Natural Policy Gradient as Doubly Smoothed Policy Iteration: A Bellman-Operator Framework
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917480828502016 |
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| author | Nanda, Phalguni Chen, Zaiwei |
| author_facet | Nanda, Phalguni Chen, Zaiwei |
| contents | In this work, we show that natural policy gradient, a core algorithm in reinforcement learning, admits an exact formulation as a smoothed and averaged form of policy iteration. Specifically, we introduce doubly smoothed policy iteration (DSPI), a Bellman-operator framework in which each policy is obtained by applying a regularized greedy step to a weighted average of past $Q$-functions. DSPI includes policy iteration, dual-averaged policy iteration, natural policy gradient, and more general policy dual averaging methods as special cases. Using only monotonicity and contraction of smoothed Bellman operators, we prove distribution-free global geometric convergence of DSPI. Consequently, standard natural policy gradient and policy dual averaging achieve an iteration complexity of $\mathcal{O}((1-γ)^{-1}\log((1-γ)^{-1}ε^{-1}))$ for computing an $ε$-optimal policy, without modifying the MDP, adding regularization beyond the mirror map inherent in the update, or using adaptive, trajectory-dependent stepsizes. For the unregularized greedy case, corresponding to dual-averaged policy iteration, we also prove finite termination. The same Bellman-operator framework further extends to discounted MDPs with linear function approximation and stochastic shortest path problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10671 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Natural Policy Gradient as Doubly Smoothed Policy Iteration: A Bellman-Operator Framework Nanda, Phalguni Chen, Zaiwei Machine Learning Optimization and Control In this work, we show that natural policy gradient, a core algorithm in reinforcement learning, admits an exact formulation as a smoothed and averaged form of policy iteration. Specifically, we introduce doubly smoothed policy iteration (DSPI), a Bellman-operator framework in which each policy is obtained by applying a regularized greedy step to a weighted average of past $Q$-functions. DSPI includes policy iteration, dual-averaged policy iteration, natural policy gradient, and more general policy dual averaging methods as special cases. Using only monotonicity and contraction of smoothed Bellman operators, we prove distribution-free global geometric convergence of DSPI. Consequently, standard natural policy gradient and policy dual averaging achieve an iteration complexity of $\mathcal{O}((1-γ)^{-1}\log((1-γ)^{-1}ε^{-1}))$ for computing an $ε$-optimal policy, without modifying the MDP, adding regularization beyond the mirror map inherent in the update, or using adaptive, trajectory-dependent stepsizes. For the unregularized greedy case, corresponding to dual-averaged policy iteration, we also prove finite termination. The same Bellman-operator framework further extends to discounted MDPs with linear function approximation and stochastic shortest path problems. |
| title | Natural Policy Gradient as Doubly Smoothed Policy Iteration: A Bellman-Operator Framework |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2605.10671 |