We study the model-based reward-free reinforcement learning with linear function approximation for episodic Markov decision processes (MDPs). In this setting, the agent works in two phases. In the exploration phase, the agent interacts with the environment and collects samples without the reward. In the planning phase, the agent is given a specific reward function and uses samples collected from the exploration phase to learn a good policy. We propose a new provably efficient algorithm, called UCRL-RFE under the Linear Mixture MDP assumption, where the transition probability kernel of the MDP can be parameterized by a linear function over certain feature mappings defined on the triplet of state, action, and next state. We show that to obtain an $\epsilon$-optimal policy for arbitrary reward function, UCRL-RFE needs to sample at most $\tilde O(H^5d^2\epsilon^{-2})$ episodes during the exploration phase. Here, $H$ is the length of the episode, $d$ is the dimension of the feature mapping. We also propose a variant of UCRL-RFE using Bernstein-type bonus and show that it needs to sample at most $\tilde O(H^4d(H + d)\epsilon^{-2})$ to achieve an $\epsilon$-optimal policy. By constructing a special class of linear Mixture MDPs, we also prove that for any reward-free algorithm, it needs to sample at least $\tilde \Omega(H^2d\epsilon^{-2})$ episodes to obtain an $\epsilon$-optimal policy. Our upper bound matches the lower bound in terms of the dependence on $\epsilon$ and the dependence on $d$ if $H \ge d$.

本文研究线性函数逼近的无奖励强化学习与马尔可夫决策过程，并提出了一种新算法UCRL-RFE，其中使用线性函数对状态、动作和下一个状态进行特征映射，能够在探索阶段最多采样$\tilde{\mathcal{O}}(H^5d^2\epsilon^{-2})$周期，用于构建奖励函数并实现任意奖励下的$\epsilon$-最优策略。

使用线性函数逼近的无奖励模型强化学习