  We consider the offline reinforcement learning problem, where the aim is to
learn a decision making policy from logged data. Offline RL -- particularly
when coupled with (value) function approximation to allow for generalization in
large or continuous state spaces -- is becoming increasingly relevant in
practice, because it avoids costly and time-consuming online data collection
and is well suited to safety-critical domains. Existing sample complexity
guarantees for offline value function approximation methods typically require
both (1) distributional assumptions (i.e., good coverage) and (2)
representational assumptions (i.e., ability to represent some or all $Q$-value
functions) stronger than what is required for supervised learning. However, the
necessity of these conditions and the fundamental limits of offline RL are not
well understood in spite of decades of research. This led Chen and Jiang (2019)
to conjecture that concentrability (the most standard notion of coverage) and
realizability (the weakest representation condition) alone are not sufficient
for sample-efficient offline RL. We resolve this conjecture in the positive by
proving that in general, even if both concentrability and realizability are
satisfied, any algorithm requires sample complexity polynomial in the size of
the state space to learn a non-trivial policy.
  Our results show that sample-efficient offline reinforcement learning
requires either restrictive coverage conditions or representation conditions
that go beyond supervised learning, and highlight a phenomenon called
over-coverage which serves as a fundamental barrier for offline value function
approximation methods. A consequence of our results for reinforcement learning
with linear function approximation is that the separation between online and
offline RL can be arbitrarily large, even in constant dimension.
