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arxiv: 2410.15335 · v2 · submitted 2024-10-20 · 📡 eess.SY · cs.SY

A Distributed Primal-Dual Method for Constrained Multi-agent Reinforcement Learning with General Parameterization

Ali Kahe , Hamed Kebriaei This is my paper

Pith reviewed 2026-05-23 19:04 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords constrained multi-agent reinforcement learningdistributed primal-dual algorithmactor-critic methodsLagrangian multipliersconsensusgeneral parameterizationCournot game
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The pith

Agents solve constrained multi-agent reinforcement learning problems using only local estimates of Lagrangian multipliers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a fully decentralized algorithm for cooperative constrained multi-agent reinforcement learning in which each agent maintains its own copies of both policy parameters and dual variables. Updates follow an actor-critic style that incorporates locally estimated multipliers, and the method is proved to drive these multiplier copies toward a common value while the joint process converges to an equilibrium. The equilibrium's performance gap relative to the best unparameterized policy is bounded, and the approach is demonstrated on a stochastic constrained Cournot game. A sympathetic reader would care because existing constrained MARL methods typically require centralized training or full state sharing, which limits use in networked systems where only local communication is feasible.

Core claim

The authors propose a distributed primal-dual actor-critic algorithm for constrained multi-agent RL with general parameterization. Each agent updates its local primal and dual variables from local information only. The algorithm establishes consensus among the agents' Lagrangian multiplier estimates and proves convergence to an equilibrium point whose sub-optimality relative to the exact solution of the unparameterized problem is characterized.

What carries the argument

Distributed primal-dual actor-critic updates that maintain and reach consensus on local Lagrangian multiplier estimates from neighbor information.

If this is right

  • Agents can perform online constrained learning with only local communication and no central coordinator.
  • The method applies to general function approximation rather than requiring linear or tabular representations.
  • Convergence holds to an equilibrium whose performance is characterized relative to the unparameterized optimum.
  • The algorithm can be evaluated on stochastic constrained Cournot games with dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local dual estimation pattern might extend to other networked constrained optimization problems outside reinforcement learning.
  • Tighter sub-optimality bounds could be obtained by choosing richer parameterizations.
  • Performance under time-varying networks or non-stationary dynamics is left open by the current analysis.

Load-bearing premise

Local multiplier estimates reach consensus under the chosen step-size schedules and network connectivity conditions.

What would settle it

A run of the algorithm on the constrained Cournot game in which the agents' multiplier estimates fail to converge to a common value or the achieved cost violates the shared constraint beyond the stated sub-optimality bound.

Figures

Figures reproduced from arXiv: 2410.15335 by Ali Kahe, Hamed Kebriaei.

Figure 1
Figure 1. Figure 1: Convergence of locally estimated Lagrangian multipliers during training. 1.60 1.52 1.44 1.36 1.28 1.20 Global Objective Cost J 0 2000 4000 6000 8000 10000 12000 step(t) 0.15 0.10 0.05 0.00 0.05 Global Constrained Cost Violation G b [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Global objective cost (J) and global constraint cost (Gˆ − b) during training. The plot illustrates how the algorithm seeks to decrease the objective cost while maintaining constraint violations near zero. potential of our method for scalable, decentralized solutions in real￾world applications, such as smart grids and autonomous systems. Future work will explore adaptations for more complex environments an… view at source ↗
read the original abstract

This paper proposes a novel distributed approach for solving a cooperative Constrained Multi-agent Reinforcement Learning (CMARL) problem, where agents seek to minimize a global objective function subject to shared constraints. Unlike existing methods that rely on centralized training or coordination, our approach enables fully decentralized online learning, with each agent maintaining local estimates of both primal and dual variables. Specifically, we develop a distributed primal-dual algorithm based on actor-critic methods, leveraging local information to estimate Lagrangian multipliers. We establish consensus among the Lagrangian multipliers across agents and prove the convergence of our algorithm to an equilibrium point, analyzing the sub-optimality of this equilibrium compared to the exact solution of the unparameterized problem. Furthermore, we introduce a constrained cooperative Cournot game with stochastic dynamics as a test environment to evaluate the algorithm's performance in complex, real-world scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes a distributed primal-dual actor-critic algorithm for cooperative constrained multi-agent reinforcement learning under general parameterization. Each agent maintains local estimates of primal (policy) and dual (Lagrangian multiplier) variables using only local information and neighbor communication. The central claims are that the local multiplier estimates reach consensus, the algorithm converges to an equilibrium point, and the equilibrium's sub-optimality relative to the unparameterized constrained optimum is bounded; a constrained cooperative Cournot game is introduced as a numerical testbed.

