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

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(d3 log(1/ε) log2(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.