Before revision:  We then establish a positive result on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above . We show that our sample complexity is tight with matching upper and lower bounds . Finally, we extend our learning results to the setting where the follower plays in a Markov Decision Process (MDP), and the setting where the leader and the follower act simultaneously .
After revision:  We then establish a positive result on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above , with matching lower bounds in the dependency on the gap, error tolerance, and the size of the action spaces. Overall, our results unveil unique challenges in learning Stackelberg equilibria under noisy bandit feedback, which we hope could shed light on future research on this topic .