Policy-based Primal-Dual Methods for Concave CMDP with Variance Reduction
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917675127537664 |
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| author | Ying, Donghao Guo, Mengzi Amy Lee, Hyunin Ding, Yuhao Lavaei, Javad Shen, Zuo-Jun Max |
| author_facet | Ying, Donghao Guo, Mengzi Amy Lee, Hyunin Ding, Yuhao Lavaei, Javad Shen, Zuo-Jun Max |
| contents | We study Concave Constrained Markov Decision Processes (Concave CMDPs) where both the objective and constraints are defined as concave functions of the state-action occupancy measure. We propose the Variance-Reduced Primal-Dual Policy Gradient Algorithm (VR-PDPG), which updates the primal variable via policy gradient ascent and the dual variable via projected sub-gradient descent. Despite the challenges posed by the loss of additivity structure and the nonconcave nature of the problem, we establish the global convergence of VR-PDPG by exploiting a form of hidden concavity. In the exact setting, we prove an $O(T^{-1/3})$ convergence rate for both the average optimality gap and constraint violation, which further improves to $O(T^{-1/2})$ under strong concavity of the objective in the occupancy measure. In the sample-based setting, we demonstrate that VR-PDPG achieves an $\widetilde{O}(ε^{-4})$ sample complexity for $ε$-global optimality. Moreover, by incorporating a diminishing pessimistic term into the constraint, we show that VR-PDPG can attain a zero constraint violation without compromising the convergence rate of the optimality gap. Finally, we validate the effectiveness of our methods through numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_10715 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Policy-based Primal-Dual Methods for Concave CMDP with Variance Reduction Ying, Donghao Guo, Mengzi Amy Lee, Hyunin Ding, Yuhao Lavaei, Javad Shen, Zuo-Jun Max Machine Learning Optimization and Control We study Concave Constrained Markov Decision Processes (Concave CMDPs) where both the objective and constraints are defined as concave functions of the state-action occupancy measure. We propose the Variance-Reduced Primal-Dual Policy Gradient Algorithm (VR-PDPG), which updates the primal variable via policy gradient ascent and the dual variable via projected sub-gradient descent. Despite the challenges posed by the loss of additivity structure and the nonconcave nature of the problem, we establish the global convergence of VR-PDPG by exploiting a form of hidden concavity. In the exact setting, we prove an $O(T^{-1/3})$ convergence rate for both the average optimality gap and constraint violation, which further improves to $O(T^{-1/2})$ under strong concavity of the objective in the occupancy measure. In the sample-based setting, we demonstrate that VR-PDPG achieves an $\widetilde{O}(ε^{-4})$ sample complexity for $ε$-global optimality. Moreover, by incorporating a diminishing pessimistic term into the constraint, we show that VR-PDPG can attain a zero constraint violation without compromising the convergence rate of the optimality gap. Finally, we validate the effectiveness of our methods through numerical experiments. |
| title | Policy-based Primal-Dual Methods for Concave CMDP with Variance Reduction |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2205.10715 |