## Abstract

We investigate the hardness of online reinforcement learning in fixed horizon, sparse linear Markov decision process (MDP), with a special focus on the high-dimensional regime where the ambient dimension is larger than the number of episodes. Our contribution is two-fold. First, we provide a lower bound showing that linear regret is generally unavoidable in this case, even if there exists a policy that collects well-conditioned data. The lower bound construction uses an MDP with a fixed number of states while the number of actions scales with the ambient dimension. Note that when the horizon is fixed to one, the case of linear stochastic bandits, the linear regret can be avoided. Second, we show that if the learner has oracle access to a policy that collects well-conditioned data then a variant of Lasso fitted Q-iteration enjoys a nearly dimension free regret of O^{e}(s^{2}/^{3}N^{2}/^{3}) where N is the number of episodes and s is the sparsity level. This shows that in the large-action setting, the difficulty of learning can be attributed to the difficulty of finding a good exploratory policy.

Original language | English (US) |
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Pages (from-to) | 316-324 |

Number of pages | 9 |

Journal | Proceedings of Machine Learning Research |

Volume | 130 |

State | Published - 2021 |

Event | 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021 - Virtual, Online, United States Duration: Apr 13 2021 → Apr 15 2021 |

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability