Density-Based Algorithms for Corruption-Robust Contextual Search and Convex Optimization

Abstract: 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 $\epsilon$-ball and the absolute loss. For the $\epsilon$-ball loss, we give a tight regret bound of $O(C + d \log(1/\epsilon))$ improving over the $O(d^3 \log(1/\epsilon) \log^2(T) + C \log(T) \log(1/\epsilon))$ bound of Krishnamurthy et al (Operations Research '23). For the absolute loss, we give an efficient algorithm with regret $O(C+d \log T)$. To tackle the absolute 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.