  While much progress has been made in understanding the minimax sample
complexity of reinforcement learning (RL) -- the complexity of learning on the
"worst-case" instance -- such measures of complexity often do not capture the
true difficulty of learning. In practice, on an "easy" instance, we might hope
to achieve a complexity far better than that achievable on the worst-case
instance. In this work we seek to understand the "instance-dependent"
complexity of learning near-optimal policies (PAC RL) in the setting of RL with
linear function approximation. We propose an algorithm, \textsc{Pedel}, which
achieves a fine-grained instance-dependent measure of complexity, the first of
its kind in the RL with function approximation setting, thereby capturing the
difficulty of learning on each particular problem instance. Through an explicit
example, we show that \textsc{Pedel} yields provable gains over low-regret,
minimax-optimal algorithms and that such algorithms are unable to hit the
instance-optimal rate. Our approach relies on a novel online experiment
design-based procedure which focuses the exploration budget on the "directions"
most relevant to learning a near-optimal policy, and may be of independent
interest.
