## Abstract

Modern reinforcement learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation trade-off. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic settingwith linear dynamics and linear rewards, forwhich only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial run time and polynomial sample complexity in this linear setting, without requiring a "simulator"or additional assumptions. Concretely, we prove that an optimistic modification of least-squares value iteration-a classical algorithm frequently studied in the linear setting-achieves O (√ d3H3T ) regret, where d is the ambient dimension of feature space, H is the length of each episode, and T is the total number of steps. Importantly, such regret is independent of the number of states and actions.

Original language | English (US) |
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Pages (from-to) | 1496-1521 |

Number of pages | 26 |

Journal | Mathematics of Operations Research |

Volume | 48 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2023 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Computer Science Applications
- Management Science and Operations Research

## Keywords

- episodic MDP
- exploration
- linear function approximation
- reinforcement learning