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REVIEW 4 major objections 6 minor 29 references

A Multi-Objective Optimization framework for Decentralized Learning with coordination constraints

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.13983 v1 pith:WTJHW27G submitted 2025-07-18 math.OC

Roberto Morales , Umberto Biccari This is my paper
classification math.OC MSC 58E1790C1590C2390C2693A14
keywords Multi-objectiveoptimizationDecentralizedlearningParetooptimalityScalarizationFederatedConvergenceanalysisNon-IIDdata
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

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.

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 (4)
  1. [Algorithm 1, lines 3 and 10] 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.
  2. [Theorem 2.4, eq. (2.14)] 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.
  3. [Theorem 2.4, definition of D] 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.
  4. [Proposition 2.2(3)] 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.
minor comments (6)
  1. [Appendix A, eq. (A.2)] 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.
  2. [Theorem 2.3 proof, Step 3] 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)'.
  3. [Assumption (A2), eqs. (2.5)-(2.6) and (2.9)] 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.
  4. [Lemma A.1 proof] 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.
  5. [Table 6] Table 6, Agent 5, t=1 shows '0000' instead of '0.0000'.
  6. [Definition 2.1(3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence and Pareto-optimality results are self-contained or cited to independent standard references.

full rationale

The paper's two central claims are (i) convergence of Algorithm 1 to a minimizer of the scalarized objective F (Theorems 2.3 and 2.4) and (ii) Pareto optimality of solutions to the weighting scalarization (Propositions 2.1 and 2.2). The convergence proofs in Appendix A derive the bounds from Assumptions (A1)-(A3) using standard inequalities (L-smoothness, Young's inequality, variance bounds, and the drift estimate Lemma A.1); no equation is defined in terms of the target result, and no fitted parameter is later renamed as a prediction. The scalarization weight lambda and the learning rate eta are user-chosen algorithm parameters, not parameters calibrated to the reported test accuracies. Pareto optimality is quoted from the external monographs of Jahn and Miettinen, and it is not used as an input to the convergence argument. The self-referential or network citations in the introduction (e.g., references [17] and [28]) are contextual and non-load-bearing. The reviewer's noted discrepancy in Theorem 2.4 — the printed term Sigma*sqrt(T)/(2LM*tau) versus the value Sigma/(2LM*tau*sqrt(T)) obtained by substituting eta into the proof's final bound, and the telescoping reference to Theta^T rather than Theta* — concerns internal consistency and correctness of the theorem statement, not circularity. For the circularity question, the derivation chain is self-contained given the stated assumptions, so the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only hand-chosen quantity is the scalarization weight lambda. The assumptions (A1)-(A3) are standard in the local SGD literature and are not checked against the experimental data. No new entities are postulated.

free parameters (1)
  • lambda (scalarization weight) = user-selected, swept in [0, 0.88]
    Controls the balance between agent objectives and coordinator objective; all experiments and convergence bounds depend on it. It is not fitted to data but is a hand-chosen hyperparameter.
assumptions (4)
  • domain assumption Objectives Ci and Sj are convex and L-smooth (Assumption A1)
    Needed for the smoothness inequalities and the Pareto optimality claims (Prop 2.1-2.2). Not verified in the CNN experiments.
  • domain assumption Stochastic gradients are unbiased with bounded variance (Assumption A2, Eqs. 2.5-2.6)
    Used in Step 3 of Theorem 2.3 proof; in practice, the CNN training uses mini-batch SGD, but no variance bounds are measured.
  • domain assumption Bounded gradient heterogeneity (Assumption A3, Eq. 2.7)
    Needed for Lemma A.1 drift bound; not verified in the non-IID experiments where heterogeneity is high.
  • domain assumption Parameter space U is a compact convex Euclidean space and updates stay in U
    Algorithm 1 does not project, so U must be the entire space or the descent step must be adjusted; used for Weierstrass existence and Prop 2.1.

