From Dueling Bandits to Episodic MDPs: Learning with Adversarial Preference Feedback

In dueling bandits, the learner compares two arms in each round and observes only which one is preferred. This paper asks what changes when the compared pair also affects the next state, as in a recommendation interface where the pair shown to a user affects the user's next context. We formulate this problem as a state-wise preference-based MDP, an episodic tabular MDP in which the learner selects a pair of arms at each visited state, observes only a noisy comparison between them, and moves to a next state that may depend on the pair. The preferences may change adversarially across episodes, and the loss of a pair is defined through Borda scores. We show that the minimax regret of this problem under known transitions is $\tilde{\Theta }(\min \{HT,(H^{3}SK{)}^{1/3}{T}^{2/3}\})$, where $H$, $S$, $K$, and $T$ are the numbers of steps per episode, states, arms, and episodes, and that this rate is achieved by an algorithm based on online linear optimization over the set of occupancy measures. If the pair did not affect the next state, each state would be a separate dueling bandit, and the $T^{2/3}$ term of the minimax regret would be smaller by a factor of $H^{1/3}$, demonstrating the cost of having action-dependent state transitions. We also give a policy optimization algorithm with closed-form updates, whose regret is $\tilde{O}((H^{4}S{K}^3{)}^{1/3}{T}^{2/3})$ up to lower-order terms, and we extend both algorithms to unknown transitions.