Significance. If the convergence and sub-optimality results hold under the stated conditions, the work provides a fully decentralized method for constrained MARL that avoids centralized training or coordination, which is relevant for large-scale networked systems. The explicit treatment of general parameterization and the introduction of a stochastic Cournot test environment are positive contributions; the analysis appears to rely on standard technical conditions (connected graph, step-size schedules, bounded gradients) rather than novel axioms.

major comments (2)
  1. [§4.3, Theorem 4.2] §4.3, Theorem 4.2: the sub-optimality bound is stated to hold with respect to the unparameterized optimum, yet the proof sketch separates the parameterization bias from the distributed estimation error only at a high level; the dependence of the constant on the actor-critic approximation error (via the general parameterization) is not made explicit enough to verify the claimed rate.
  2. [Assumption 3.2 and §5.1] Assumption 3.2 and §5.1: the bounded-gradient and Lipschitz conditions required for consensus of the dual variables are listed, but it is unclear whether they remain uniform when the policy parameterization is nonlinear and the constraint functions depend on the joint state-action distribution; a counter-example or explicit verification for the Cournot game would strengthen the claim.
minor comments (3)
  1. [§3] Notation for the local multiplier estimates λ_i and the consensus variable is introduced in §3 but used interchangeably in the convergence statements of §4; a single consistent symbol or explicit equivalence would improve readability.
  2. [Figure 2] Figure 2 (Cournot game trajectories) lacks error bars or multiple random seeds; reporting mean and standard deviation over 10 independent runs would make the empirical comparison clearer.
  3. [Assumption 3.3] The step-size schedule η_t = 1/t^α is stated without specifying the admissible range of α that simultaneously guarantees consensus and convergence; adding this interval explicitly after Assumption 3.3 would help readers replicate the experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the minor revision recommendation. We address each major comment below.

read point-by-point responses

  1. Referee: [§4.3, Theorem 4.2] §4.3, Theorem 4.2: the sub-optimality bound is stated to hold with respect to the unparameterized optimum, yet the proof sketch separates the parameterization bias from the distributed estimation error only at a high level; the dependence of the constant on the actor-critic approximation error (via the general parameterization) is not made explicit enough to verify the claimed rate.

    Authors: We agree the dependence on the approximation error should be stated more explicitly for verification. The proof sketch in Theorem 4.2 already separates the parameterization bias from the distributed error at a high level, but we will revise the discussion immediately following the theorem to explicitly derive and display how the sub-optimality constant depends on the actor-critic error bound. revision: yes

  2. Referee: [Assumption 3.2 and §5.1] Assumption 3.2 and §5.1: the bounded-gradient and Lipschitz conditions required for consensus of the dual variables are listed, but it is unclear whether they remain uniform when the policy parameterization is nonlinear and the constraint functions depend on the joint state-action distribution; a counter-example or explicit verification for the Cournot game would strengthen the claim.

    Authors: Assumption 3.2 is stated to hold uniformly over the considered parameterization class. For the Cournot testbed the constraint functions are linear in actions and the compact action spaces together with the continuous parameterization ensure the gradient and Lipschitz bounds remain uniform. To strengthen the claim we will insert a short verification paragraph in §5.1 confirming that the conditions of Assumption 3.2 hold for this environment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proposes a distributed primal-dual actor-critic algorithm for constrained MARL, establishes consensus on multipliers, proves convergence to an equilibrium, and bounds sub-optimality relative to the unparameterized optimum. These are standard convergence results resting on technical conditions (connected graph, step-size schedules, bounded gradients, parameterization properties) that are external to the target claims and not defined in terms of the equilibrium or bounds themselves. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described claims. The central results are independent of the algorithm's own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Mastering the game of go with deep neural networks and tree search,

    D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V . Panneershelvam, M. Lanctot et al., “Mastering the game of go with deep neural networks and tree search,” Nature, vol. 529, no. 7587, pp. 484–489, 2016

    work page 2016

  2. [2]

    Reinforcement learning in robotics: A survey,

    J. Kober, J. A. Bagnell, and J. Peters, “Reinforcement learning in robotics: A survey,” The International Journal of Robotics Research , vol. 32, no. 11, pp. 1238–1274, 2013

    work page 2013

  3. [3]

