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 $H \ge d$.
翻译:我们研究基于模型的无奖赏强化学习,其直线函数近似于Sindic Markov 决策程序(MDPs) 。 在这种环境下,代理器分两个阶段工作。 在勘探阶段,代理器与环境互动,收集样本而没有奖赏。 在规划阶段,代理器被赋予特定奖赏功能,并使用从勘探阶段收集的样本来学习良好的政策。 我们在线性混合 mixture 假设下, 名为 UCRL-RFE 的新型高效算法, 称为UCRL- RFE, 其中MDP的过渡概率内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 等内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 等内核内核内核内核内核内核内核内核内核内核内核内 等内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内