Gap-Dependent Unsupervised Exploration for Reinforcement Learning
Jingfeng Wu, Vladimir Braverman, Lin Yang
Abstract
For the problem of task-agnostic reinforcement learning (RL), an agent first collects samples from an unknown environment without the supervision of reward signals, then is revealed with a reward and is asked to compute a corresponding near-optimal policy. Existing approaches mainly concern the worst-case scenarios, in which no structural information of the reward/transition-dynamics is utilized. Therefore the best sample upper bound is $\propto\widetilde{\mathcal{O}}(1/\epsilon^2)$, where $\epsilon>0$ is the target accuracy of the obtained policy, and can be overly pessimistic. To tackle this issue, we provide an efficient algorithm that utilizes a gap parameter, $\rho>0$, to reduce the amount of exploration. In particular, for an unknown finite-horizon Markov decision process, the algorithm takes only $\widetilde{\mathcal{O}} (1/\epsilon \cdot (H^3SA / \rho + H^4 S^2 A) )$ episodes of exploration, and is able to obtain an $\epsilon$-optimal policy for a post-revealed reward with sub-optimality gap at least $\rho$, where $S$ is the number of states, $A$ is the number of actions, and $H$ is the length of the horizon, obtaining a nearly quadratic saving in terms of $\epsilon$. We show that, information-theoretically, this bound is nearly tight for $\rho < \Theta(1/(HS))$ and $H>1$. We further show that $\propto\widetilde{\mathcal{O}}(1)$ sample bound is possible for $H=1$ (i.e., multi-armed bandit) or with a sampling simulator, establishing a stark separation between those settings and the RL setting.
BibTeX
@InProceedings{pmlr-v151-wu22b,
title = { Gap-Dependent Unsupervised Exploration for Reinforcement Learning },
author = {Wu, Jingfeng and Braverman, Vladimir and Yang, Lin},
booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics},
pages = {4109--4131},
year = {2022},
editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel},
volume = {151},
series = {Proceedings of Machine Learning Research},
month = {28--30 Mar},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v151/wu22b/wu22b.pdf},
url = {https://proceedings.mlr.press/v151/wu22b.html},
abstract = { For the problem of task-agnostic reinforcement learning (RL), an agent first collects samples from an unknown environment without the supervision of reward signals, then is revealed with a reward and is asked to compute a corresponding near-optimal policy. Existing approaches mainly concern the worst-case scenarios, in which no structural information of the reward/transition-dynamics is utilized. Therefore the best sample upper bound is $\propto\widetilde{\mathcal{O}}(1/\epsilon^2)$, where $\epsilon>0$ is the target accuracy of the obtained policy, and can be overly pessimistic. To tackle this issue, we provide an efficient algorithm that utilizes a gap parameter, $\rho>0$, to reduce the amount of exploration. In particular, for an unknown finite-horizon Markov decision process, the algorithm takes only $\widetilde{\mathcal{O}} (1/\epsilon \cdot (H^3SA / \rho + H^4 S^2 A) )$ episodes of exploration, and is able to obtain an $\epsilon$-optimal policy for a post-revealed reward with sub-optimality gap at least $\rho$, where $S$ is the number of states, $A$ is the number of actions, and $H$ is the length of the horizon, obtaining a nearly quadratic saving in terms of $\epsilon$. We show that, information-theoretically, this bound is nearly tight for $\rho < \Theta(1/(HS))$ and $H>1$. We further show that $\propto\widetilde{\mathcal{O}}(1)$ sample bound is possible for $H=1$ (i.e., multi-armed bandit) or with a sampling simulator, establishing a stark separation between those settings and the RL setting. }
}