Finite-time lower bounds for the two-armed bandit problem

We obtain minimax lower bounds on the regret for the classical two-armed bandit problem. We provide a finite-sample minimax version of the well-known log n asymptotic lower bound of Lai and Robbins (1985). The finite-time lower bound allows us to derive conditions for the amount of time necessary to make any significant gain over a random guessing strategy. These bounds depend on the class of possible distributions of the rewards associated with the arms. For example, in contrast to the log n asymptotic results on the regret, we show that the minimax regret is achieved by mere random guessing under fairly mild conditions on the set of allowable configurations of the two arms. That is, we show that for every allocation rule and for every n, there is a configuration such that the regret at time n is at least 1-ε times the regret of random guessing, where ε is any small positive constant