有偏反馈线性强盗系统中不公平的代价
The price of unfairness in linear bandits with biased feedback
摘要：
本文研究了带有偏线性反馈的公平序贯决策问题。在每一轮，玩家选择一个由协变量和敏感属性描述的动作。感知的奖励是所选动作的协变量的线性组合，但玩家只观察到对这种奖励的有偏见的评价，这取决于敏感属性。为了描述该问题的难度，我们设计了一个阶段消除算法，纠正不公平的评价，并建立其遗憾的上界。我们表明，最坏情况下的遗憾是小于$\mathcal{O}(κ_*^{1/3}\log(T)^{1/3}T^{2/3})$，其中$κ_*$是一个明确的几何常数，其特征在于偏差估计的难度。我们证明了下界的最坏情况下的遗憾的一些行动表明，这一速度是紧到一个可能的子对数因子。我们还得到了间隙依赖的上界的遗憾，和匹配的下界的一些问题instance.Interestingly，这些结果揭示了一个政权之间的过渡，该政权的问题是一样困难，因为它的无偏对应，和政权，它可以更难。
英文摘要：
In this paper, we study the problem of fair sequential decision making with biased linear bandit feedback. At each round, a player selects an action described by a covariate and by a sensitive attribute. The perceived reward is a linear combination of the covariates of the chosen action, but the player only observes a biased evaluation of this reward, depending on the sensitive attribute. To characterize the difficulty of this problem, we design a phased elimination algorithm that corrects the unfair evaluations, and establish upper bounds on its regret. We show that the worst-case regret is smaller than $\mathcal{O}(κ_*^{1/3}\log(T)^{1/3}T^{2/3})$, where $κ_*$ is an explicit geometrical constant characterizing the difficulty of bias estimation. We prove lower bounds on the worst-case regret for some sets of actions showing that this rate is tight up to a possible sub-logarithmic factor. We also derive gap-dependent upper bounds on the regret, and matching lower bounds for some problem instance.Interestingly, these results reveal a transition between a regime where the problem is as difficult as its unbiased counterpart, and a regime where it can be much harder.
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