{"id":"5abf8044-a299-45de-be7c-19153298ffc5","arxiv_id":"2507.13983","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A weighted-sum scalarization of agent and coordinator objectives yields a FedAvg-like decentralized algorithm with O(1/sqrt(T)) convergence under convexity and bounded heterogeneity.","lead":"The paper frames decentralized learning as a multi-objective problem where agents and a central coordinator each bring separate objectives, then solves it by merging everything into one weighted sum. It provides convergence guarantees for the resulting averaging algorithm and tests how the weighting parameter changes MNIST accuracy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.4 as printed is not a valid convergence guarantee: the proof gives a 1/sqrt(T) noise term, while (2.14) prints sqrt(T), and the telescoping distance is to the output Theta^T, not to Theta*. Read literally, the bound does not vanish and does not follow from the derivation.","rationale":"The paper's intended contribution is a convergence rate for a decentralized, scalarized multi-objective algorithm. The proof architecture is recognizable and mostly salvageable: Theorem 2.3 is a one-round descent inequality, Lemma A.1 bounds agent drift under the heterogeneity assumption (A3), and Theorem 2.4 sums these bounds over rounds. The reader's identified weakest assumption, (A3), is standard in federated optimization and does not threaten the argument: even if zeta is large, the zeta-dependent term is 5*zeta^2/(8*L*T), which vanishes as T grows. The more serious problem is that the central theorem is mis-stated. The proof's algebra yields a vanishing Sigma/(2*L*M*tau*sqrt(T)) term, while the printed statement has Sigma*sqrt(T)/(2*L*M*tau), which grows without bound. Separately, the proof's telescoping sum uses the squared distance to the final output Theta^T, but the theorem defines D as the distance to an optimizer Theta*. These two reference points are not interchangeable, and the displayed bound as written does not follow from the derivation. This is a concrete, checkable inconsistency in the strongest claim of the paper, not merely a matter of presentation. The pseudocode inconsistency flagged by the reader (Algorithm 1, line 3 resetting every round to Theta^0, and line 10 omitting the factor 1/M in the aggregation) compounds the problem, because the proof relies on the clock property ThetaBar^{t,tau} = ThetaBar^{t+1,0}, which requires each new round to start from the averaged global model. If the experiments used the algorithm as written rather than the intended FedAvg-style update, the experiments may not instantiate the analyzed method. These issues are all fixable, and they do not establish that the framework is wrong; the intended bound can likely be repaired by correcting the typo and choosing one consistent reference point. I therefore agree with the reader's CONDITIONAL verdict rather than moving to accept or reject.","tokens_in":20756,"tokens_out":11939,"duration_ms":155200,"concrete_test":"Analytically re-derive the final line of the proof of Theorem 2.4: substitute (2.13) into 1/(2*eta*tau*T)*||ThetaBar^{0,0} - Theta^T||^2 + 2*Sigma*eta/M + 10*L*tau^2*eta^2*zeta^2 + 4*L*tau*eta^2*Sigma and compare the result with (2.14). If the second term is Sigma/(2*L*M*tau*sqrt(T)) rather than Sigma*sqrt(T)/(2*L*M*tau), or if the telescoping distance is ||Theta^{0,0} - Theta^T|| rather than D^2 = ||Theta^{0,0} - Theta*||^2, then Theorem 2.4 as stated is incorrect and must be revised before the convergence claim can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence result, Theorem 2.4, is internally inconsistent. Substituting eta = 1/(4L*tau*sqrt(T)) into the proof's final bound gives D^2/(2*eta*tau*T) + 2*Sigma*eta/M + 10*L*tau^2*eta^2*zeta^2 + 4*L*tau*eta^2*Sigma = 2*L*D^2/sqrt(T) + Sigma/(2*L*M*tau*sqrt(T)) + 5*zeta^2/(8*L*T) + Sigma/(4*L*tau*T). But (2.14) displays Sigma*sqrt(T)/(2*L*M*tau) as the second term, which grows with T and makes the bound vacuous as a convergence statement. Moreover, the proof telescopes the quantity ||ThetaBar^{0,0} - Theta^T||^2, whereas the theorem defines D := ||Theta^{0,0} - Theta*||. Unless Theta^T = Theta*, these are different; the displayed bound therefore does not follow from the proof. A one-line correction may repair the rate, but the theorem needs restating with a consistent reference point. This is more load-bearing than Assumption (A3): bounded heterogeneity is standard, and the zeta^2/T term vanishes even for large zeta, so A3 affects constants rather than the validity of the convergence claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a decentralized learning framework in which M agents minimize local empirical risks while a