We study the problem of contextual search, a generalization of binary search in higher dimensions, in the adversarial noise model. Let $d$ be the dimension of the problem, $T$ be the time horizon and $C$ be the total amount of adversarial noise in the system. We focus on the $ε$-ball and the symmetric loss. For the $ε$-ball loss, we give a tight regret bound of $O(C + d \log(1/ε))$ improving over the $O(d^3 \log(1/ε) \log^2(T) + C \log(T) \log(1/ε))$ bound of Krishnamurthy et al (Operations Research '23). For the symmetric loss, we give an efficient algorithm with regret $O(C+d \log T)$. To tackle the symmetric loss case, we study the more general setting of Corruption-Robust Convex Optimization with Subgradient feedback, which is of independent interest. Our techniques are a significant departure from prior approaches. Specifically, we keep track of density functions over the candidate target vectors instead of a knowledge set consisting of the candidate target vectors consistent with the feedback obtained.

翻译：本文研究上下文搜索问题——即高维空间中二分搜索的推广——在对抗性噪声模型下的表现。设$d$为问题维度，$T$为时间范围，$C$为系统中对抗性噪声总量。我们重点关注$ε$-球损失与对称损失两种情况。对于$ε$-球损失，我们给出了$O(C + d \log(1/ε))$的严格遗憾界，改进了Krishnamurthy等人（Operations Research '23）提出的$O(d^3 \log(1/ε) \log^2(T) + C \log(T) \log(1/ε))$界限。针对对称损失，我们提出了一种高效算法，其遗憾界为$O(C+d \log T)$。为处理对称损失情形，我们进一步研究了具有次梯度反馈的抗噪声凸优化这一更普适的设定，该研究本身具有独立价值。我们的技术方法与先前研究存在显著差异：具体而言，我们通过跟踪候选目标向量上的密度函数来推进算法，而非维护与已获反馈一致的候选目标向量所构成的知识集合。