$QD$-Learning: A Collaborative Distributed Strategy for Multi-Agent Reinforcement Learning Through Consensus + Innovations (1205.0047v2)

Abstract: The paper considers a class of multi-agent Markov decision processes (MDPs), in which the network agents respond differently (as manifested by the instantaneous one-stage random costs) to a global controlled state and the control actions of a remote controller. The paper investigates a distributed reinforcement learning setup with no prior information on the global state transition and local agent cost statistics. Specifically, with the agents' objective consisting of minimizing a network-averaged infinite horizon discounted cost, the paper proposes a distributed version of $Q$-learning, $\mathcal{QD}$-learning, in which the network agents collaborate by means of local processing and mutual information exchange over a sparse (possibly stochastic) communication network to achieve the network goal. Under the assumption that each agent is only aware of its local online cost data and the inter-agent communication network is \emph{weakly} connected, the proposed distributed scheme is almost surely (a.s.) shown to yield asymptotically the desired value function and the optimal stationary control policy at each network agent. The analytical techniques developed in the paper to address the mixed time-scale stochastic dynamics of the \emph{consensus + innovations} form, which arise as a result of the proposed interactive distributed scheme, are of independent interest.

• The paper presents a consensus plus innovations approach that enables decentralized Q-learning with guaranteed asymptotic convergence.
• It employs iterative updates over a sparse communication network, ensuring individual agent estimates match centralized solutions.
• Strong analytical and numerical results validate its effectiveness for optimizing multi-agent MDPs in dynamic, uncertain environments.

Overview of Collaborative $\mathcal{QD}$-Learning for Multi-Agent Reinforcement Learning

The paper presents a detailed investigation into collaborative multi-agent reinforcement learning utilizing distributed $Q$-learning, dubbed $\mathcal{QD}$-learning. It addresses challenges inherent in multi-agent Markov decision processes (MDPs) where agents, operating in a dynamic and uncertain environment, must optimize a network-averaged infinite horizon discounted cost devoid of prior knowledge of state transitions and reward statistics. Unlike centralized alternatives that necessitate constant transmission of instantaneous costs to a central controller, $\mathcal{QD}$-learning facilitates decentralized learning across agents connected through a sparse communication network.

The key innovation of $\mathcal{QD}$-learning lies in its consensus + innovations approach, integrating local cost information sensed by each agent with inter-agent communications to collaboratively reach an optimal control strategy. The learning process is articulated through an intricate interplay between consensus dynamics and local innovation updates, which are weighted appropriately over time to ensure convergence. This approach allows each agent to autonomously learn the optimal value function $\mathbf{V}^{\ast}$ and the corresponding control policy $\pi^{\ast}$ with guarantees of asymptotic correctness.

Strong Numerical and Analytical Results

The paper introduces an algorithmic scheme where each agent updates its $Q$-matrix iteratively based on state-action transitions and exchanges information with neighbors. With minimal connectivity assumptions, the approach guarantees convergence with high probability, demonstrated by the bounded consensus value $\mathbf{Q}^{\ast}$ that aligns with $\mathbf{V}^{\ast}$, the optimal network objective. The convergence results indicate that distributed approximations at individual agents are asymptotically equivalent to centralized calculations. Analytical methods, including stochastic approximations, are leveraged to manage mixed time-scale evolutions arising from temporal dependencies in state-action trajectories.

Implications and Future Directions

The implications of this research fundamentally enrich distributed learning in multi-agent systems, providing pathways for efficiently computing optimal strategies in decentralized architectures. This aligns with applications spanning smart building control to financial networks, substantiating the practical utility of $\mathcal{QD}$-learning within diverse contexts. Theoretical advancements in consensus mechanisms further position this paper as a step toward more robust understanding and implementation of collaborative reinforcement learning.

Future avenues include exploration of decentralized actuation wherein agents are empowered to independently influence the global state, accommodating partial state observability challenges where the global signal is accessible at limited fidelity. Moreover, convergence rates under specific probabilistic modeling of state-action pairs may provide deeper insights into performance loss scenarios comparing centralized and distributed alternatives.

Overall, the paper provides a meticulous development of $\mathcal{QD}$-learning, advocating for its deployment in complex distributed environments, emphasizing both practical significance and theoretical robustness.