    Multi-agent deep reinforcement learning for large-scale traffic signal control,

    T. Chu, J. Wang, L. Codec `a, and Z. Li, “Multi-agent deep reinforcement learning for large-scale traffic signal control,” IEEE Transactions on Intelligent Transportation Systems, vol. 21, no. 3, pp. 1086–1095, 2020

    work page 2020

  4. [4]

    Multi-agent reinforcement learning approach for hedging portfolio problem,

    U. Pham, Q. Luu, and H. Tran, “Multi-agent reinforcement learning approach for hedging portfolio problem,” Soft Computing , vol. 25, no. 12, pp. 7877–7885, 2021

    work page 2021

  5. [5]

    Smart grid for industry using multi-agent reinforcement learning,

    M. Roesch, C. Linder, R. Zimmermann, A. Rudolf, A. Hohmann, and G. Reinhart, “Smart grid for industry using multi-agent reinforcement learning,” Applied Sciences, vol. 10, no. 19, p. 6900, 2020

    work page 2020

  6. [6]

    Multi-agent reinforcement learning: A selective overview of theories and algorithms,

    K. Zhang, Z. Yang, and T. Bas ¸ar, “Multi-agent reinforcement learning: A selective overview of theories and algorithms,” Handbook of Rein- forcement Learning and Control , 2021

    work page 2021

  7. [7]

    Deep reinforcement learning for multiagent systems: A review of challenges, solutions, and applications,

    T. T. Nguyen, N. D. Nguyen, and S. Nahavandi, “Deep reinforcement learning for multiagent systems: A review of challenges, solutions, and applications,”IEEE transactions on cybernetics, vol. 50, no. 9, pp. 3826– 3839, 2020

    work page 2020

  8. [8]

    A review of cooperative multi-agent deep reinforcement learning,

    A. Oroojlooy and D. Hajinezhad, “A review of cooperative multi-agent deep reinforcement learning,” Applied Intelligence, vol. 53, no. 11, pp. 13 677–13 722, 2023

    work page 2023

  9. [9]

    Cooperative multiagent reinforcement learning with partial observations,

    Y . Zhang and M. M. Zavlanos, “Cooperative multiagent reinforcement learning with partial observations,” IEEE Transactions on Automatic Control, vol. 69, no. 2, pp. 968–981, 2023

    work page 2023

  10. [10]

    Multiagent fully decentralized value function learning with linear convergence rates,

    L. Cassano, K. Yuan, and A. H. Sayed, “Multiagent fully decentralized value function learning with linear convergence rates,” IEEE Transac- tions on Automatic Control , vol. 66, no. 4, pp. 1497–1512, 2020

    work page 2020

  11. [11]

    Fully decentralized multi-agent reinforcement learning with networked agents,

    K. Zhang, Z. Yang, H. Liu, T. Zhang, and T. Basar, “Fully decentralized multi-agent reinforcement learning with networked agents,” in Proceed- ings of the 35th International Conference on Machine Learning (ICML) , vol. 80. PMLR, 2018, pp. 5872–5881

    work page 2018

  12. [12]

    Decentralized learning for optimality in stochastic dynamic teams and games with local control and global state information,

    B. Yongacoglu, G. Arslan, and S. Y ¨uksel, “Decentralized learning for optimality in stochastic dynamic teams and games with local control and global state information,” IEEE Transactions on Automatic Control, vol. 67, no. 10, pp. 5230–5245, 2021

    work page 2021

  13. [13]

    Multi-agent safe policy learning for power management of networked microgrids,

    Q. Zhang, K. Dehghanpour, Z. Wang, F. Qiu, and D. Zhao, “Multi-agent safe policy learning for power management of networked microgrids,” IEEE Transactions on Smart Grid , vol. 12, no. 2, pp. 1048–1062, 2021

    work page 2021

  14. [14]

    A robust and constrained multi-agent reinforcement learning electric vehicle rebalancing method in amod systems,

    S. He, Y . Wang, S. Han, S. Zou, and F. Miao, “A robust and constrained multi-agent reinforcement learning electric vehicle rebalancing method in amod systems,” in 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) , 2023, pp. 5637–5644

    work page 2023

  15. [15]

    Constrained markov games: Nash equilib- ria,

    E. Altman and A. Shwartz, “Constrained markov games: Nash equilib- ria,” in Advances in Dynamic Games and Applications . Springer, 2000, pp. 213–221

    work page 2000

  16. [16]

    Con- strained reinforcement learning has zero duality gap,

    S. Paternain, L. Chamon, M. Calvo-Fullana, and A. Ribeiro, “Con- strained reinforcement learning has zero duality gap,” in Advances in Neural Information Processing Systems (NeurIPS) , vol. 32, 2019

    work page 2019

  17. [17]

