Stabilizing Q-learning with Linear Architectures for Provably Efficient Learning

The $Q$-learning algorithm is a simple and widely-used stochastic approximation scheme for reinforcement learning, but the basic protocol can exhibit instability in conjunction with function approximation. Such instability can be observed even with linear function approximation. In practice, tools such as target networks and experience replay appear to be essential, but the individual contribution of each of these mechanisms is not well understood theoretically. This work proposes an exploration variant of the basic $Q$-learning protocol with linear function approximation. Our modular analysis illustrates the role played by each algorithmic tool that we adopt: a second order update rule, a set of target networks, and a mechanism akin to experience replay. Together, they enable state of the art regret bounds on linear MDPs while preserving the most prominent feature of the algorithm, namely a space complexity independent of the number of step elapsed. We show that the performance of the algorithm degrades very gracefully under a novel and more permissive notion of approximation error. The algorithm also exhibits a form of instance-dependence, in that its performance depends on the "effective" feature dimension.