arXiv:2206.07528v3 Announce Type: replace
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 symmetric 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 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.