Efficient Opportunistic Approachability

We study the problem of opportunistic approachability: a generalization of Blackwell approachability where the learner would like to obtain stronger guarantees (i.e., approach a smaller set) when their adversary limits themselves to a subset of their possible action space. Bernstein et al. (2014) introduced this problem in 2014 and presented an algorithm that guarantees sublinear approachability rates for opportunistic approachability. However, this algorithm requires the ability to produce calibrated online predictions of the adversary's actions, a problem whose standard implementations require time exponential in the ambient dimension and result in approachability rates that scale as $T^{-O(1/d)}$. In this paper, we present an efficient algorithm for opportunistic approachability that achieves a rate of $O(T^{-1/4})$ (and an inefficient one that achieves a rate of $O(T^{-1/3})$), bypassing the need for an online calibration subroutine. Moreover, in the case where the dimension of the adversary's action set is at most two, we show it is possible to obtain the optimal rate of $O(T^{-1/2})$.