in closed form: the server's optimal frequency coefficients are $\alpha_n^* = c_n\hat D_n / (t_{\mathrm{total}} f_S)$ when resources suffice, and $\alpha_n^* = c_n\hat D_n / \sum_n c_n\hat D_n$ when the server is fully loaded; the clients' optimal mapping ratio is $v_n^*=v_n^{\max}$, the optimal local frequency is $f_n^*=\max\{\tilde f_n, f_n^{\min}\}$, and the optimal transmit power comes from a Dinkelbach-based successive solution. The paper further claims that the reputation-based selection of clients — combining accuracy contribution, model staleness, and positive interactions detected by RONI — keeps global accuracy high under label-flipping poisoners, and that the whole update converges because the aggregated model has the form $\Gamma(w^{t-1}-\eta\nabla L(w^{t-1}))$ with $\Gamma=1+\varepsilon N/D$.

Load-bearing premise

The convergence claim rests on treating the global update $w^t = \Gamma(w^{t-1}-\eta\nabla L(w^{t-1}))$, with $\Gamma=1+\varepsilon N/D>1$, as if it were ordinary gradient descent; if this scaled iterate does not actually converge to the intended model, the accuracy results lose their foundation.

Editorial extensions

If this is right

Deploying a digital twin that mirrors only insensitive client data lets weak clients offload most training to the server, cutting local compute latency and energy without exposing private data.
Reputation-based selection that includes positive interactions keeps accuracy high even with 30–50% label-flipping poisoners, where selection based only on data size and staleness degrades sharply.
Closed-form frequency and power rules mean per-round resource allocation can be computed cheaply, so the scheme is practical in dynamic wireless settings.
NOMA's simultaneous transmissions allow more clients per round at the same bandwidth, so total cost grows slowly as the number of selected clients increases, unlike OMA.
Larger digital-twin mapping deviations reduce accuracy, more on CIFAR-10 than MNIST, so the benefit of the twin depends on keeping its estimated data close to real data.

Reading between the lines

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

Inference (not in the paper): the update in (4) suggests a concrete correction — rescale the aggregate by $D/(D+\varepsilon N)$ or shrink $\eta$ by $\Gamma$ — to recover standard gradient descent; whether the closed-form solutions survive that correction is untested.
Inference (not in the paper): the poisoning-mitigation claim inherits RONI's assumptions; an adaptive attacker whose updates pass the influence check while still biasing the global model is outside the paper's scope and would stress the reputation metric.
Inference (not in the paper): the leader is a coalition minimizing total energy, which assumes clients are willing to follow the coalition-optimal $v_n^* = v_n^{\max}$; individually rational clients could deviate if that choice shifts energy onto them, so voluntary participation is an implicit assumption.

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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

The paper is a competent engineering combination of DT-assisted FL, NOMA, and reputation-based client selection, with a clean Stackelberg formulation. The genuinely new bits are the joint problem and the positive-interaction term in the reputation score, plus the closed-form alpha* solutions in (26) and (29). The simulations on MNIST/CIFAR-10, IID and non-IID, are reasonably extensive.

The soft spot the reader and stress-test flag is real. Moving from Eq. (3) to Eq. (4) contains a normalization error: the numerator sums over N clients with an extra epsilon per client, so the denominator should be D + N*epsilon, not D. As written, the update is Gamma*(w^{t-1} - eta*grad L) with Gamma = 1 + N*epsilon/D > 1, which is not standard gradient descent and may converge to a shifted point or diverge. The one-line appeal to the 'gradient descent property' does not cover this. This is load-bearing because the accuracy plots in Figs. 5-8 are presented as validation of the whole scheme. The fix is straightforward - normalize by D + N*epsilon - and I suspect the simulations actually used the correct average, but the derivation as written is wrong and the proof is unsupported.

The second concern is the leader-level optimality claim. The problem is non-convex; the proposed decomposition into v, f, p subproblems is a block coordinate descent, and there is no proof that it reaches a global optimum or a true Stackelberg equilibrium. The paper calls it an equilibrium, but that overstates what was shown. This is common in this literature, so it is a presentation issue as much as a technical one.

Minor points: the Fig. 5 benchmark removes PI by design, which stacks the deck; there are no error bars; and the Weibull parameters and reputation weights are not specified in the simulation table.