central coordinator contributes N additional objectives, formalized as the multi-objective problem (2.1). The authors scalarize this problem with a weighted sum controlled by a single parameter λ, rewrite the surrogate as min (1−λ)/M Σ_i F_i with F_i = C_i + α Σ_j S_j, and propose a FedAvg-like Algorithm 1. The main theoretical results are Theorem 2.3, a per-round convergence estimate under smoothness, convexity and bounded stochastic gradients, and Theorem 2.4, an O(1/√T) bound on the average gap to the scalarized objective under an additional bounded-heterogeneity condition. The paper closes with MNIST experiments in IID and non-IID settings and a discussion of open problems.","tokens_in":21089,"tokens_out":13666,"duration_ms":144998,"significance":"If the technical issues are repaired, the paper offers a clean formalization of coordinator-side objectives in decentralized optimization and correctly identifies that weighted-sum scalarization transfers standard local-SGD analyses to this setting. The convergence proof is self-contained, the bound is the expected stochastic first-order rate, and the link to reproducible code is a useful addition. The Pareto-optimality discussion is standard material, and the framework's distinctiveness is mainly the active role of a coordinator objective embedded in local updates; the experiments illustrate but do not certify the theory (they do not verify Assumption (A3)). The contribution is incremental but publishable after the load-bearing corrections.","major_comments":[{"comment":"Algorithm 1, lines 3 and 10, is not the recursion analyzed in the proofs. Line 3 sets Θ_i^{t−1,0} ← Θ0 at the start of every outer iteration, and line 10 sets Θ_t = Σ_{i=1}^M Θ_i^{t−1,τ} without the factor 1/M. The proofs define the averaged iterate Θbar^{t,k} in (2.11) and, in the proof of Theorem 2.4, use the identity Θbar^{t,τ} = Θbar^{t+1,0}. With the pseudocode as printed, all outer rounds start from the same Θ0 and the aggregate is never averaged, so the telescoping sum over t in the proof of Theorem 2.4 is not implemented. Please modify the pseudocode so that round t+1 starts from the averaged output of round t (or define the algorithm accordingly) and output Θ_T = (1/M) Σ_i Θ_i^{T−1,τ}, matching Θ^T in the theorems.","section":"Algorithm 1, lines 3 and 10"},{"comment":"The right-hand side of (2.14) as printed has Σ√T/(2LMτ) as its second term, which grows with T and makes the bound vacuous as a convergence statement. The proof of Theorem 2.4 ends with D²/(2ητT) + 2Ση/M + 10Lτ²η²ζ² + 4Lτη²Σ; substituting η = 1/(4Lτ√T) gives Σ/(2LMτ√T), a decaying term. The printed formula therefore does not follow from the derivation and should be corrected.","section":"Theorem 2.4, eq. (2.14)"},{"comment":"Theorem 2.4 sets D := ∥Θ^{0,0} − Θ*∥, but the proof telescopes the quantities ∥Θbar^{t,0} − Θ^T∥² and the final bound contains ∥Θ^{0,0} − Θ^T∥²/(2ητT). Replacing the latter by D²/(2ητT) requires Θ^T = Θ*, which is not established anywhere. The theorem needs to be restated with a consistent reference point: either bound the gap to F(Θ*) with an explicit control of F(Θ^T) − F(Θ*), or define D with respect to the output Θ^T.","section":"Theorem 2.4, definition of D"},{"comment":"Proposition 2.2(3) is not correct as written. The clause 'there exists Θ∗ is also a solution of (2.2)' is grammatically incomplete, and the claimed converse cannot hold for the one-parameter scalarization (2.3): the weight vector has all agent weights equal and all coordinator weights equal, whereas a general convex multi-objective problem may have Pareto-optimal points that require unequal weights. This statement should be replaced by the standard weighted-sum characterization with unrestricted nonnegative weights, or qualified to the equal-weight subfamily.","section":"Proposition 2.2(3)"}],"minor_comments":[{"comment":"Eq. (A.2) writes √(2η)/M ΣG where the surrounding algebra requires √2 η/M ΣG, since the squared norm is used as 2η²/M²∥ΣG∥²; please correct the typo.","section":"Appendix A, eq. (A.2)"},{"comment":"In Step 3 of the proof of Theorem 2.3, the term involving ∇F_i(Θ^{t,k}_i) − G^{t,k}_i should carry the index k−1 in view of (A.6), and the reference to 'Assumption (A3)' for condition (2.5) should be to 'Assumption (A2)'.","section":"Theorem 2.3 proof, Step 3"},{"comment":"The stochastic gradient h is described as a stochastic gradient of Σ_j S_j, but (2.9) contains α²N²σ_S², which suggests h is meant to be N separate stochastic gradients; the notation and variance bound should be aligned.","section":"Assumption (A2), eqs. (2.5)-(2.6) and (2.9)"},{"comment":"In the proof of Lemma A.1, 'i ∈ JM − 1K' should be stated more precisely, and the final display '10τ²η²τ ζ²' contains