    Safe policies for reinforcement learning via primal-dual methods,

    S. Paternain, M. Calvo-Fullana, L. F. Chamon, and A. Ribeiro, “Safe policies for reinforcement learning via primal-dual methods,” IEEE Transactions on Automatic Control, vol. 68, no. 3, pp. 1321–1336, 2022

    work page 2022

  18. [18]

    On the hardness of constrained cooperative multi-agent reinforcement learning,

    Z. Chen, Y . Zhou, and H. Huang, “On the hardness of constrained cooperative multi-agent reinforcement learning,” in Proceedings of the 12th International Conference on Learning Representations (ICLR) , 2024

    work page 2024

  19. [19]

    An online actor-critic algorithm with function approximation for constrained markov decision processes,

    S. Bhatnagar and K. Lakshmanan, “An online actor-critic algorithm with function approximation for constrained markov decision processes,” Journal of Optimization Theory and Applications, vol. 153, pp. 688–708, 2012

    work page 2012

  20. [20]

    Actor- critic algorithms for constrained multi-agent reinforcement learning,

    R. B. Diddigi, D. S. K. Reddy, P. KJ, and S. Bhatnagar, “Actor- critic algorithms for constrained multi-agent reinforcement learning,” in Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems , 2019, pp. 1931–1933

    work page 2019

  21. [21]

    Decentralized policy gradient descent ascent for safe multi-agent reinforcement learning,

    S. Lu, K. Zhang, T. Chen, T. Bas ¸ar, and L. Horesh, “Decentralized policy gradient descent ascent for safe multi-agent reinforcement learning,” in Proceedings of the AAAI Conference on Artificial Intelligence , vol. 35, no. 10, 2021, pp. 8767–8775

    work page 2021

  22. [22]

    Multi-agent first order constrained optimization in policy space,

    Y . Zhao, Y . Yang, Z. Lu, W. Zhou, and H. Li, “Multi-agent first order constrained optimization in policy space,” in Advances in Neural Information Processing Systems (NeurIPS) , vol. 36, 2023, pp. 39 189– 39 211

    work page 2023

  23. [23]

    Depaint: A decentralized safe multi-agent reinforcement learning algorithm con- sidering peak and average constraints,

    R. Hassan, K. S. Wadith, M. M. Rashid, and M. M. Khan, “Depaint: A decentralized safe multi-agent reinforcement learning algorithm con- sidering peak and average constraints,” Applied Intelligence , pp. 1–17, 2024

    work page 2024

  24. [24]

    Decom: Decomposed policy for constrained cooperative multi-agent reinforcement learning,

    Z. Yang, H. Jin, R. Ding, H. You, G. Fan, X. Wang, and C. Zhou, “Decom: Decomposed policy for constrained cooperative multi-agent reinforcement learning,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 9, 2023, pp. 10 861–10 870

    work page 2023

  25. [25]

    Scalable primal- dual actor-critic method for safe multi-agent reinforcement learning with general utilities,

    D. Ying, Y . Zhang, Y . Ding, A. Koppel, and J. Lavaei, “Scalable primal- dual actor-critic method for safe multi-agent reinforcement learning with general utilities,” Advances in Neural Information Processing Systems (NeurIPS), vol. 36, 2024

    work page 2024

  26. [26]

    Distributed constrained op- timization by consensus-based primal-dual perturbation method,

    T.-H. Chang, A. Nedi ´c, and A. Scaglione, “Distributed constrained op- timization by consensus-based primal-dual perturbation method,” IEEE Transactions on Automatic Control, vol. 59, no. 6, pp. 1524–1538, 2014

    work page 2014

  27. [27]

    Distributed subgradient methods for multi- agent optimization,

    A. Nedic and A. Ozdaglar, “Distributed subgradient methods for multi- agent optimization,” IEEE Transactions on Automatic Control , vol. 54, no. 1, pp. 48–61, 2009

    work page 2009

  28. [28]

    Reinforcement learning: An introduction,

    R. S. Sutton, “Reinforcement learning: An introduction,” 2018

    work page 2018

  29. [29]

    H. J. Kushner and D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems . Springer, 1978

    work page 1978

  30. [30]

    On-policy deep reinforcement learning for the average-reward criterion,

    Y . Zhang and K. W. Ross, “On-policy deep reinforcement learning for the average-reward criterion,” in Proceedings of the 38th International Conference on Machine Learning (ICML) , vol. 139. PMLR, 2021, pp. 12 535–12 545

    work page 2021