I think the reader's reject is too harsh. The resource allocation portion is standard but solid, the convergence error is concrete and fixable, and the rest is typical for the field. I would send it to a serious referee with instructions to require the Eq. (4) fix and a more careful statement about the leader-level solution. With those edits, it could become a passable conference paper or a borderline journal paper.

Referee Report

4 major / 6 minor

Summary. This paper proposes a digital twin (DT)-assisted federated learning (FL) system over a NOMA uplink in which a server deploys a DT network to compensate for clients with poor computation and communication resources. The authors introduce a reputation-based client selection rule based on accuracy contribution, model staleness, and positive interactions, and model the energy-vs-latency trade-off as a Stackelberg game with clients as leader and server as follower. The server's frequency allocation is derived in closed form via Theorem 1, and the leader's mapping ratio, local CPU frequency, and transmit power are obtained by problem decomposition and a Dinkelbach-type procedure. The paper reports simulations on MNIST and CIFAR-10 comparing the proposed scheme with benchmarks that omit DT, use OMA, or use random allocation.

Significance. The paper addresses a timely combination of DT, NOMA, reputation-based client selection, and game-theoretic resource allocation for FL under poisoning attacks. The system model is clearly presented and the derivations are mostly self-contained; the authors provide explicit algorithms and evaluate on two datasets with IID and non-IID splits. However, the current manuscript contains load-bearing mathematical errors: the global aggregation step in Eq. (4) is not properly normalized, the power-allocation latency constraint in Eq. (35b) has the wrong inequality direction, and the decomposition ignores the common transmission time of NOMA clients. In addition, the claimed Stackelberg equilibrium and global optimality of the leader-level solution are not proved. Because these issues directly affect the convergence and cost results, the significance of the work cannot be assessed until they are resolved.

major comments (4)

[II-A, Eq. (4)] The global aggregation update is incorrectly normalized. Substituting w_t^n = w_t^S = w^{t-1} - eta grad L(w^{t-1}) into Eq. (3) gives sum over n of [(1-v_n)D_n + v_nD_n + epsilon] = D + N epsilon, not D. Hence Eq. (4) defines w^t = Gamma (w^{t-1} - eta grad L(w^{t-1})) with Gamma = 1 + epsilon N / D > 1. This is not a standard gradient-descent iterate. A fixed point of this update satisfies grad L(w) = ((Gamma-1)/(Gamma eta)) w, so even when it converges it approaches a shifted point rather than a minimizer of L, and the convergence condition becomes |Gamma (1 - eta lambda)| < 1 rather than |1 - eta lambda| < 1. The sentence 'following the gradient descent property, the global model wt can converge to the expected accuracy' is therefore unsupported. Since the accuracy curves in Figs. 5-8 are produced by this update, the FL validation is premised on an unproved and likely incorrect convergence statement. The likely intended normalization is division by D + N epsilon, which would cancel Gamma; the paper should correct the aggregation rule and re-run the simulations, or provide a valid convergence analysis of the scaled update.
[V-B, Eq. (35b)] The latency constraint has the wrong direction. From (19b), t_cmp_n + t_com_n <= T_max, so with G_n = T_max - c_n(1-v_n)D_n/f_n, the requirement is R_n >= d_n/G_n, i.e., B log2(1 + p_n F_n) >= d_n/G_n. Eq. (35b) states the opposite inequality, B log2(1 + p_n F_n) <= d_n/G_n. This reversed sign is carried into the Lagrangian (40) and the multiplier update (45a), so the power solution p_n^* in (43) can violate the maximum-latency constraint. The energy results in Fig. 9, which depend on these power values, are therefore not reliable.
[V-B, Eqs. (22), (31), (35)] The decomposition into per-client subproblems is inconsistent with the NOMA transmission model. Eq. (22) asserts t_com_n = t_com for all n, so constraint (19b) is a joint constraint t_cmp_n + t_com <= T_max with a common t_com. In the subproblems, A_n = T_max - t_com_n is treated as a client-specific constant, and the power subproblem (35) imposes only a per-client rate bound without any equality constraint linking the rates to a common t_com. Consequently, the per-client solutions may not correspond to any feasible NOMA schedule. The paper should solve the coupled problem or justify why the decomposition preserves feasibility and optimality.
[V-C, Algorithm 2 and Eq. (21)] The claim that Algorithm 2 attains the Stackelberg equilibrium is not established. Problem (30) is non-convex and the variables are coupled; Algorithm 2 is a cyclic coordinate descent that stops 'until E converges' with no convergence criterion or optimality certificate. The Dinkelbach procedure (Algorithm 1) only solves the scalar power subproblem (35) for fixed v_n and f_n, so it cannot certify a solution of the joint problem. The statements in the final paragraph of Section V that 'the optimal solutions ... can be obtained effectively' and that the equilibrium is 'realized' are therefore overclaims. The authors should provide a convergence and optimality analysis, or clearly present the algorithm as a heuristic and adjust the wording of the contributions accordingly.

