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Learning to Act and Cooperate for Distributed Black-Box Consensus Optimization
Zi-Bo Qin, Feng-Feng Wei, Tai-You Chen, Wei-Neng Chen
TL;DR
Distributed black-box consensus optimization must coordinate agents using local queries and neighbor communication, but existing methods provide limited trajectory-driven adaptation. LAC-MAS combines adaptive swarm dynamics with sparse LLM guidance for local actions and cooperation, scheduled through phased cognitive guidance. Across benchmarks and ablations, the coordinated approach delivers stable, consistently strong performance while improving solution quality, consensus formation, and communication cost.
Problem
Existing distributed black-box optimization methods have limited ability to adapt both local agent behavior and inter-agent coordination under decentralized information constraints.
Method
LAC-MAS combines adaptive internal swarm dynamics with sparse LLM guidance that learns agent actions and cooperation from historical trajectories, using phased cognitive scheduling.
Results
LAC-MAS consistently achieves stable and strong performance across diverse landscapes, with internal learning improving solution quality and escape capability while cooperative learning accelerates consensus and reduces communication cost.
Takeaways & Limitations
The results support trajectory-driven self-design as a practical direction for moving beyond handcrafted distributed coordination in multi-agent black-box optimization.
Takeaways & Limitations
On narrow-valley or strongly directional functions such as F3 and F6, LAC-MAS matches strong baselines but does not show statistically significant improvement over MASOIE.
Abstract
from arXiv · showhide
Distributed blackbox consensus optimization is a fundamental problem in multi-agent systems, where agents must improve a global objective using only local objective queries and limited neighbor communication. Existing methods largely rely on handcrafted update rules and static cooperation patterns, which often struggle to balance local adaptation, global coordination, and communication efficiency in heterogeneous nonconvex environments. In this paper, we take an initial step toward trajectory-driven self-design for distributed black-box consensus optimization. We first redesign the agent-level swarm dynamics with an adaptive internal mechanism tailored to decentralized consensus settings, improving the balance between exploration, convergence, and local escape. Built on top of this adaptive execution layer, we propose Learning to Act and Cooperate (LACMAS), a trajectorydriven framework in which large language models provide sparse highlevel guidance for shaping both agentinternal action behaviors and agentexternal cooperation patterns from historical optimization trajectories. We further introduce a phased cognitive scheduling strategy to activate different forms of adaptation in a resource-aware manner. Experiments on standard distributed black-box benchmarks and real-world distributed tasks show that LAC-MAS consistently improves solution quality, convergence efficiency, and communication efficiency over strong baselines, suggesting a practical route from handcrafted distributed coordination toward self-designing multi-agent optimization systems.
1. Introduction
Distributed black-box consensus optimization requires agents to coordinate using local information and limited communication, while existing approaches struggle with black-box nonconvex adaptation. LAC-MAS addresses this gap through adaptive agent dynamics, trajectory-driven LLM guidance, and phased cognitive coordination.
- Distributed agents must improve a global objective using local observability, limited communication, and heterogeneous feedback.
- Existing gradient-based methods require explicit objective structure, while reinforcement-learning approaches face unstable training, weak scalability, and difficult multi-agent credit assignment.
- LAC-MAS jointly adapts agent-internal action behaviors and agent-external cooperation patterns from local and neighbor historical trajectories.
- The framework redesigns swarm dynamics with an adaptive internal mechanism balancing exploration, convergence, and local escape under decentralized consensus constraints.
- A phased cognitive guidance strategy activates stage-aware action and cooperation adaptation, making high-level guidance more resource-aware for decentralized optimization.
2. Related Work
Related work spans decentralized black-box optimization, learning-driven algorithm design, and adaptive coordination. These lines motivate combining trajectory-based strategy adaptation with communication-aware cooperation in distributed optimization.
- Distributed Black-Box Consensus Optimization: Decentralized black-box optimization studies cooperative global-objective optimization using local function evaluations and neighbor communication.
- Learning-Driven Optimization Design: Learning-driven optimization design adapts strategies from feedback, historical trajectories, and higher-level performance signals, including LLM-based iterative refinement and trajectory-aware adaptation.
- Adaptive Coordination in Decentralized Systems: Adaptive coordination research examines learned communication graphs, network heterogeneity, and communication-efficient mixing strategies in decentralized optimization.
3. Problem Formulation
The paper formulates optimization over a fixed connected communication graph, where agents query only local black-box objectives and exchange information with neighbors. They seek simultaneous objective improvement and consensus without access to global optimization state.
- The problem is defined over a connected graph G=(V,E), with agents communicating only through their neighbor sets.
- The communication topology remains fixed and connected, although the cooperation mechanism may adapt weights assigned to existing neighbors.
- Each agent has a local black-box objective accessible only through function evaluation, so gradients are unavailable.
