{"id":"8234d95c-75cf-407f-80bb-7ab88a12cee4","arxiv_id":"2508.00341","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A low-complexity greedy device scheduling policy based on channel strength and alignment reduces air-aggregation error and improves federated learning accuracy in low-SNR mobile IoT settings.","lead":"This paper picks which mobile devices join each round of over-the-air federated learning and how the base station steers its receiver, to make model averaging accurate when wireless links are noisy. The main practical result is a fast, channel-aware scheduling rule that simulations show cuts aggregation error and improves MNIST accuracy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 3 does not actually select the candidate with the largest beamforming objective as claimed; it ranks candidates by a proxy (max-min pairwise inner product) that is never shown to equal or bound problem (9), so Policy_greedy's reported advantage is not tied to its stated selection rule.","rationale":"The paper's central claim is that the low-complexity Policy_greedy achieves the best MSE and best test accuracy among the compared methods. The strongest support for this is empirical, but the algorithm that produces the reported results is described inconsistently: the prose claims selection of the best objective among G candidates, while the pseudocode selects by tmp, a pairwise-alignment proxy. The reader's weakest_assumption focused on Theorems 5.1 and 5.2 not proving that greedy selection is near-optimal in the general unequal-norm case. I agree with that concern and would add a more concrete internal defect: even the limited 'best among G candidates' claim is not implemented, because Algorithm 3 never evaluates the actual objective function (9) for the candidates. Theorems 5.1 and 5.2 only provide lower bounds or monotonicity statements for fixed subsets; they do not establish that the tmp ranking preserves the ranking of the beamformed objective. The proof of Theorem 5.1 also contains an incorrect equality in (19), though the theorem statement may still be salvageable. These issues do not by themselves disprove the empirical results: the proxy-based heuristic may genuinely perform well in the tested configurations, and the figures are consistent with that. But they mean the paper currently overstates its theoretical support and does not fully specify the criterion that actually drives the reported gains. The appropriate disposition remains conditional: the authors should fix Algorithm 3 to either compute the actual objective for each candidate or explicitly state that tmp is a heuristic proxy, correct the Theorem 5.1 proof, and provide the reproducibility artifacts (code, seeds, error bars) needed to verify the central empirical claim. I therefore leave the reader's CONDITIONAL verdict unchanged.","tokens_in":19718,"tokens_out":11921,"duration_ms":139007,"concrete_test":"Instrument Algorithm 3: for each of the G stored candidate subsets S_c (from lines 12-18), run Algorithm 1 on S_c and compute the actual objective obj_c = P * min_{s in S_c} |m_c^H h_s|^2 (equivalently the normalized MSE 1/(P min_s |m_c^H h_s|^2)). Over 100 random channel realizations (K=100, N_r=8, S=10, G=5), compare the subset chosen by the current proxy (max tmp) with the subset having the largest obj_c. Also check whether the proxy rank order agrees with the true objective rank order. If the proxy-selected subset is not the one with the largest objective (or lowest MSE), the algorithm is mis-specified; then rerun the MNIST i.i.d./non-i.i.d. comparisons with the corrected best-of-G rule and see whether Policy_greedy still beats the baselines.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weakness is an internal mismatch between the stated selection rule and Algorithm 3. The text before Algorithm 3 says 'the selected users subset S is obtained with the largest obj value among the G considered objective values.' But Algorithm 3 never computes the objective of problem (9) (or the equivalent MSE objective of (7)). Its stored quantity tmp (Algorithm 3, lines 5, 12, 18) is the max-min absolute inner product between the newly added user and the current partial subset. Lines 20-22 then select the candidate with the largest such tmp. Nothing in Theorems 5.1-5.2 implies that this pairwise-alignment proxy is order-equivalent to the true max-min beamforming objective: Theorem 5.1 is an equal-norm lower bound for a fixed subset, and Theorem 5.2 is a same-direction scaling monotonicity. Neither covers the unequal-norm candidate ranking used in Algorithm 3. Moreover, the proof of Theorem 5.1 has an incorrect equality at (19): with m = h_p, min_i |m^H h_i| equals cos(max_i alpha_{p,i}), which is generally strictly larger than cos alpha. The theorem's conclusion is probably repairable, but the result still does not validate the greedy proxy. Consequently, the central empirical claim that Policy_greedy achieves the best MSE/accuracy is not backed by the algorithm as described: the simulations exercise a proxy-based heuristic, while the paper explains the result as best-of-G by objective. If the proxy and the true objective disagree, the reported best performance may be configuration-specific and the design rationale collapses.