minor comments (6)

[III, Eq. (12)] The Weibull parameters omega_{1,n}, omega_{2,n}, omega_{3,n} are described as predefined but their values are never specified in the simulation setup, and the RONI threshold in Section III.3 is also not given; this makes the reputation values and poisoning-detection results non-reproducible.
[VI] The accuracy curves in Figs. 5-8 and the cost curves in Fig. 9 are presented without error bars, confidence intervals, or the number of random seeds; the claimed superior performance of the proposed scheme cannot be statistically assessed.
[VI-A] The benchmark in Fig. 5 deliberately omits the positive-interaction factor from the reputation score, so its degradation under poisoners is expected by construction; a sensitivity analysis over the reputation weights xi_1, xi_2, xi_3 would better isolate the contribution of the proposed selection rule.
[II] The notation 'M clients indexed by M in {1,2,...,M}' and 'N clients N in {1,2,...,N}' is confusing; please use distinct symbols for the sets and their cardinalities.
[III, Eqs. (13)-(16)] The normalized staleness MS_n^t in Eq. (14) is later written as MS_n in Eq. (16), conflating the raw and normalized values; please make the notation consistent.
[Fig. 4] The convergence plot of Algorithm 1 shows a single run without specifying the random channel realization or the termination threshold delta; please provide these details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Stackelberg and closed-form optimization results are derived from the stated model equations, and the main weakness is a non-circular convergence proof gap at Eq. (4).

full rationale

I walked the claimed derivation chain and found no step where a fitted parameter is renamed as a prediction or where a result equals its input by construction. The leader/follower solutions in Section V follow from the model equations of Sections II-B and II-C via monotonicity, equalization arguments, KKT conditions, and the Dinkelbach algorithm; the simulated accuracy and cost comparisons are not used as inputs to these derivations. The reputation weights and Weibull coefficients are chosen parameters, not fit to the reported outcomes, and the benchmark's omission of PI is a deliberate ablation rather than a circular construction. The self-citations ([1], [18], [21]) are background or earlier work and are not load-bearing for the Stackelberg/optimization claim, so they do not raise the circularity score under the review rules. The manuscript does contain a serious non-circular correctness gap: Section II-A derives the global update as wt = Gamma*(w^{t-1} - eta*grad L(w^{t-1})) with Gamma = 1 + epsilon*N/D, then asserts convergence "following the gradient descent property." Because the Gamma-scaled iterate is not plain gradient descent, the convergence statement is unsupported; this is an omitted or invalid proof, not circular reasoning, and should be weighed as a correctness risk.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The optimization derivation relies on standard convex analysis and known results (KKT, Dinkelbach). Domain assumptions include the SIC decoding order, identical NOMA transmission time, ignored server energy, and DT fidelity. The ad hoc assumption is that the scaled global update in Eq (4) converges despite a factor Γ>1. Free parameters include hand-chosen reputation weights, unspecified Weibull parameters, and the RONI threshold.

free parameters (3)

Reputation weights ξ1, ξ2, ξ3 = 0.3, 0.5, 0.2
Hand-set in Section VI-A; not optimized or cross-validated, yet they directly determine client selection and reported FL accuracy.
Weibull model parameters (ω1_n, ω2_n, ω3_n) = not specified
Eq (12) calls them predefined but never lists values or a fitting procedure; AC scores, and thus selection, depend on them.
RONI threshold for classifying negative influence = not specified
Section III-3 says updates are NI if they degrade performance 'beyond a specified threshold'; the threshold is not given, affecting PI and reputation.

