Q-Measure-Learning for Continuous State RL: Efficient Implementation and Convergence
arXiv:2603.03523v1 Announce Type: new Abstract: We study reinforcement learning in infinite-horizon discounted Markov decision processes with continuous state spaces, where data are generated online from a single trajectory under a Markovian behavior policy. To avoid maintaining an infinite-dimensional, function-valued estimate, we propose the novel Q-Measure-Learning, which learns a signed empirical measure supported on visited state-action pairs and reconstructs an action-value estimate via kernel integration. The method jointly estimates the stationary distribution of the behavior chain and the Q-measure through coupled stochastic approximation, leading to an efficient weight-based implementation with $O(n)$ memory and $O(n)$ computation cost per iteration. Under uniform ergodicity of the behavior chain, we prove almost sure sup-norm convergence of the induced Q-function to the fixed point of a kernel-smoothed Bellman operator. We also bound the approximation error between this lim
arXiv:2603.03523v1 Announce Type: new Abstract: We study reinforcement learning in infinite-horizon discounted Markov decision processes with continuous state spaces, where data are generated online from a single trajectory under a Markovian behavior policy. To avoid maintaining an infinite-dimensional, function-valued estimate, we propose the novel Q-Measure-Learning, which learns a signed empirical measure supported on visited state-action pairs and reconstructs an action-value estimate via kernel integration. The method jointly estimates the stationary distribution of the behavior chain and the Q-measure through coupled stochastic approximation, leading to an efficient weight-based implementation with $O(n)$ memory and $O(n)$ computation cost per iteration. Under uniform ergodicity of the behavior chain, we prove almost sure sup-norm convergence of the induced Q-function to the fixed point of a kernel-smoothed Bellman operator. We also bound the approximation error between this limit and the optimal $Q^*$ as a function of the kernel bandwidth. To assess the performance of our proposed algorithm, we conduct RL experiments in a two-item inventory control setting.
Executive Summary
This article proposes Q-Measure-Learning, a novel approach to reinforcement learning in continuous state spaces. It learns a signed empirical measure supported on visited state-action pairs and reconstructs an action-value estimate via kernel integration. The method achieves efficient implementation and convergence, with a proven almost sure sup-norm convergence of the induced Q-function to the fixed point of a kernel-smoothed Bellman operator. The article also conducts experiments in a two-item inventory control setting to assess the algorithm's performance.
Key Points
- ▸ Q-Measure-Learning approach for continuous state RL
- ▸ Efficient implementation with O(n) memory and computation cost
- ▸ Proven almost sure sup-norm convergence of the induced Q-function
Merits
Efficient Implementation
The proposed algorithm has a low memory and computation cost, making it suitable for large-scale applications.
Theoretical Guarantees
The article provides theoretical guarantees for the convergence of the algorithm, which is essential for trustworthiness and reliability.
Demerits
Limited Exploration
The algorithm relies on a single trajectory under a Markovian behavior policy, which may limit exploration and lead to suboptimal solutions.
Kernel Bandwidth Selection
The choice of kernel bandwidth can significantly affect the algorithm's performance, and selecting an optimal value can be challenging.
Expert Commentary
The proposed Q-Measure-Learning approach is a significant contribution to the field of reinforcement learning, as it provides an efficient and theoretically grounded method for learning in continuous state spaces. The algorithm's ability to learn from a single trajectory and its low computational cost make it an attractive option for real-world applications. However, further research is needed to address the limitations of the algorithm, such as limited exploration and kernel bandwidth selection. Additionally, combining Q-Measure-Learning with other techniques, such as deep learning, can potentially lead to even more powerful and efficient algorithms.
Recommendations
- ✓ Further research on kernel bandwidth selection and exploration strategies to improve the algorithm's performance.
- ✓ Application of Q-Measure-Learning to more complex environments and real-world problems to demonstrate its practicality and effectiveness.