We consider a best arm identification (BAI) problem for stochastic bandits with adversarial corruptions in the fixed-budget setting of $T$ steps. We design a novel randomized algorithm, Probabilistic Sequential Shrinking$(u)$ (PSS$(u)$), which is agnostic to the amount of corruptions. When the amount of corruptions per step (CPS) is below a threshold, PSS$(u)$ identifies the best arm or item with probability tending to $1$ as $T\rightarrow\infty$. Otherwise, the optimality gap of the identified item degrades gracefully with the CPS. We argue that such a bifurcation is necessary. In addition, we show that when the CPS is sufficiently large, no algorithm can achieve a BAI probability tending to $1$ as $T\rightarrow \infty$. In PSS$(u)$, the parameter $u$ serves to balance between the optimality gap and success probability. En route, the injection of randomization is shown to be essential to mitigate the impact of corruptions. Indeed, we show that PSS$(u)$ has a better performance than its deterministic analogue, the Successive Halving (SH) algorithm by Karnin et al. (2013). PSS$(2)$'s performance guarantee matches SH's when there is no corruption. Finally, we identify a term in the exponent of the failure probability of PSS$(u)$ that generalizes the common $H_2$ term for BAI under the fixed-budget setting.

翻译：我们认为,在固定预算($T$)的设置中,对有对抗性腐败的暴徒来说,最好的武器识别(BAI)是最好的武器识别(BAI)问题。我们设计了一种新的随机算法,即概率序列递减(u)美元(PSS$(u),这与腐败的数额是不可知的。当每一步(CPS)的腐败数额低于阈值时,PSS$(u)确定最好的武器或物品的概率为1美元($),或者说很可能是1美元($T\rightarrowle\infty $)。否则,所查明的项目的最佳性差会与CPS相比优优优于概率。我们证明,当CPS(美元)的常规性平价程(PSS)期比稳定性平价程(美元)的准确性能效果要好一些。