an extra τ; the intended constant is 10τ²η²ζ².","section":"Lemma A.1 proof"},{"comment":"Table 6, Agent 5, t=1 shows '0000' instead of '0.0000'.","section":"Table 6"},{"comment":"Definition 2.1(3) as written defines local weak Pareto optimality rather than global weak Pareto optimality; consider renaming the notion or removing the neighborhood δ from the weak Pareto definition.","section":"Definition 2.1(3)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript is within the journal's scope and the core idea is sound, but the main convergence theorem and the algorithm pseudocode must be repaired before the claims can be trusted. The issues are local and repairable, so I do not recommend rejection. I also note that the contribution is incremental relative to the local-SGD and MOO scalarization literature; the authors should moderate the novelty claims in the abstract, since 'provably Pareto optimal' applies only to the equal-weight scalarized solutions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the framing is sensible and the paper is honest about its own limits, but the central convergence theorem and the algorithm pseudocode are inconsistent with the proof as printed. The defects look fixable, but they are load-bearing, so I would not rely on the current version.\n\nWhat is actually new: the coordinator's objective enters as an explicit separate term in the scalarized problem, rather than being folded into the average loss. That is a modest extension of FedAvg—the update is exactly FedAvg with an added global regularizer—but the MOO framing and the Pareto-optimality discussion are clean and could be useful for people who want a principled way to talk about coordinator constraints. Credit where due: the paper ships code, runs both IID and non-IID MNIST experiments, and explicitly flags the lambda-to-1 blow-up in Remark 2.1 and the open problems in Section 4. That is more honest than much of what crosses my desk.\n\nThe soft spots, in order of severity. First, Algorithm 1 does not match the proof. Line 3 sets Theta^{t-1,0}_i = Theta^0, so every round restarts from the initial guess; line 10 sets Theta^t = sum_i Theta^{t-1,tau}_i without dividing by M. The proof analyzes averaged iterates with Theta^{t,0} equal to the previous round's average. The printed algorithm is not the analyzed algorithm. This must be fixed, and it is not a one-character typo—it changes what the experiments actually ran.\n\nSecond, Theorem 2.4 as printed cannot be true. Substituting eta = 1/(4L*tau*sqrt(T)) into the proof's final bound gives the second term Sigma/(2LM*tau*sqrt(T)), but (2.14) displays Sigma*sqrt(T)/(2LM*tau), which grows with T. That is an algebraic typo, but it makes the claimed convergence vacuous.\n\nThird, and more structural: the theorem defines D := ||Theta^{0,0}-Theta*||, but the proof telescopes ||Theta^{0,0}-Theta^T||. The displayed bound therefore does not follow from the derivation. The theorem needs restating with a consistent reference point, and the authors should say plainly what the bound means—as written it says the average iterate catches up to the output, not that the output reaches an optimum.\n\nMinor: the experiments have no baselines or repeated runs, so they are illustrative only. Assumption (A3) is standard and only affects constants; the reader's worry about zeta is not the real issue.\n\nWho this is for: people working on fairness and personalization in federated learning who want a formal MOO framing. It is not new optimization machinery. I would send it to review because the idea is coherent and the defects are identifiable and repairable, but the current version needs major revision. I would not cite it until the theorem and pseudocode are repaired.","headline":"A sensible MOO framing of coordinated decentralized learning, undermined by an algorithm–proof mismatch and a convergence theorem that does not follow from its own derivation.","tokens_in":21584,"tokens_out":4210,"would_cite":false,"duration_ms":47521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58E17","90C15","90C23","90C26","93A14"],"pacs":[],"model":"deepseek-v4-flash","headline":"Decentralized learning with a central coordinator can be recast as a multi-objective optimization problem, and the paper's scalarized algorithm provably converges to Pareto-optimal solutions.","keywords":["Multi-objective optimization","Decentralized learning","Pareto optimality","Scalarization","Federated learning","Convergence analysis","Non-IID data"],"falsifier":"Run the non-IID MNIST setup of Section 3.2 and measure zeta = max_i sup_Theta ||nabla F_i(Theta) - nabla F(Theta)|| over the actual training trajectory (for instance by evaluating all five agents at the averaged iterates). If