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers over-the-air federated learning (AirComp FL) in low-SNR mobile IoT, where only a subset of users can transmit in each round. It formulates an MSE-minimization problem with fixed selection size and per-user power constraints, reduces it to a max-min beamforming alignment problem (problem (9)), and proposes two solution approaches: an iterative DC-based alternating method (Algorithms 1-2) and a low-complexity channel-based greedy scheduling policy (Policy and Policy_greedy, Algorithm 3). The empirical section reports MSE, CDF, runtime, and MNIST accuracy comparisons against several baselines, claiming that Policy_greedy achieves the best MSE and test accuracy in both i.i.d. and non-i.i.d. settings.","tokens_in":20128,"tokens_out":10386,"duration_ms":107539,"significance":"If the claims were fully supported, the paper would provide a practically useful low-complexity scheduling heuristic for AirComp FL, together with a DC-based beamforming solver. The problem formulation is clean, the transmit coefficient follows prior work [42] without parameter fitting, and the simulation study is reasonably broad: it includes MSE versus K, CDFs over 50 runs, complexity/runtime comparisons, and i.i.d./non-i.i.d. MNIST tasks. The main weaknesses are in the supporting theory: the NP-hardness proof in Appendix A is not a valid reduction, the linear-convergence claim in Theorem 4.1 is a generic performance-estimation argument not tied to the proposed algorithm, and the validation of the greedy proxy in Theorem 5.1 contains a false equality and does not cover the actual selection rule of Algorithm 3. These issues are load-bearing for the paper's stated contributions, though they appear repairable in a revision.","major_comments":[{"comment":"The text preceding Algorithm 3 states that 'the selected users subset S is obtained with the largest obj value among the G considered objective values,' and the paper attributes the empirical advantage of Policy_greedy to this selection. However, Algorithm 3 never computes the objective of problem (9) or the equivalent MSE objective of (7). The quantity tmp computed at lines 5, 12, and 18 is max_i min_s |h_s^H h_i|, a pairwise-alignment proxy, and lines 20-22 select the subset with the largest stored value of this proxy. Theorems 5.1 and 5.2 do not establish that this proxy is order-equivalent to the true max-min beamforming objective: Theorem 5.1 is an equal-norm lower bound for a fixed subset, and Theorem 5.2 is a same-direction scaling monotonicity. Neither covers the unequal-norm candidate ranking executed by Algorithm 3. The reported MSE and accuracy advantages of Policy_greedy are therefore not backed by the selection rule as described. The authors should either modify Algorithm 3 to evaluate the true objective of (9) for each candidate subset, prove that the proxy is order-equivalent to (or bounds) the true objective, or present the method explicitly as an unproven heuristic and temper the corresponding claims.","section":"Sec. 5, Algorithm 3"},{"comment":"In the proof of Theorem 5.1, the equality min_{i in S} |h_p^H h_i| = cos alpha in Eq. (19) is generally false. Since alpha is the maximum angle between any two channels in S, every channel satisfies |h_p^H h_i| >= cos alpha, and the minimum is usually strictly larger than cos alpha; the member h_p itself has inner product 1 with itself. The contradiction only needs the inequality min_i |h_p^H h_i| >= cos alpha, so the theorem's conclusion is likely repairable, but the proof as written is incorrect. Please correct this step and state the valid inequality explicitly.","section":"Sec. 5, Theorem 5.1 proof, Eq. (19)"},{"comment":"Theorem 4.1 claims linear convergence of the DC algorithm, but the proof in Appendix B is a generic performance-estimation argument for a difference-of-convex decomposition with abstract functions f1, f2 and parameters mu1, mu2, L1, L2. The proof never identifies these functions with the iterates of Algorithm 1 or with problem (14), and it assumes the PL inequality for a single-user quadratic f(m)=|m^H h_k|^2, whereas Algorithm 1 solves the max-min problem (10) over all k in S. The PL property for the max-min objective is never verified. As written, the result does not establish linear convergence of the proposed algorithm. Please either provide a proof specific to the algorithm's update rule or state a corrected convergence result with explicitly verified assumptions.","section":"Sec. 4.1, Theorem 4.1 and Appendix B"},{"comment":"The claimed NP-hardness proof of problem (9) is not a valid polynomial-time reduction from Maximum Clique. It asserts without proof that problem (9) 'can be transformed' to selecting a subset maximizing