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This research presents a theoretical advancement in policy optimization by establishing global linear convergence for Policy Mirror Descent when coupled with simple one-step temporal-difference updates. The analysis demonstrates that high sample efficiency can be achieved without relying on exact policy evaluation, trajectory resets, or generative models, offering a more practical framework for reinforcement learning algorithms.
Fast Regularized Policy Mirror Descent with One-Step TD Updates
arXiv:2609.39837v1 Announce Type: new Abstract: Policy mirror descent (PMD) enjoys fast convergence in regularized Markov decision processes (MDPs), but existing guarantees often rely on exact or increasingly accurate policy evaluation. We analyze PMD coupled with a persistent critic advanced by one temporal-difference (TD) update. For finite discounted MDPs, we establish global linear convergence in value for exact coordinate-wise Bellman updates, with any positive constant actor stepsize and arbitrary finite critic initialization. The proof combines a resolvent-based auxiliary distribution with a decaying Bellman-violation correction and a potential weighted by inverse coordinate weights. We then study stochastic TD-PMD with general strongly convex mirror maps under a single off-policy Markov trajectory. With suitably chosen constant stepsizes and a finite-batch TD update, the method achieves an expected value gap of $\epsilon$ after $\widetilde{O}(1/((1-\gamma)^5 \widetilde{\sigma}_b \epsilon))$ transitions. The stochastic analysis relies on the trajectory-wise Lipschitz continuity of the regularizer, derived from uniform bounds on vertex Bregman divergences, together with a visitation-weighted resolvent estimate for signed critic-error propagation that yields an inverse-linear dependence on behavior coverage $\widetilde{\sigma}_b$. In contrast to many prior guarantees for regularized policy optimization, our sample-complexity guarantee holds without trajectory resets, generative-model access, or nested policy-evaluation loops. Numerical results are consistent with the theoretical convergence analysis.
Abstract:Policy mirror descent (PMD) enjoys fast convergence in regularized Markov decision processes (MDPs), but existing guarantees often rely on exact or increasingly accurate policy evaluation. We analyze PMD coupled with a persistent critic advanced by one temporal-difference (TD) update. For finite discounted MDPs, we establish global linear convergence in value for exact coordinate-wise Bellman updates, with any positive constant actor stepsize and arbitrary finite critic initialization. The proof combines a resolvent-based auxiliary distribution with a decaying Bellman-violation correction and a potential weighted by inverse coordinate weights. We then study stochastic TD-PMD with general strongly convex mirror maps under a single off-policy Markov trajectory. With suitably chosen constant stepsizes and a finite-batch TD update, the method achieves an expected value gap of $\epsilon$ after $\widetilde{O}(1/((1-\gamma)^5 \widetilde{\sigma}_b \epsilon))$ transitions. The stochastic analysis relies on the trajectory-wise Lipschitz continuity of the regularizer, derived from uniform bounds on vertex Bregman divergences, together with a visitation-weighted resolvent estimate for signed critic-error propagation that yields an inverse-linear dependence on behavior coverage $\widetilde{\sigma}_b$. In contrast to many prior guarantees for regularized policy optimization, our sample-complexity guarantee holds without trajectory resets, generative-model access, or nested policy-evaluation loops. Numerical results are consistent with the theoretical convergence analysis.
Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2609.39837 [cs.LG]
(or arXiv:2609.39837v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.39837 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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