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Cite this review

Pith. "Pith review of A Multi-Objective Optimization framework for Decentralized Learning with coordination constraints." pith.science (2026). https://pith.science/paper/WTJHW27G

@misc{pith2026250713983,
  author       = {Pith},
  title        = {Pith review of: A Multi-Objective Optimization framework for Decentralized Learning with coordination constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTJHW27G}},
  note         = {Machine review of arXiv:2507.13983}
}
read the original abstract

This article introduces a generalized framework for Decentralized Learning formulated as a Multi-Objective Optimization problem, in which both distributed agents and a central coordinator contribute independent, potentially conflicting objectives over a shared model parameter space. Unlike traditional approaches that merge local losses under a common goal, our formulation explicitly incorporates coordinator-side criteria, enabling more flexible and structured training dynamics. To navigate the resulting trade-offs, we explore scalarization strategies, particularly weighted sums, to construct tractable surrogate problems. These yield solutions that are provably Pareto optimal under standard convexity and smoothness assumptions, while embedding global preferences directly into local updates. We propose a decentralized optimization algorithm with convergence guarantees, and demonstrate its empirical performance through simulations, highlighting the impact of the coordinator's influence on local agent behavior. The proposed approach offers a principled and customizable strategy for balancing personalization, fairness, and coordination in decentralized learning systems.

Figures

Figures reproduced from arXiv: 2507.13983 by the authors.

Figure 1
Figure 1. Evolution over 50 rounds of the agents’ training performances with λ = 0.87 and IID data distribution. Moreover, these classification performances can be improved by reducing the value of λ, so that to give more importance to the agents’ contribution. This phenomenon is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Evolution over 50 rounds of the agents’ training performances with λ = 0.87 and non-IID data distribution. in accordance with the data distribution, since the agents are all training over one fifth of the labels. Despite this behavior, the global model is capable of achieving 0.8056 accuracy and 0.7999 F1 score on the test dataset after 50 rounds. Finally, in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

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Works this paper leans on

29 extracted references · 21 canonical work pages

  1. [1]

    Agrachev and A

    A. Agrachev and A. Sarychev. Control on the manifolds of mappings as a setting for deep learning. arXiv preprint arXiv:2008.12702, 2020

    arXiv 2008

  2. [2]

    M. H. Alsharif, R. Kannadasan, W. Wei, K. S. Nisar, and A.-H. Abdel-Aty. A contemporary survey of recent advances in federated learning: Taxonomies, applications, and challenges. Internet of Things, 27:101251, 2024

    work page 2024

  3. [3]

    Bottou, F

    L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. SIAM Rev., 60(2):223--311, 2018

    work page 2018

  4. [4]

    S. P. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004

    2004

  5. [5]

    Floridi, J

    L. Floridi, J. Cowls, M. Beltrametti, R. Chatila, P. Chazerand, V. Dignum, C. Luetge, R. Madelin, U. Pagallo, F. Rossi, et al. Ai4people - an ethical framework for a good ai society: Opportunities, risks, principles, and recommendations. Minds Mach., 28(4):689--707, 2018

    work page 2018

  6. [6]

    Geshkovski and E

    B. Geshkovski and E. Zuazua. Turnpike in optimal control of pdes, resnets, and beyond. Acta Num., 31:135--263, 2022

    work page 2022

  7. [7]

    https://github.com/robertomoralesp/Decentralized-Learning.git, 2025

    Github:. https://github.com/robertomoralesp/Decentralized-Learning.git, 2025

    work page 2025

  8. [8]

    Jahn et al

    J. Jahn et al. Vector optimization. Springer, 2009

    work page 2009

Show all 29 references
  1. [9]

    Kairouz, H

    P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings, et al. Advances and open problems in federated learning. Found. Trends Mach. Learn., 14(1-- 2):1--210, 2021

    2021

  2. [10]

    S. P. Karimireddy, S. Kale, M. Mohri, S. J. Reddi, S. U. Stich, and A. T. Suresh. Scaffold: Stochastic controlled averaging for on-device federated learning. arXiv preprint arXiv:1910.06378, 2(6), 2019