the empirical drift term 10 $tau^{2}$ $eta^{2}$ $zeta^{2}$ exceeds the observed gap between the averaged objective and its value at the final iterate, then Theorem 2.4 does not explain the convergence; likewise, if a point that strictly improves every C_i and S_1 is found within the training envelope, Proposition 2.2's guarantee is violated.","tokens_in":20572,"feed_emoji":"⚖️","tokens_out":4575,"duration_ms":45575,"temperature":0.7,"pith_summary":"The paper claims that decentralized learning with a central coordinator can be recast as a multi-objective optimization problem, where each agent minimizes its own empirical loss and the coordinator contributes structural criteria such as regularization or fairness. Scalarizing these competing objectives with a weight lambda yields a surrogate problem whose solutions are provably Pareto optimal under convexity, and whose local form embeds the coordinator's preferences directly into each agent's gradient. The authors propose a decentralized algorithm that runs tau local stochastic gradient steps per round and averages the results, and they prove a convergence rate of O(1/sqrt(T)) plus a heterogeneity-dependent drift term. Experiments on MNIST show that increasing the coordinator's weight harms accuracy when data is IID but helps when data is non-IID, because it counteracts agent drift. If correct, the framework offers a principled dial for trading local personalization against global coordination.","feed_headline":"Decentralized learning with a coordinator converges to Pareto optima","feed_subtitle":"Scalarizing agent and coordinator losses yields an O(1/√T) bound and a tunable fairness knob.","key_machinery":"The load-bearing mechanism is the scalarized local objective F_i(Theta) = C_i(Theta) + $\\alpha$ sum_{j=1}^N S_j(Theta), with $\\alpha$ = $\\lambda$/((1-$\\lambda$)N). Each agent performs tau local stochastic gradient steps on F_i and then the parameters are averaged across agents, mimicking FedAvg but with the coordinator's criteria built into every local gradient. The factor $\\alpha$ encodes the coordinator's influence: $\\lambda$ = 0 removes it, $\\lambda$ approaching 1 makes it dominate. The convergence proof rests on Lemma A.1, a drift bound showing that the expected divergence between a local iterate and the averaged parameter is controlled by the heterogeneity constant zeta and the noise variance Sigma, and on Theorem 2.3's per-round descent estimate, which telescopes over rounds once the drift term is bounded.","core_discovery":"The central discovery is that the weighted-sum scalarization min_{Theta} [(1-$\\lambda$)/M sum_i C_i(Theta) + $\\lambda$/N sum_j S_j(Theta)] can be rewritten as min_{Theta} (1/M) sum_i F_i(Theta) with F_i(Theta) = C_i(Theta) + $\\alpha$ sum_j S_j(Theta) and $\\alpha$ = $\\lambda$/((1-$\\lambda$)N). In this form the coordinator's objectives are folded into every agent's local objective, so decentralized averaging of local SGD steps solves a scalarized multi-objective problem rather than a single common-loss problem. Under convexity and L-smoothness of all objectives plus a bounded-heterogeneity assumption (A3), Theorem 2.4 guarantees that, with learning rate eta = 1/(4L tau $\\sqrt$(T)), the averaged iterates satisfy E[(1/(tau T)) sum_{t,k}(F(bar{Theta}^{t,k}) - F(Theta^T))] <= $2D^{2}$ L/$\\sqrt$(T) + Sigma/(2LM tau) $\\sqrt$(T) + 5 $zeta^{2}$/(8LT) + Sigma/(4L tau T). By Proposition 2.2, solutions of the scalarized problem are weakly Pareto optimal, and Pareto optimal under uniqueness or convexity. The algorithm therefore provably converges to a point on the Pareto front of the original multi-agent, coordinator-augmented problem.","pith_inferences":["A direct corollary the paper does not spell out: the same scalarization should extend to non-convex objectives in the sense of convergence to stationary points of F, since Theorem 2.3's proof only uses smoothness and unbiased gradients; the Pareto optimality claim, however, would no longer hold.","The framework suggests a principled way to choose lambda from data: monitor the gradient heterogeneity zeta along training and set lambda just large enough to keep the drift term 10 tau^2 eta^2 zeta^2 below the desired accuracy gap, which would make the convergence bound non-vacuous in practice.","The coordinator objectives need not be fixed; one could test adaptive S_j that tighten as training proceeds, e.g., increasing the weight on a fairness criterion once per-agent accuracies stabilize, which the proof would permit as long as smoothness and convexity hold at each stage.","Because the algorithm is exactly FedAvg when lambda = 0, the paper's convergence result can be read as a parameterized family of local-SGD guarantees where the