- Agents seek to reduce the global objective while driving their local states toward agreement.
- Agents access local query results, particle states, and aggregated neighbor trajectory statistics, but not other objectives, gradients, or global optimization state.
4. Methodology
LAC-MAS combines adaptive swarm execution with trajectory-driven LLM guidance to shape agents’ internal search and neighbor cooperation. Phased Cognitive Guidance schedules these adaptations sparsely while preserving the decentralized consensus structure.
- Framework Overview: Each agent couples an adaptive swarm-based execution layer with a trajectory-driven guidance layer that updates internal behavior and external cooperation from optimization trajectories.The execution layer performs local black-box optimization, while the guidance layer periodically refreshes both forms of adaptation.
- Adaptive Swarm Execution: The adaptive execution layer uses particle-population divergence to distinguish exploratory and locally convergent regimes, regulating internal swarm variation accordingly.Larger divergence indicates broader exploration, whereas smaller divergence indicates stronger concentration and local convergence.
- Learning to Act: LLM trajectory guidance infers internal coefficient sets from recent optimization history, replacing fixed handcrafted coefficients with refreshed, trajectory-informed search behavior.The learned coefficients are instantiated online through the divergence-based execution rule.
- Learning to Cooperate: Trajectory-driven cooperation learns the relative value of existing neighbors’ information without changing the communication topology.The mechanism reweights neighbor influence according to historical optimization utility while preserving decentralized communication constraints.
- Phased Cognitive Guidance: Phased Cognitive Guidance schedules refreshes of internal action and external cooperation guidance according to distinct functional roles, avoiding continuous iteration-level intervention.The schedule decouples continuous optimization from sparse guidance updates without requiring strict iteration-level synchronization.
- Consensus Guarantees: Under standard assumptions, internal adaptation, trajectory-driven cooperation, and phased refresh preserve the consensus structure of the decentralized swarm optimizer.Normalization and projection preserve admissible mixing conditions, while bounded coefficient modulation and finite-stage refresh constrain the adaptive dynamics.
5. Experimental Setup and Results
Experiments evaluate LAC-MAS on distributed black-box benchmarks and a cooperative multi-target localization task, with ablations separating action and cooperation learning. Across these settings, LAC-MAS generally improves fitness, consensus efficiency, and communication efficiency, while matching strong baselines on some specialized landscapes.
- Experimental Setup: The benchmark suite contains ten diverse functions with 100 decision variables distributed across 20 agents using only local black-box queries.The global objective is the average of local objectives and remains inaccessible during optimization.
- Experimental Setup: LAC-MAS is compared with multi-agent swarm, consensus-driven population, and classical distributed optimization baselines under repeated independent runs.Convergence is declared when disagreement falls below 10^-7, and evaluation uses final fitness, cumulative communication cost, and disagreement.
- Benchmark Results: Across benchmarks, LAC-MAS matches or outperforms strong baselines on most functions, with lower mean and median fitness and stable disagreement reduction.The gains are particularly pronounced when flexible regulation of exploration and convergence is required.
- Benchmark Results: On narrow-valley or strongly directional functions such as F3 and F6, LAC-MAS matches MASOIE without statistically significant differences.These functions provide limited room for further improvement over specialized search dynamics.
- Ablation Experiments: Act learning improves final fitness, whereas cooperation learning accelerates disagreement reduction and lowers communication cost across most benchmarks.The full framework jointly learns both components and coordinates them through phased cognitive guidance.
- Transfer Validation: On multi-target localization, LAC-MAS achieves the lowest estimation error across all tested target numbers despite error increases as the target count grows.The results support transfer beyond synthetic functions to a realistic distributed black-box task under limited communication.
6. Conclusion
The paper presents LAC-MAS as an LLM-assisted framework that jointly learns agent actions and cooperation through phased cognitive guidance. Experiments and ablations find complementary roles for the two learning components, with their integration producing stable performance across diverse landscapes.
- LAC-MAS jointly learns agent-internal behaviors and agent-external coordination through phased cognitive guidance to balance exploration, convergence, and communication efficiency.
- Internal behavioral learning improves solution quality and escape capability, while cooperative learning accelerates consensus formation and reduces communication cost.Their coordinated integration leads to stable and consistently strong performance across diverse problem landscapes.
Impact Statement
The work targets more efficient and robust distributed black-box optimization in multi-agent systems, with applications including cooperative sensing, resource allocation, and distributed control. It does not involve human subjects or personal data and is not expected to introduce significant additional ethical or societal risks.
- The framework focuses on improving efficiency and robustness in distributed black-box optimization for multi-agent systems.
- Potential applications include cooperative sensing, resource allocation, and distributed control, which may contribute to more efficient and resilient large-scale systems.
