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 $O(T^{-1/2})$.

翻译：我们研究机会性逼近性问题：这是Blackwell逼近性的一种推广，其中学习器希望在对手将其行动空间限制于某个子集时获得更强的保证（即逼近一个更小的集合）。Bernstein等人（2014）于2014年提出了该问题，并给出了一种保证机会性逼近性具有次线性逼近速率的算法。然而，该算法需要能够生成对手行动的校准在线预测，而该问题的标准实现需要时间维度呈指数级增长，并导致逼近速率按$T^{-O(1/d)}$缩放。本文提出了一种高效的机会性逼近性算法，其达到$O(T^{-1/4})$的速率（以及一种达到$O(T^{-1/3})$速率的非高效算法），从而绕过了对在线校准子程序的需求。此外，在对手行动集维度不超过二的情况下，我们证明可以获得$O(T^{-1/2})$的最优速率。