pairwise inner products, but Theorems 5.1 and 5.2 provide only bounds, not an equivalence between problem (9) and the pairwise-inner-product objective. The proof then says the threshold tau is adjusted iteratively until |S| = S, which means the construction is not a single polynomial-time mapping, and no correctness or termination proof is provided. The reduction also fails to map clique-size decision instances to fixed-S instances of problem (9). The heuristic methods do not depend on this claim, but the formal NP-hardness statement should be removed or replaced by a correct reduction.","section":"Appendix A"}],"minor_comments":[{"comment":"The desired signal y_des in (2) is the sum over the selected subset S, while the global model update in (1) averages over all K users. Since only S users participate, please clarify the normalization (or state that the scaling factor eta absorbs the 1/|S| factor), as this affects the interpretation of the aggregation error.","section":"Sec. 3.1, Eqs. (1)-(2)"},{"comment":"The paragraph on learning performance says 'only the remaining seven methods were included' but then lists six methods ('Iterative', 'Policy', 'Policy_greedy', 'Subgradient', 'Iterative reweighted', and 'RL') and reports seven average MSE/sigma^2 values. Please correct the enumeration and the mapping of values to methods.","section":"Sec. 6.6"},{"comment":"The line 'Select the user with the largest S projection values' should read 'Select the users with the largest S projection values', since a subset of size S is being chosen.","section":"Algorithm 2, line 6"},{"comment":"There are small typographical errors: 'parametcers' should be 'parameters' in Table 1, and 'lager modulus' should be 'larger modulus' in Remark 5.2.1. Please proofread the final text.","section":"Table 1 and Sec. 5"}],"recommendation":"major_revision","confidential_remarks":"The main empirical claim that Policy_greedy achieves the best MSE and accuracy may be valid, but the paper's supporting theory needs substantial rework. The most serious issue is the internal mismatch between the description of Algorithm 3 and its actual selection rule; this should be addressed before publication. The NP-hardness appendix and the linear-convergence theorem are not rigorous as written and should be corrected or toned down. I do not see concerns about duplicate publication or problematic citation behavior; the self-citation [39] is used as a baseline."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Start with the good: the paper empirically shows that a greedy channel-based user selection, choosing users with small pairwise angles and large channel norms, reduces MSE and improves FL accuracy in low-SNR AirComp. The simulations are consistent and the comparison set is plausible. Policy_greedy is genuinely new, though it builds on known DC/SDR machinery.\n\nBut the theory as written does not support the algorithm. The stress-test note is right: Algorithm 3 stores tmp = max over candidates of min over selected users of |h_s^H h_i|, and selects the candidate with the largest tmp. The text says it selects the subset with the largest objective value of problem (9). Those are not the same. Theorems 5.1 and 5.2 give a lower bound for a fixed subset under equal norms, and a monotonicity for same-direction scaling. Neither proves that pairwise-angle proxy ranks subsets the same way as the max-min beamforming objective. So the reported advantage of Policy_greedy is not actually tied to the stated selection rationale. That is a load-bearing mismatch.\n\nThe NP-hardness appendix is also not a valid reduction: it maps to max clique but then adjusts the threshold through an iterative procedure, which is not a polynomial-time reduction. Theorem 4.1's convergence proof is a generic performance-estimation argument that does not demonstrate linear convergence of their specific DC algorithm. And the proof of Theorem 5.1 contains an incorrect equality at (19); the conclusion might be repairable, but as written it's wrong.\n\nThese are not minor typos. The paper's central claim is 'rigorously proved to yield a superior objective value,' and the proofs don't deliver that. That said, the heuristic itself is plausible and the empirical evidence is respectable. With a proper rewrite of the theory—or a demotion of those claims to conjectures—the paper could be a useful subfield contribution. As it stands, the manuscript oversells its analysis.