Density-Based Algorithms for Corruption-Robust Contextual Search and Convex Optimization
Renato Paes Leme, Chara Podimata, Jon Schneider
TL;DR
This work tackles corruption-robust contextual search and corruption-robust convex optimization under adversarial noise. It introduces density-based algorithms that maintain a log-concave distribution over the set of candidate targets and use centroid queries, contrasting with prior knowledge-set approaches. The results yield tight regret bounds: for the $\varepsilon$-ball loss, $O(C + d\log(1/\varepsilon))$; for the symmetric loss, $O(C + d\log T)$; and a CRoCO framework achieving $O(C + d\log T)$ regret with subgradient feedback. The approach offers a principled, robust alternative to cutting-plane methods, with potential practical and theoretical impact in online learning under corruption.
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 $ε$-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.