Low-complexity models such as linear function representation play a pivotal role in enabling sample-efficient reinforcement learning (RL). The current paper pertains to a scenario with value-based linear representation, which postulates the linear realizability of the optimal Q-function (also called the linear $Q^{star}$ problem). While linear realizability alone does not allow for sample-efficient solutions in general, the presence of a large sub-optimality gap is a potential game changer, depending on the sampling mechanism in use. Informally, sample efficiency is achievable with a large sub-optimality gap when a generative model is available but is unfortunately infeasible when we turn to standard online RL settings. In this paper, we make progress towards understanding this linear $Q^{star}$ problem by investigating a new sampling protocol, which draws samples in an online/exploratory fashion but allows one to backtrack and revisit previous states in a controlled and infrequent manner. This protocol is more flexible than the standard online RL setting, while being practically relevant and far more restrictive than the generative model. We develop an algorithm tailored to this setting, achieving a sample complexity that scales polynomially with the feature dimension, the horizon, and the inverse sub-optimality gap, but not the size of the state/action space. Our findings underscore the fundamental interplay between sampling protocols and low-complexity structural representation in RL.
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