  We consider offline reinforcement learning (RL) with preference feedback in
which the implicit reward is a linear function of an unknown parameter. Given
an offline dataset, our objective consists in ascertaining the optimal action
for each state, with the ultimate goal of minimizing the {\em simple regret}.
We propose an algorithm, \underline{RL} with \underline{L}ocally
\underline{O}ptimal \underline{W}eights or {\sc RL-LOW}, which yields a simple
regret of $\exp ( - \Omega(n/H) )$ where $n$ is the number of data samples and
$H$ denotes an instance-dependent hardness quantity that depends explicitly on
the suboptimality gap of each action. Furthermore, we derive a
first-of-its-kind instance-dependent lower bound in offline RL with preference
feedback. Interestingly, we observe that the lower and upper bounds on the
simple regret match order-wise in the exponent, demonstrating order-wise
optimality of {\sc RL-LOW}. In view of privacy considerations in practical
applications, we also extend {\sc RL-LOW} to the setting of
$(\varepsilon,\delta)$-differential privacy and show, somewhat surprisingly,
that the hardness parameter $H$ is unchanged in the asymptotic regime as $n$
tends to infinity; this underscores the inherent efficiency of {\sc RL-LOW} in
terms of preserving the privacy of the observed rewards. Given our focus on
establishing instance-dependent bounds, our work stands in stark contrast to
previous works that focus on establishing worst-case regrets for offline RL
with preference feedback.