    1910 arXiv

  3. [11]

    Khaled, K

    A. Khaled, K. Mishchenko, and P. Richt´ arik. Tighter theory for local SGD on identical and heterogeneous data. In International conference on artificial intelligence and statistics, pages 4519--4529. PMLR, 2020

    2020

  4. [12]

    Khorram, K

    E. Khorram, K. Khaledian, and M. Khaledyan. A numerical method for constructing the pareto front of multi-objective optimization problems. J. Comput. Appl. Math., 261:158--171, 2014

    2014

  5. [13]

    Kurilovas

    E. Kurilovas. Advanced machine learning approaches to personalise learning: learning analytics and decision making. Behav. Inf. Technol., 38(4):410--421, 2019

    2019

  6. [14]

    T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. Federated optimization in heterogeneous networks. Proc. Mach. Learn. Syst., 2:429--450, 2020

    2020

  7. [15]

    T. Li, M. Sanjabi, A. Beirami, and V. Smith. Fair resource allocation in federated learning. arXiv preprint arXiv:1905.10497, 2019

    1905 arXiv

  8. [16]

    B. Liu, N. Lv, Y. Guo, and Y. Li. Recent advances on federated learning: A systematic survey. Neurocomputing, page 128019, 2024

    2024

  9. [17]

    K. Liu, Z. Wang, and E. Zuazua. Game theory in federated learning: A potential game perspective. submitted, under review, 2024

    2024

  10. [18]

    P. Ma, T. Du, and W. Matusik. Efficient continuous pareto exploration in multi-task learning. In International Conference on Machine Learning, pages 6522--6531. PMLR, 2020

    2020

  11. [19]

    J. R. Marden, G. Arslan, and J. S. Shamma. Cooperative control and potential games. IEEE Trans. Syst. Man Cybern. B Cybern., 39(6):1393--1407, 2009

    2009

  12. [20]

    McMahan, E

    B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics, pages 1273--1282. PMLR, 2017

    2017

  13. [21]

    Miettinen

    K. Miettinen. Nonlinear multiobjective optimization, volume 12. Springer Science & Business Media, 1999

    1999

  14. [22]

    I. D. Raji and J. Buolamwini. Actionable auditing revisited: Investigating the impact of publicly naming biased performance results of commercial ai products. Commun. ACM, 66(1):101--108, 2022

    2022

  15. [23]

    L. J. Ratliff, S. A. Burden, and S. S. Sastry. Characterization and computation of local nash equilibria in continuous games. SIAM J. Optim., 26(1):646--672, 2016. 15

    2016

  16. [24]

    Reddi, Z

    S. Reddi, Z. Charles, M. Zaheer, Z. Garrett, K. Rush, J. Koneˇ cn` y, S. Kumar, and H. B. McMahan. Adaptive federated optimization. arXiv preprint arXiv:2003.00295, 2020

    2003 arXiv

  17. [25]

    J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, pages 520--534, 1965

    1965

  18. [26]

    S. U. Stich. Local SGD converges fast and communicates little. arXiv preprint arXiv:1805.09767, 2018

    2018 arXiv

  19. [27]

    A. Z. Tan, H. Yu, L. Cui, and Q. Yang. Towards personalized federated learning. IEEE Trans. Neural Netw. Learn. Syst., 34(12):9587--9603, 2022

    2022

  20. [28]

    Tr´ elat and E

    E. Tr´ elat and E. Zuazua. Turnpike in optimal control and beyond: a survey. arXiv preprint arXiv:2503.20342, 2025

    2025 arXiv

  21. [29]

    1 M MX i=1 D ∇Fi(Θt,k i ) − Gt,k i , Θ t,k − ΘT E F t,k−1 # = E

    Q. Yang, Y. Liu, T. Chen, and Y. Tong. Federated machine learning: Concept and applications. ACM Trans. Intell. Systems Technol., 10(2):1--19, 2019. Appendix A. Proof of the convergence results We present here the complete proofs for the two main theoretical result of the pres...

    2019

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