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

TL;DR

This paper introduces density-based algorithms for adversarially noisy contextual search and convex optimization, achieving tight regret bounds and improving robustness over previous methods.

Contribution

It presents a novel density-tracking approach for corruption-robust contextual search and convex optimization, with improved regret bounds and efficiency.

Findings

Achieves tight regret bound of O(C + d log(1/ε)) for ε-ball loss.

Provides an efficient algorithm with regret O(C + d log T) for symmetric loss.

Introduces a density function tracking technique as a departure from prior knowledge set methods.

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.