assumptions (6)

domain assumption The receiver performs SIC with decoding order aligned with descending channel gains, |h1|^2 ≥ ... ≥ |hN|^2.
Section II-C: the achievable rate in Eq (9) depends on this order; a different decoding order changes the rate expressions and the optimization.
domain assumption All selected clients transmit for the same duration tcom in the NOMA phase, and local training times are equal to a common tcmp.
Section V-A: Eqs (22)-(24) assume identical tcom and tcmp; this is a modeling simplification, not derived from an optimality condition.
domain assumption The server has sufficient power, so its energy consumption is ignored.
Section II-B: 'Due to the sufficient power of the server, we reasonably ignore the corresponding energy consumption.'
ad hoc to paper The global model update wt = Γ(w^{t-1} - η∇L(w^{t-1})) with Γ=1+εN/D converges to the expected accuracy by gradient descent properties.
Section II-A, Eq (4): the convergence is asserted, not proven; the scaling by Γ is not standard gradient descent and may break convergence.
domain assumption The DT network can estimate the insensitive data of each client with bounded deviation ε, and this estimate is a faithful proxy for training.
Section II: 'we consider that the DT network can only reflect the insensitive data of client to guarantee data privacy.'
domain assumption The server knows all clients' reputation values before selection.
Section III: 'We assume that the server has knowledge of all clients' reputation values prior to selecting clients.'

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Cite this review

Pith. "Pith review of Stackelberg Game Based Performance Optimization in Digital Twin Assisted Federated Learning over NOMA Networks." pith.science (2026). https://pith.science/paper/HEZLV5TE

@misc{pith2026250101584,
  author       = {Pith},
  title        = {Pith review of: Stackelberg Game Based Performance Optimization in Digital Twin Assisted Federated Learning over NOMA Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEZLV5TE}},
  note         = {Machine review of arXiv:2501.01584}
}

read the original abstract

Despite the advantage of preserving data privacy, federated learning (FL) still suffers from the straggler issue due to the limited computing resources of distributed clients and the unreliable wireless communication environment. By effectively imitating the distributed resources, digital twin (DT) shows great potential in alleviating this issue. In this paper, we leverage DT in the FL framework over non-orthogonal multiple access (NOMA) network to assist FL training process, considering malicious attacks on model updates from clients. A reputationbased client selection scheme is proposed, which accounts for client heterogeneity in multiple aspects and effectively mitigates the risks of poisoning attacks in FL systems. To minimize the total latency and energy consumption in the proposed system, we then formulate a Stackelberg game by considering clients and the server as the leader and the follower, respectively. Specifically, the leader aims to minimize the energy consumption while the objective of the follower is to minimize the total latency during FL training. The Stackelberg equilibrium is achieved to obtain the optimal solutions. We first derive the strategies for the followerlevel problem and include them in the leader-level problem which is then solved via problem decomposition. Simulation results verify the superior performance of the proposed scheme.

Figures

Figures reproduced from arXiv: 2501.01584 by the authors.

**Figure 2.** Analysis of t total and t S n. at the server side, the transmission time for all selected clients is assumed to be identical [33], which can be expressed as t com n = t com , ∀n ∈ N . (22) In this case, it is required that the selected clients complete their local model training within a time period t cmp before the NOMA transmission phase. By integrating (5) into (6), the energy consumption of client n for local tr… view at source ↗

**Figure 3.** Solution analysis of fn. and we can obtain the optimal solution of fn by analyzing following two cases: If ˜fn ≤ f min n , the optimal solution is derived by f ∗ n = f min n ; Otherwise, if f min n < ˜fn ≤ f max n , the optimal solution is derived by f ∗ n = ˜fn. The analysis is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Convergence of Algorithm 1. VI. SIMULATION AND ANALYSIS In this section, extensive numerical simulations are conducted to evaluate the performance of the proposed scheme. We consider a circular area of 500 radius, where the server is deployed at the center and 20 clie…

**Figure 5.** Figure 5: FL performance with different number of poisoners. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 7.** Figure 7: FL performance on MNIST dataset (30% poisoners). CIFAR-10 datasets, respectively. Note that the DT deviation needs to be multiplied by a random value between −1 and 1 before applying it to each mapping data. Intuitively, as the DT deviation for mapping data increases, …

**Figure 9.** Figure 9: Performance of total cost. and OMA schemes. In particular, the OMA scheme has the worst performance and is not robust, due to the insufficient selected clients at each round. Hence, NOMA is promising in improving FL performance given the inherent communication constrai…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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