A Duality Approach for Regret Minimization in Average-Award Ergodic Markov Decision Processes

Hao Gong, Mengdi Wang
Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:862-883, 2020.

Abstract

In light of the Bellman duality, we propose a novel value-policy gradient algorithm to explore and act in infinite-horizon Average-reward Markov Decision Process (AMDP) and show that it has sublinear regret. The algorithm is motivated by the Bellman saddle point formulation. It learns the optimal state-action distribution, which encodes a randomized policy, by interacting with the environment along a single trajectory and making primal-dual updates. The key to the analysis is to establish a connection between the min-max duality gap of Bellman saddle point and the cumulative regret of the learning agent. We show that, for ergodic AMDPs with finite state space $\mathcal{S}$ and action space $\mathcal{A}$ and uniformly bounded mixing times, the algorithm’s $T$-time step regret is $$ R(T)=\tilde{\mathcal{O}}\left( \left(t_{mix}^*\right)^2 \tau^{\frac{3}{2}} \sqrt{(\tau^3 + |\mathcal{A}|) |\mathcal{S}| T} \right), $$ where $t_{mix}^*$ is the worst-case mixing time, $\tau$ is an ergodicity parameter, $T$ is the number of time steps and $\tilde{\mathcal{O}}$ hides polylog factors.

Cite this Paper

BibTeX
@InProceedings{pmlr-v120-gong20a, title = {A Duality Approach for Regret Minimization in Average-Award Ergodic Markov Decision Processes}, author = {Gong, Hao and Wang, Mengdi}, booktitle = {Proceedings of the 2nd Conference on Learning for Dynamics and Control}, pages = {862--883}, year = {2020}, editor = {Bayen, Alexandre M. and Jadbabaie, Ali and Pappas, George and Parrilo, Pablo A. and Recht, Benjamin and Tomlin, Claire and Zeilinger, Melanie}, volume = {120}, series = {Proceedings of Machine Learning Research}, month = {10--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v120/gong20a/gong20a.pdf}, url = {https://proceedings.mlr.press/v120/gong20a.html}, abstract = {In light of the Bellman duality, we propose a novel value-policy gradient algorithm to explore and act in infinite-horizon Average-reward Markov Decision Process (AMDP) and show that it has sublinear regret. The algorithm is motivated by the Bellman saddle point formulation. It learns the optimal state-action distribution, which encodes a randomized policy, by interacting with the environment along a single trajectory and making primal-dual updates. The key to the analysis is to establish a connection between the min-max duality gap of Bellman saddle point and the cumulative regret of the learning agent. We show that, for ergodic AMDPs with finite state space $\mathcal{S}$ and action space $\mathcal{A}$ and uniformly bounded mixing times, the algorithm’s $T$-time step regret is $$ R(T)=\tilde{\mathcal{O}}\left( \left(t_{mix}^*\right)^2 \tau^{\frac{3}{2}} \sqrt{(\tau^3 + |\mathcal{A}|) |\mathcal{S}| T} \right), $$ where $t_{mix}^*$ is the worst-case mixing time, $\tau$ is an ergodicity parameter, $T$ is the number of time steps and $\tilde{\mathcal{O}}$ hides polylog factors.} }
Endnote
%0 Conference Paper %T A Duality Approach for Regret Minimization in Average-Award Ergodic Markov Decision Processes %A Hao Gong %A Mengdi Wang %B Proceedings of the 2nd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2020 %E Alexandre M. Bayen %E Ali Jadbabaie %E George Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire Tomlin %E Melanie Zeilinger %F pmlr-v120-gong20a %I PMLR %P 862--883 %U https://proceedings.mlr.press/v120/gong20a.html %V 120 %X In light of the Bellman duality, we propose a novel value-policy gradient algorithm to explore and act in infinite-horizon Average-reward Markov Decision Process (AMDP) and show that it has sublinear regret. The algorithm is motivated by the Bellman saddle point formulation. It learns the optimal state-action distribution, which encodes a randomized policy, by interacting with the environment along a single trajectory and making primal-dual updates. The key to the analysis is to establish a connection between the min-max duality gap of Bellman saddle point and the cumulative regret of the learning agent. We show that, for ergodic AMDPs with finite state space $\mathcal{S}$ and action space $\mathcal{A}$ and uniformly bounded mixing times, the algorithm’s $T$-time step regret is $$ R(T)=\tilde{\mathcal{O}}\left( \left(t_{mix}^*\right)^2 \tau^{\frac{3}{2}} \sqrt{(\tau^3 + |\mathcal{A}|) |\mathcal{S}| T} \right), $$ where $t_{mix}^*$ is the worst-case mixing time, $\tau$ is an ergodicity parameter, $T$ is the number of time steps and $\tilde{\mathcal{O}}$ hides polylog factors.
APA
Gong, H. & Wang, M.. (2020). A Duality Approach for Regret Minimization in Average-Award Ergodic Markov Decision Processes. Proceedings of the 2nd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 120:862-883 Available from https://proceedings.mlr.press/v120/gong20a.html.

Related Material