  We consider the problem of preference based reinforcement learning (PbRL),
where, unlike traditional reinforcement learning, an agent receives feedback
only in terms of a 1 bit (0/1) preference over a trajectory pair instead of
absolute rewards for them. The success of the traditional RL framework
crucially relies on the underlying agent-reward model, which, however, depends
on how accurately a system designer can express an appropriate reward function
and often a non-trivial task. The main novelty of our framework is the ability
to learn from preference-based trajectory feedback that eliminates the need to
hand-craft numeric reward models. This paper sets up a formal framework for the
PbRL problem with non-markovian rewards, where the trajectory preferences are
encoded by a generalized linear model of dimension $d$. Assuming the transition
model is known, we then propose an algorithm with almost optimal regret
guarantee of $\tilde {\mathcal{O}}\left( SH d \log (T / \delta) \sqrt{T}
\right)$. We further, extend the above algorithm to the case of unknown
transition dynamics, and provide an algorithm with near optimal regret
guarantee $\widetilde{\mathcal{O}}((\sqrt{d} + H^2 + |\mathcal{S}|)\sqrt{dT}
+\sqrt{|\mathcal{S}||\mathcal{A}|TH} )$. To the best of our knowledge, our work
is one of the first to give tight regret guarantees for preference based RL
problems with trajectory preferences.