A. Appendix A: Stage-wise Interpretation of Phased Cognitive Guidance
Phased Cognitive Guidance organizes trajectory-driven adaptation into interpretable stages without imposing rigid temporal boundaries. Learning emphasis shifts from trajectory accumulation and action adaptation toward joint adaptation and final consensus-oriented cooperation.
- Overview: PCG creates implicit stages through internal and external cognitive gates, providing an interpretable abstraction for how learning emphasis evolves over time.
- Stage I: Trajectory Accumulation: Stage I keeps both cognitive gates inactive while agents explore and collect foundational trajectories for later trajectory-conditioned learning.
- Stage II: Learning to Act: Stage II activates internal behavioral learning while suppressing external cooperation learning, refining local search after sufficient exploration.
- Stage III: Learning to Act and Cooperate: Stage III periodically activates both mechanisms so agents adapt internal behaviors and reassess neighbor cooperation, mitigating premature convergence from misleading local information.
- Stage IV: Consensus-Oriented Cooperation: Stage IV deactivates internal adaptation while continuing cooperation learning, prioritizing stable coordination and consensus formation.
B. Appendix B: Detailed Ablation Results
The ablation results compare MASOIE with internal action learning, external cooperation learning, and the full LAC-MAS configuration across benchmark metrics. The full configuration generally provides the strongest overall trade-off, while the components contribute differently to fitness and communication cost.
- Ablation findings: The full LAC-MAS achieves the best or near-best performance on most benchmarks, demonstrating complementary effects from internal action and external cooperation learning.Single-component variants show partial improvements but underperform the combined configuration.
- Ablation findings: Internal action learning tends to lower fitness values, indicating more effective local optimization.The ablation discussion attributes this pattern to the Act variants.
- Ablation findings: External cooperation learning consistently reduces communication cost by reallocating neighbor influence more efficiently.The Coop variants improve communication efficiency even when their fitness gains are more limited.
C. Appendix C: Additional Implementation Details for Learning to Act and Cooperate
LAC-MAS uses sparse, trajectory-driven LLM guidance at two levels: agents select internal behavior modes and adapt external neighbor weights. Fixed short guidance intervals and decentralized local information preserve stable execution.
- Learning to act: LLMs select discrete internal behavior modes from recent divergence statistics, keeping each mode fixed over a short interval.This separates high-level behavioral learning from iteration-level responsiveness.
- Learning to cooperate: LLMs map trajectory-based summaries of neighboring agents to normalized relative importance scores for adaptive consensus weights.The weighting remains within the existing sparse communication topology.
- Decentralized execution: All LLM-assisted outputs are applied locally and remain fixed within short intervals, preventing oscillatory behavior without centralized supervision.Agents use only locally available trajectories and statistics exchanged with neighbors.
- Prompt interfaces: Internal prompts adapt parameters from recent fitness and disagreement trajectories, while cooperation prompts update neighbor weights using aggregated historical statistics.The prompts encode lightweight rules and normalization constraints rather than explicit objective models.
E. Appendix E. Additional Details for Consensus Preservation
The consensus-preservation argument models LAC-MAS as a connected row-stochastic switching system with bounded finite-stage adaptation and asymptotically vanishing perturbations. Standard switching-consensus reasoning then yields vanishing disagreement.
- Consensus model: The cooperative update is represented by a mixing matrix A(t) acting on stacked agent states, plus a perturbation ξ(t) from local swarm evolution.This form isolates consensus fusion from the local black-box search dynamics.
- Admissibility: Normalization over existing neighbors makes A(t) nonnegative, graph-compatible, and row-stochastic while preserving the fixed connected communication graph.These properties establish admissibility for the switching consensus analysis.
- Finite-stage adaptation: The internal action mechanism contributes bounded, finitely refreshed modulation rather than persistent high-frequency switching.Internal-guidance refresh occurs only at scheduled times and is deactivated after the calibrated horizon.
- Vanishing perturbation: After the final guidance refresh, the perturbation from local black-box search vanishes asymptotically in the stabilized execution regime.The remaining variation comes from decaying local swarm adjustment.
- Consensus conclusion: Because the disagreement component is contracted by connected row-stochastic switching dynamics and ξ(t) tends to zero, disagreement asymptotically vanishes.This is the basis of the consensus result in Theorem 4.1.
F. Appendix F. Symbol Table
The symbol table defines notation used for the distributed optimization and WSN localization formulations, including sensor locations, target counts, RSS quantities, and system-level estimation error.
- WSN localization symbols: yi denotes the known location of sensor i in the WSN localization task.
- WSN localization symbols: Nt denotes the number of targets in the WSN localization task, while ϕit denotes the RSS measurement from target t to sensor i.
- System-level symbols: Err(k) denotes the WSN estimation error, defined as F(¯x(k)), where ¯x(k) is the system-level estimate averaged across sensors at communication round k.