$Q$-learning with function approximation is one of the most popular methods in reinforcement learning. Though the idea of using function approximation was proposed at least $60$ years ago, even in the simplest setup, i.e, approximating $Q$-functions with linear functions, it is still an open problem how to design a provably efficient algorithm that learns a near-optimal policy. The key challenges are how to efficiently explore the state space and how to decide when to stop exploring in conjunction with the function approximation scheme. The current paper presents a provably efficient algorithm for $Q$-learning with linear function approximation. Under certain regularity assumptions, our algorithm, Difference Maximization $Q$-learning (DMQ), combined with linear function approximation, returns a near optimal policy using polynomial number of trajectories. Our algorithm introduces a new notion, the Distribution Shift Error Checking (DSEC) oracle. This oracle tests whether there exists a function in the function class that predicts well on a distribution $\mathcal{D}_1$, but predicts poorly on another distribution $\mathcal{D}_2$, where $\mathcal{D}_1$ and $\mathcal{D}_2$ are distributions over states induced by two different exploration policies. For the linear function class, this oracle is equivalent to solving a top eigenvalue problem. We believe our algorithmic insights, especially the DSEC oracle, are also useful in designing and analyzing reinforcement learning algorithms with general function approximation.

本文介绍了一种可证明有效的基于线性函数逼近的 $Q$-learning 算法，提出了 DSEC oracle 概念，该算法使用多项式数量的轨迹返回近似最优策略，并可用于设计和分析具有一般函数逼近的强化学习算法。

通过分布偏移误差检查预示着的函数逼近$Q$学习的可证明高效性