Corruption-Robust Lipschitz Contextual Search
Shiliang Zuo
TL;DR
This work addresses learning an $L$-Lipschitz function $f:[0,1]^d\to[0,L]$ with adversarial, corrupted binary feedback over $T$ rounds. It introduces agnostic checking, a technique that robustifies adaptive discretization against unknown corruption budget $C$, and develops corruption-robust algorithms for both absolute and pricing losses. In 1D, the absolute-loss method achieves regret $L\cdot O(C\log T)$, while in higher dimensions it scales as $L\cdot O_d\big(C\log T + T^{(d-1)/d}\big)$; for the pricing loss, the bound is $\widetilde{O}\big(T^{d/(d+1)} + C T^{1/(d+1)}\big)$. The paper also provides lower bounds showing near-tightness and extends the framework to higher dimensions, with implications for robust contextual search and dynamic pricing under adversarial noise.
Abstract
I study the problem of learning a Lipschitz function with corrupted binary signals. The learner tries to learn a $L$-Lipschitz function $f: [0,1]^d \rightarrow [0, L]$ that the adversary chooses. There is a total of $T$ rounds. In each round $t$, the adversary selects a context vector $x_t$ in the input space, and the learner makes a guess to the true function value $f(x_t)$ and receives a binary signal indicating whether the guess is high or low. In a total of $C$ rounds, the signal may be corrupted, though the value of $C$ is \emph{unknown} to the learner. The learner's goal is to incur a small cumulative loss. This work introduces the new algorithmic technique \emph{agnostic checking} as well as new analysis techniques. I design algorithms which: for the symmetric loss, the learner achieves regret $L\cdot O(C\log T)$ with $d = 1$ and $L\cdot O_d(C\log T + T^{(d-1)/d})$ with $d > 1$; for the pricing loss, the learner achieves regret $L\cdot \widetilde{O} (T^{d/(d+1)} + C\cdot T^{1/(d+1)})$.