coordinator's presence is folded into the heterogeneity constant, unifying personalization and federation in one rate."],"forward_implications":["Any convex decentralized problem with a coordinator criterion admits a distributed algorithm whose iterates converge to a Pareto-optimal point of the full multi-objective problem, at the same O(1/sqrt(T)) rate as standard centralized SGD.","The scalarization weight lambda becomes a tunable fairness knob: pushing lambda up biases the solution toward the coordinator's criteria, and the theory quantifies the cost in extra iterations through the blow-up of L and Sigma as lambda approaches 1.","The drift lemma suggests that the number of local steps tau can be chosen to balance communication cost against heterogeneity: fewer local steps shrink the zeta^2 tau^2 term, which matters when agents hold very different data.","The non-IID experiments indicate that coordinator-side objectives act as a regularizer against agent overfitting, reversing the usual accuracy decline when data becomes heterogeneous."],"supporting_citations":[{"why":"McMahan et al.'s FedAvg: the averaging scheme Algorithm 1 is modeled on, and the baseline the framework generalizes.","marker":"[20]"},{"why":"Miettinen: source of Proposition 2.2 on scalarization yielding (weakly) Pareto optimal solutions.","marker":"[21]"},{"why":"Jahn: source of Proposition 2.1 and the non-uniqueness example, grounding Pareto optimality facts.","marker":"[8]"},{"why":"Karimireddy et al., SCAFFOLD: proof techniques for drift control used in Lemma A.1.","marker":"[10]"},{"why":"Khaled et al.: tighter local-SGD theory on identical and heterogeneous data, the template for Theorem 2.4.","marker":"[11]"},{"why":"Stich: local SGD convergence rate that the O(1/sqrt(T)) bound follows, referenced as the insight source.","marker":"[26]"}],"fun_headline_variants":["Coordinator-folded objectives achieve Pareto-optimal decentralized learning","Folding coordinator objectives into local updates yields Pareto-optimal learning","Decentralized learning with coordinator objectives lands on Pareto front","Scalarized multi-objective decentralized learning converges to Pareto optimum","Coordinator-aware decentralized learning reaches Pareto-optimal trade-offs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence bound relies on Assumption (A3), which says the local objectives F_i are uniformly close to their average F in gradient: max_i sup_Theta ||nabla F_i(Theta) - nabla F(Theta)|| <= zeta. If agents' data are so different that this heterogeneity constant is large, the drift term 5 $zeta^{2}$/(8LT) dominates and the O(1/$\\sqrt$(T)) guarantee is vacuous, yet the paper's non-IID experiment satisfies this assumption only by asserting it rather than measuring it.","fun_headline_variants_meta":{"raw":{"variants":["Coordinator-folded objectives achieve Pareto-optimal decentralized learning","Folding coordinator objectives into local updates yields Pareto-optimal learning","Decentralized learning with coordinator objectives lands on Pareto front","Scalarized multi-objective decentralized learning converges to Pareto optimum","Coordinator-aware decentralized learning reaches Pareto-optimal trade-offs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001123,"raw_usage":{"total_tokens":4691,"prompt_tokens":987,"completion_tokens":3704,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":3621}},"tokens_in":603,"tokens_out":3704,"duration_ms":27879,"temperature":1.0,"reasoning_tokens":3621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:12:00.272129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the non-IID MNIST setup of Section 3.2 and measure zeta = max_i sup_Theta ||nabla F_i(Theta) - nabla F(Theta)|| over the actual training trajectory (for instance by evaluating all five agents at the averaged iterates). If the empirical drift term 10 $tau^{2}$ $eta^{2}$ $zeta^{2}$ exceeds the observed gap between the averaged objective and its value at the final iterate, then Theorem 2.4 does not explain the convergence; likewise, if a point that strictly improves every C_i and S_1 is found within the training envelope, Proposition 2.2's guarantee is violated.","supporting_citations":[{"cited_title":"Miettinen","cited_arxiv_id":null,"evidence_quote":"Miettinen: source of Proposition 2.2 on scalarization yielding (weakly) Pareto optimal solutions."},{"cited_title":"Jahn et al","cited_arxiv_id":null,"evidence_quote":"Jahn: source of Proposition 2.1 and the non-uniqueness example, grounding Pareto optimality facts."},{"cited_title":"Khaled, K","cited_arxiv_id":null,"evidence_quote":"Khaled et al.: tighter local-SGD theory on identical and heterogeneous data, the template for Theorem 2.4."}],"review_version":1}