\n\nAudience: wireless comm and AirComp FL researchers. It could be sent to review, but only with the expectation of heavy revision. I'd tell the editor to ask for corrected theory, a clear statement of what Algorithm 3 actually optimizes, and ideally code or error bars.","headline":"Plausible AirComp scheduling heuristic and solid simulations, but the theory misrepresents what the algorithm does and several proofs are invalid.","tokens_in":20609,"tokens_out":3491,"would_cite":false,"duration_ms":31819,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a low-complexity greedy scheduling policy based on channel angles and channel strengths achieves lower over-the-air aggregation error and higher MNIST test accuracy than iterative joint optimization and existing…","keywords":["over-the-air computation","federated learning","user scheduling","receive beamforming","difference-of-convex","low-SNR mobile IoT","MSE minimization","MNIST"],"falsifier":"A concrete check would be to construct a channel set in which the strongest-norm user lies far in angle from the other strong users, run the greedy policy and an exhaustive search over all subsets of the required size, and compare the resulting mean-square aggregation error: if exhaustive search materially beats the greedy subset, the paper's central scheduling criterion is incomplete.","tokens_in":19533,"feed_emoji":"📡","tokens_out":8507,"duration_ms":78608,"temperature":0.7,"pith_summary":"This paper tries to establish that user scheduling and receive beam steering in over-the-air federated learning can be decided jointly and cheaply, with a greedy channel-based policy outperforming heavier iterative optimization. The authors formulate the low-SNR aggregation problem as a max-min beamforming problem over which users to select and which receive direction to use, prove that users with smaller mutual channel angles and larger channel gains are preferable, and turn that insight into the low-complexity scheduling policy called Policy_greedy. In experiments, this policy achieves the lowest mean-square aggregation error and the highest MNIST test accuracy among all compared schemes, under both i.i.d. and non-i.i.d. data. If the claim holds, mobile IoT systems running over-the-air federated learning do not need the computational cost of iterative combinatorial optimization to obtain good aggregation.","feed_headline":"Greedy user scheduling beats iterative beam steering in over-the-air FL","feed_subtitle":"A low-complexity channel-based policy cuts aggregation MSE and lifts MNIST accuracy under i.i.d. and non-i.i.d. data.","key_machinery":"The carrying object is the max-min beamforming objective, written $\\max_{m,S}\\min_{k\\in S} P|m^H h_k|^2$, which the MSE-minimization problem reduces to after the optimal transmit coefficients are substituted in. The argument runs through two channel-geometry facts: users whose channels have a smaller maximum pairwise angle raise the lower bound on the objective, and users with larger channel modulus strengthen it. These facts justify a greedy subset construction (Algorithm 3) that starts from one of the $G$ strongest channel users and repeatedly adds the unselected user minimizing the maximum inner-product deviation from the current subset, after which the beamformer is computed once with a difference-of-convex rank-one method.","core_discovery":"The central claim is that the receive beam steering vector and the set of scheduled users can be designed together to minimize the MSE of over-the-air aggregation, and that the optimality-relevant property of a user subset is captured by the maximum angle between its channel vectors and by their moduli. The paper proves that for users with equal-norm channels, the best achievable objective is lower bounded by $P\\cos^2\\alpha$ with $\\alpha$ the largest pairwise channel angle, and that scaling channel vectors in the same direction only improves the objective; it then builds a greedy user selection policy on these two facts. The authors report that this policy, Policy_greedy, achieves the best MSE and the best test accuracy among the considered methods in both i.i.d. and non-i.i.d. MNIST experiments, making it the recommended low-complexity choice for low-SNR over-the-air federated learning.","pith_inferences":["A natural extension the paper leaves implicit is applying the same angle-and-modulus criterion per subcarrier in frequency-selective channels, which would broaden the policy to wideband AirComp and could be tested without changing the core optimizer.","Because problem (9) is reduced to maximum clique, the greedy policy's suboptimality gap likely grows with the total user count; measuring that gap statistically for large $K$ would indicate when the greedy parameter $G$ must be enlarged.","The MSE-to-accuracy link is demonstrated on MNIST only; testing the policy on tasks with larger models and stronger data heterogeneity would show how much of the accuracy gain transfers beyond this benchmark."],"forward_implications":["In low-SNR mobile IoT settings, Policy_greedy is the recommended scheduling strategy because it matches or beats the iterative method while replacing repeated $O(N_r^6)$ beamformer solves with a one-shot $O(K^2)$ subset search plus a single DC beamformer computation.","Larger user pools lower per-round MSE for the proposed policies but not for random selection, so user diversity helps only when scheduling actively picks aligned, high-gain users.","Increasing the number of scheduled users $S$ raises aggregation error, making participation count a