  A fundamental question in the theory of reinforcement learning is: suppose
the optimal $Q$-function lies in the linear span of a given $d$ dimensional
feature mapping, is sample-efficient reinforcement learning (RL) possible? The
recent and remarkable result of Weisz et al. (2020) resolved this question in
the negative, providing an exponential (in $d$) sample size lower bound, which
holds even if the agent has access to a generative model of the environment.
One may hope that this information theoretic barrier for RL can be circumvented
by further supposing an even more favorable assumption: there exists a
\emph{constant suboptimality gap} between the optimal $Q$-value of the best
action and that of the second-best action (for all states). The hope is that
having a large suboptimality gap would permit easier identification of optimal
actions themselves, thus making the problem tractable; indeed, provided the
agent has access to a generative model, sample-efficient RL is in fact possible
with the addition of this more favorable assumption.
  This work focuses on this question in the standard online reinforcement
learning setting, where our main result resolves this question in the negative:
our hardness result shows that an exponential sample complexity lower bound
still holds even if a constant suboptimality gap is assumed in addition to
having a linearly realizable optimal $Q$-function. Perhaps surprisingly, this
implies an exponential separation between the online RL setting and the
generative model setting. Complementing our negative hardness result, we give
two positive results showing that provably sample-efficient RL is possible
either under an additional low-variance assumption or under a novel
hypercontractivity assumption (both implicitly place stronger conditions on the
underlying dynamics model).
