Q-Measure-Learning for Continuous State RL: Efficient Implementation and Convergence

• URL: http://arxiv.org/abs/2603.03523v1
• Date: Tue, 03 Mar 2026 21:15:22 GMT
• Title: Q-Measure-Learning for Continuous State RL: Efficient Implementation and Convergence
• Authors: Shengbo Wang,
• Abstract summary: We study reinforcement learning in infinite-horizon discounted Markov decision processes with continuous state spaces.<n>We propose the novel Q-Measure-Learning, which learns a signed empirical measure supported on visited state-action pairs.
• Score: 10.189658648290257
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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