tunable trade-off against communication-side model quality; more receive antennas relax that trade.","Lower MSE and higher test accuracy go together in the reported MNIST experiments, so aggregation-error reduction is itself a learning-performance strategy in low-SNR over-the-air federated learning."],"supporting_citations":[{"why":"Supplies the optimal transmit coefficient formula and the reduction of the joint problem to the max-min beamforming objective in problem (9).","marker":"[42]"},{"why":"Provides approximation bounds for semidefinite relaxation, motivating the DC-based rank-one reformulation for the beamforming subproblem.","marker":"[23]"},{"why":"Underpins the difference-of-convex treatment of the rank-one constraint used in Algorithm 1.","marker":"[22]"},{"why":"Supplies the projected subgradient baseline that the proposed methods are compared against.","marker":"[17]"},{"why":"Supplies the iterative reweighted beamforming baseline used in the experiments.","marker":"[18]"},{"why":"Provides the random aggregate beamforming baseline whose performance motivates deterministic beam steering.","marker":"[39]"},{"why":"Provides the actor-critic reinforcement-learning baseline adapted to the joint scheduling and beamforming action space.","marker":"[46]"},{"why":"Supplies the max-clique NP-hardness result used to establish hardness of the scheduling problem in Appendix A.","marker":"[13]"},{"why":"Supplies the performance-estimation theory used to prove linear convergence of the DC algorithm in Theorem 4.1.","marker":"[2]"}],"fun_headline_variants":["Greedy scheduling beats iterative beam steering in OTA-FL","Channel-aware greedy user scheduling wins in over-the-air FL","Low-complexity scheduling beats iterative methods for OTA-FL","Joint scheduling and beam steering: greedy is best for FL","Greedy channel-based scheduling outperforms iterative OTA-FL"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The policy recommendation rests on the assumption that the angle-and-modulus rules proved for idealized equal-norm and same-direction channel cases also govern the unequal-norm case that the greedy algorithm actually faces; the paper does not prove near-optimality of the sequential greedy selection in that general setting.","fun_headline_variants_meta":{"raw":{"variants":["Greedy scheduling beats iterative beam steering in OTA-FL","Channel-aware greedy user scheduling wins in over-the-air FL","Low-complexity scheduling beats iterative methods for OTA-FL","Joint scheduling and beam steering: greedy is best for FL","Greedy channel-based scheduling outperforms iterative OTA-FL"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000386,"raw_usage":{"total_tokens":2052,"prompt_tokens":974,"completion_tokens":1078,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":1005}},"tokens_in":590,"tokens_out":1078,"duration_ms":8660,"temperature":1.0,"reasoning_tokens":1005,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:12:54.458017+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to construct a channel set in which the strongest-norm user lies far in angle from the other strong users, run the greedy policy and an exhaustive search over all subsets of the required size, and compare the resulting mean-square aggregation error: if exhaustive search materially beats the greedy subset, the paper's central scheduling criterion is incomplete.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the optimal transmit coefficient formula and the reduction of the joint problem to the max-min beamforming objective in problem (9)."},{"cited_title":"Sidiropoulos, Paul Tseng, and Shuzhong Zhang","cited_arxiv_id":null,"evidence_quote":"Provides approximation bounds for semidefinite relaxation, motivating the DC-based rank-one reformulation for the beamforming subproblem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underpins the difference-of-convex treatment of the rank-one constraint used in Algorithm 1."},{"cited_title":"Lee Swindlehurst, and Daeyoung Park","cited_arxiv_id":null,"evidence_quote":"Supplies the projected subgradient baseline that the proposed methods are compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the iterative reweighted beamforming baseline used in the experiments."},{"cited_title":"Random Aggregate Beamforming for Over-the-Air Federated Learning in Large-Scale Networks","cited_arxiv_id":"2403.18946","evidence_quote":"Provides the random aggregate beamforming baseline whose performance motivates deterministic beam steering."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the actor-critic reinforcement-learning baseline adapted to the joint scheduling and beamforming action space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the max-clique NP-hardness result used to establish hardness of the scheduling problem in Appendix A."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the performance-estimation theory used to prove linear convergence of the DC algorithm in Theorem 4.1."}],"review_version":1}