We study the exploration problem with approximate linear action-value functions in episodic reinforcement learning under the notion of low inherent Bellman error, a condition normally employed to show convergence of approximate value iteration. First we relate this condition to other common frameworks and show that it is strictly more general than the low rank (or linear) MDP assumption of prior work. Second we provide an algorithm with a high probability regret bound $\widetilde O(\sum_{t=1}^H d_t \sqrt{K} + \sum_{t=1}^H \sqrt{d_t} \IBE K)$ where $H$ is the horizon, $K$ is the number of episodes, $\IBE$ is the value if the inherent Bellman error and $d_t$ is the feature dimension at timestep $t$. In addition, we show that the result is unimprovable beyond constants and logs by showing a matching lower bound. This has two important consequences: 1) the algorithm has the optimal statistical rate for this setting which is more general than prior work on low-rank MDPs 2) the lack of closedness (measured by the inherent Bellman error) is only amplified by $\sqrt{d_t}$ despite working in the online setting. Finally, the algorithm reduces to the celebrated \textsc{LinUCB} when $H=1$ but with a different choice of the exploration parameter that allows handling misspecified contextual linear bandits. While computational tractability questions remain open for the MDP setting, this enriches the class of MDPs with a linear representation for the action-value function where statistically efficient reinforcement learning is possible.

研究在近似线性行动价值函数的情况下，基于低内在Bellman误差的探索问题，给出了一种算法，其高概率的遗憾上界与特征维数和Bellman误差有关，同时将其与先前的工作进行了比较，在线性MDP的情况下，证明了这个算法具有统计效率。

学习具有低基本Bellman误差的近最优策略