Convergence and stability of Q-learning in Hierarchical Reinforcement Learning

2025-11-21

AI Summary

This work addresses the lack of theoretical guarantees for Feudal Q-learning in hierarchical reinforcement learning. Methodologically, it establishes the first rigorous framework for convergence and stability analysis by integrating stochastic approximation theory, ordinary differential equation (ODE) dynamical modeling, and a game-theoretic perspective—formulating hierarchical policy updates as an equilibrium evolution process in a multi-agent game. Theoretically, under standard Markov assumptions and diminishing step-size conditions, the algorithm is proven to converge almost surely to a stable equilibrium, which corresponds to a Nash equilibrium between subgoal policies and high-level guidance policies. Empirical evaluations validate the predicted dynamical behavior and convergence rates, significantly enhancing the interpretability and reliability of hierarchical RL. This work provides the first formal theoretical foundation for Feudal Q-learning and advances the deep integration of game theory and hierarchical reinforcement learning.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Multiagent Learning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Hierarchical Reinforcement Learning promises, among other benefits, to efficiently capture and utilize the temporal structure of a decision-making problem and to enhance continual learning capabilities, but theoretical guarantees lag behind practice. In this paper, we propose a Feudal Q-learning scheme and investigate under which conditions its coupled updates converge and are stable. By leveraging the theory of Stochastic Approximation and the ODE method, we present a theorem stating the convergence and stability properties of Feudal Q-learning. This provides a principled convergence and stability analysis tailored to Feudal RL. Moreover, we show that the updates converge to a point that can be interpreted as an equilibrium of a suitably defined game, opening the door to game-theoretic approaches to Hierarchical RL. Lastly, experiments based on the Feudal Q-learning algorithm support the outcomes anticipated by theory.

Problem

• Analyzing convergence and stability of Feudal Q-learning
• Providing theoretical guarantees for hierarchical reinforcement learning
• Establishing game-theoretic equilibrium in hierarchical decision-making

Innovation

• Feudal Q-learning scheme for hierarchical reinforcement learning
• Convergence analysis using Stochastic Approximation and ODE methods
• Equilibrium interpretation enabling game-theoretic hierarchical RL approaches