  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.
