  We study human-in-the-loop reinforcement learning (RL) with trajectory
preferences, where instead of receiving a numeric reward at each step, the
agent only receives preferences over trajectory pairs from a human overseer.
The goal of the agent is to learn the optimal policy which is most preferred by
the human overseer. Despite the empirical successes, the theoretical
understanding of preference-based RL (PbRL) is only limited to the tabular
case. In this paper, we propose the first optimistic model-based algorithm for
PbRL with general function approximation, which estimates the model using
value-targeted regression and calculates the exploratory policies by solving an
optimistic planning problem. Our algorithm achieves the regret of $\tilde{O}
(\operatorname{poly}(d H) \sqrt{K} )$, where $d$ is the complexity measure of
the transition and preference model depending on the Eluder dimension and
log-covering numbers, $H$ is the planning horizon, $K$ is the number of
episodes, and $\tilde O(\cdot)$ omits logarithmic terms. Our lower bound
indicates that our algorithm is near-optimal when specialized to the linear
setting. Furthermore, we extend the PbRL problem by formulating a novel problem
called RL with $n$-wise comparisons, and provide the first sample-efficient
algorithm for this new setting. To the best of our knowledge, this is the first
theoretical result for PbRL with (general) function approximation.
