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Approximate Global Convergence of Independent Learning in Multi-Agent Systems

Ruiyang Jin, Zaiwei Chen, Yiheng Lin, Jie Song, Adam Wierman

TL;DR

This work tackles the challenge of achieving global convergence for independent learning in cooperative multi-agent reinforcement learning (MARL). It introduces a novel dependence level $\mathcal{E}$ and constructs a separable MDP $\hat{\mathcal{M}}$ to quantify and bound the non-separable dynamics, enabling finite-sample analysis of two representative IL algorithms: independent Q-learning (IQL) and independent natural actor-critic (INAC). The authors establish last-iterate convergence bounds showing that, up to a model-difference error proportional to $\mathcal{E}$, the remaining terms yield a sample complexity of $\tilde{O}(\epsilon^{-2})$ to reach $\epsilon$-optimal performance. Numerical experiments on a synthetic MDP and an EV charging application verify the theoretical guarantees and demonstrate practical efficacy of IL under approximations, while highlighting the fundamental limitation imposed by $\mathcal{E}$ on global convergence. The proposed separable-MDP analysis framework offers a general tool for studying non-Markovian stochastic iterative algorithms and suggests that careful coordination could reduce $\mathcal{E}$ without sacrificing scalability.

Abstract

Independent learning (IL), despite being a popular approach in practice to achieve scalability in large-scale multi-agent systems, usually lacks global convergence guarantees. In this paper, we study two representative algorithms, independent $Q$-learning and independent natural actor-critic, within value-based and policy-based frameworks, and provide the first finite-sample analysis for approximate global convergence. The results imply a sample complexity of $\tilde{\mathcal{O}}(ε^{-2})$ up to an error term that captures the dependence among agents and characterizes the fundamental limit of IL in achieving global convergence. To establish the result, we develop a novel approach for analyzing IL by constructing a separable Markov decision process (MDP) for convergence analysis and then bounding the gap due to model difference between the separable MDP and the original one. Moreover, we conduct numerical experiments using a synthetic MDP and an electric vehicle charging example to verify our theoretical findings and to demonstrate the practical applicability of IL.

Approximate Global Convergence of Independent Learning in Multi-Agent Systems

TL;DR

This work tackles the challenge of achieving global convergence for independent learning in cooperative multi-agent reinforcement learning (MARL). It introduces a novel dependence level and constructs a separable MDP to quantify and bound the non-separable dynamics, enabling finite-sample analysis of two representative IL algorithms: independent Q-learning (IQL) and independent natural actor-critic (INAC). The authors establish last-iterate convergence bounds showing that, up to a model-difference error proportional to , the remaining terms yield a sample complexity of to reach -optimal performance. Numerical experiments on a synthetic MDP and an EV charging application verify the theoretical guarantees and demonstrate practical efficacy of IL under approximations, while highlighting the fundamental limitation imposed by on global convergence. The proposed separable-MDP analysis framework offers a general tool for studying non-Markovian stochastic iterative algorithms and suggests that careful coordination could reduce without sacrificing scalability.

Abstract

Independent learning (IL), despite being a popular approach in practice to achieve scalability in large-scale multi-agent systems, usually lacks global convergence guarantees. In this paper, we study two representative algorithms, independent -learning and independent natural actor-critic, within value-based and policy-based frameworks, and provide the first finite-sample analysis for approximate global convergence. The results imply a sample complexity of up to an error term that captures the dependence among agents and characterizes the fundamental limit of IL in achieving global convergence. To establish the result, we develop a novel approach for analyzing IL by constructing a separable Markov decision process (MDP) for convergence analysis and then bounding the gap due to model difference between the separable MDP and the original one. Moreover, we conduct numerical experiments using a synthetic MDP and an electric vehicle charging example to verify our theoretical findings and to demonstrate the practical applicability of IL.
Paper Structure (49 sections, 17 theorems, 81 equations, 7 figures, 2 tables, 4 algorithms)

This paper contains 49 sections, 17 theorems, 81 equations, 7 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. $Q$-Learning.
  4. Natural Actor-Critic.
  5. Independent Learning in MARL.
  6. Networked MARL.
  7. Problem Formulation
  8. The Dependence Level
  9. An Illustrative Example
  10. Main Results
  11. Independent $Q$-Learning
  12. Algorithm Details.
  13. Independent Natural Actor-Critic
  14. Independent Natural Policy Gradient for the Actor.
  15. Independent TD-Learning for the Critic.
  16. ...and 34 more sections

Key Result

Theorem 3.1

Consider $\{Q_k\}_{k\geq 0}$ generated by Algorithm algorithm:IQL. Suppose that Assumption assum_markov_chain is satisfied and $\alpha_k=\frac{\alpha}{k+k_0}$ with $k_0=\max (4\alpha, 2M_2\log K)$ and $\alpha\geq \frac{2}{\sigma'(1-\gamma)}$. Then, for any $\delta'\in (0,1)$, with probability at lea where $\pi_*$ is an optimal policy, and $\pi_K=(\pi_k^1,\pi_k^2,\cdots,\pi_k^n)$ is the policy gree

Figures (7)

  • Figure 2: Illustration of the example.
  • Figure 3: Performance of IQL and INAC
  • Figure 4: The separable MDPs for different grouping options
  • Figure 5: Illustration of a binary tree to allocate the charging capacity.
  • Figure 6: The structures of the 3-agent system (left) and the 15-agent system (right). The square nodes represent the charging stations and the circle nodes represent the agents. Note that our results do not require the interaction structure to be full binary trees.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Remark
  • Remark
  • Example 2.1
  • Remark
  • Theorem 3.1
  • Remark
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4
  • ...and 18 more