{"id":"b98ef5c9-fe34-42e9-ba03-7b2a430a2ee3","arxiv_id":"2501.01584","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"For digital twin assisted federated learning over NOMA, the paper derives a Stackelberg equilibrium for mapping ratio, local frequency, and transmit power, with reputation-based selection to resist poisoning attacks.","lead":"This paper proposes a scheme for wireless federated learning that combines digital twin assistance, NOMA transmission, and reputation-based client selection to cut latency and energy use. It models the client-server interaction as a Stackelberg game and derives resource allocation formulas, testing them on MNIST and CIFAR-10.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4)'s global update divides by D instead of D+Nε, creating the spurious factor Γ=1+εN/D; the claimed convergence to 'expected accuracy' is therefore not established and the FL accuracy results rest on it.","rationale":"The reader's weakest assumption pointed at Eq. (4); I agree and make the concern more specific. The issue is not merely that Γ>1 'may diverge'—the derivation itself divides by the wrong total, so the stated convergence result is unsupported as written and the factor Γ is an artifact of a normalization error. This is load-bearing because every accuracy result (Figs. 5–8) and the claim of superior FL performance depend on the training algorithm converging to the expected model. The remaining weaknesses—lack of a global optimality proof for the nonconvex leader problem, heuristic coordinate descent, and absent error bars—are real but secondary; a paper can still present a useful heuristic resource-allocation scheme without a global optimality certificate, but an FL paper whose training update is not shown to converge cannot support accuracy claims. I therefore keep the reader's REJECT verdict unchanged.","tokens_in":952,"tokens_out":1094,"duration_ms":71589,"concrete_test":"Take the simulation setup of Section VI (MNIST, η=0.01, N=5) and implement the global update exactly as in Eq. (4) with the DT deviation ε actually used (for example ε=0.2·D/N, matching the 0.2 DT deviation curve in Fig. 6). Plot training loss and accuracy against the properly normalized update w_t = w_{t−1} − η∇L(w_{t−1}) obtained by dividing Eq. (3)'s numerator by D+Nε instead of D. If the Eq. (4) trajectory diverges, oscillates, or converges to an accuracy outside the error bar of the normalized trajectory, the convergence premise behind Figs. 5–8 fails; as an algebraic check, confirm that replacing D by D+Nε in Eq. (3) removes Γ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central training claim is the convergence of the global model update, established in Section II-A by Eq. (4). The step from Eq. (3) to Eq. (4) is not a harmless approximation: the numerator sums N client local updates of weight (1−vn)Dn and N DT updates of weight vnDn+ε, whose total weight is Σ[(1−vn)Dn + vnDn + ε] = D + Nε, while the denominator is defined as D = ΣDn. Dividing by D yields Γ = 1 + εN/D > 1. Hence the actual update is w_t = Γ(w_{t−1} − η∇L(w_{t−1})), which is a gradient-descent iterate multiplied by a constant larger than 1. This is not a standard FedAvg update: its fixed point satisfies ∇L(w*) = ((Γ−1)/(Γη))w*, so even under convexity it converges, when it does, to a shifted point rather than the minimizer of L; for losses where Γ|1−ηL| exceeds 1 it can diverge. The proof's one-line recourse to the 'gradient descent property' is exactly the unsupported step. If the intended weighted average over all training and DT data were normalized by D+Nε, the Γ factor cancels and the standard update is recovered, which indicates a normalization error. Since the accuracy plots in Figs. 5–8 are generated by this update, the paper's central validation is premised on an unproved and likely incorrect convergence statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19526,"tokens_out":15183,"duration_ms":138212,"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":[{"comment":"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.","section":"II-A, Eq. (4)"},{"comment":"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.","section":"V-B, Eq. (35b)"},{"comment":"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.","section":"V-B, Eqs. (22), (31), (35)"},{"comment":"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.","section":"V-C, Algorithm 2 and Eq. (21)"}],"minor_comments":[{"comment":"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.","section":"III, Eq. (12)"},{"comment":"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.","section":"VI"},{"comment":"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.","section":"VI-A"},{"comment":"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.","section":"II"},{"comment":"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.","section":"III, Eqs. (13)-(16)"},{"comment":"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.","section":"Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"The technical concerns above align with the skeptical reader's stress test: the normalization error in Eq. (4) is the most serious issue because it undermines the FL convergence claims. The errors are fixable in principle, but they require re-deriving the aggregation and power-constraint steps and re-running the simulations. The paper also overlaps substantially with the authors' prior VTC paper [1] and with references [18] and [21]; the editor may wish to verify that the journal version adds sufficient new material."