Fair Exploration and Exploitation

In this paper we consider the contextual bandit problem with a finite (or infinite and clustered) context set. We consider the fully adversarial problem in which, apart from having bounded losses, there are no assumptions whatsoever on the generation of the contexts and losses. In our problem we assume that the context set is partitioned into a set of protected groups. At the start of each trial we are given a probability distribution over the context set and are required (on that trial) to be fair with respect to that distribution, in that if the context (for that trial) was drawn from the distribution then our choice of action would be unbiased towards any protected group. We develop an algorithm FexEx for this problem which has remarkable efficiency, having a space and per-trial time complexity at most linear in the dimensionality of the policy space. FexEx can handle non-stationarity, in that its regret can be bounded with respect to any sequence of policies satisfying the fairness constraints. For such a sequence the regret bound of FexEx is essentially the same as that of running Exp3.S for each context independently (an approach that does not satisfy the fairness constraints).