G : Bandits , Experts and Games 09 / 19 / 16 Lecture 3 : Lower Bounds for Bandit Algorithms

Note that (2) implies (1) since: if regret is high in expectation over problem instances, then there exists at least one problem instance with high regret. Also, (1) implies (2) if |F| is a constant. This can be seen as follows: suppose we know that for any algorithm we have high regret (say H) with one problem instance in F and low regret with all other instances in F , then, taking a uniform distribution over F , we can say that any algorithm has expected regret at least H/|F|. (So this argument breaks if |F| is large.) If we prove a stronger version of (1) that says that for any algorithm, regret is high for a constant fraction of the problem instances in F , then, considering a uniform distribution over F , this implies (2) regardless of whether |F| is large or not. In this lecture, for proving lower bounds, we consider 0-1 rewards and the following family of problem instances (with fixed to be adjusted in the analysis):