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nMinor 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.\n\nI 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.","headline":"A useful DT-NOMA-FL resource allocation paper whose main convergence proof has a concrete normalization error; fixable with major revision.","tokens_in":20041,"tokens_out":4139,"would_cite":false,"duration_ms":42189,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","91A65","90C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["federated learning","digital twin","NOMA","Stackelberg game","client selection","poisoning attacks","resource allocation","latency-energy tradeoff"],"falsifier":"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 $1$.","tokens_in":18949,"feed_emoji":"📡","tokens_out":6795,"duration_ms":64321,"temperature":0.7,"pith_summary":"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.","feed_headline":"Digital twin plus NOMA cuts federated learning's cost","feed_subtitle":"Clients and server play leader-follower: optimal data split, compute, and power are derived in closed form.","key_machinery":"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.","core_discovery":"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 $\\Gamma=1+\\varepsilon N/D$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the DT-enabled FL system model and the convergence-by-similarity argument used for the global update.","marker":"[13]"},{"why":"Supplies the RONI scheme that labels positive and negative interactions in the reputation score.","marker":"[31]"},{"why":"Gives the Stackelberg latency-energy formulation for NOMA MEC that the leader-follower structure builds on.","marker":"[32]"},{"why":"Justifies the identical transmission time for all NOMA clients used to simplify the follower-level problem.","marker":"[33]"},{"why":"Establishes the monotone energy-transmission-time tradeoff behind Remark 1.","marker":"[34]"},{"why":"Provides the Dinkelbach convergence result that underlies Algorithm 1 for transmit power.","marker":"[38]"}],"fun_headline_variants":["Stackelberg game optimizes digital twin FL over NOMA","Closed-form solution for DT-FL latency and energy tradeoff","Game theory cuts cost in digital twin federated learning","Reputation-based client selection thwarts poisoning in DT-FL","DT and NOMA: optimal resource split via Stackelberg game"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Stackelberg game optimizes digital twin FL over NOMA","Closed-form solution for DT-FL latency and energy tradeoff","Game theory cuts cost in digital twin federated learning","Reputation-based client selection thwarts poisoning in DT-FL","DT and NOMA: optimal resource split via Stackelberg game"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000356,"raw_usage":{"total_tokens":1975,"prompt_tokens":1030,"completion_tokens":945,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":860}},"tokens_in":646,"tokens_out":945,"duration_ms":10181,"temperature":1.0,"reasoning_tokens":860,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:25:40.921982+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $1$.","supporting_citations":[{"cited_title":"Resource allocati on based on digital twin-enabled federated learning framework in he terogeneous cellular network,","cited_arxiv_id":null,"evidence_quote":"Provides the DT-enabled FL system model and the convergence-by-similarity argument used for the global update."},{"cited_title":"St ackelberg game of energy consumption and latency in MEC systems with NO MA,","cited_arxiv_id":null,"evidence_quote":"Gives the Stackelberg latency-energy formulation for NOMA MEC that the leader-follower structure builds on."},{"cited_title":"Wireless federated learning (WFL) for 6G networks—part II: The compu te-then- transmit NOMA paradigm,","cited_arxiv_id":null,"evidence_quote":"Justifies the identical transmission time for all NOMA clients used to simplify the follower-level problem."},{"cited_title":"Joint optimization of tas k assignment and power allocation for NOMA-aided MEC systems,","cited_arxiv_id":null,"evidence_quote":"Establishes the monotone energy-transmission-time tradeoff behind Remark 1."}],"review_version":1}