Corruption-Robust Lipschitz Contextual Search

TL;DR

This paper develops algorithms for learning Lipschitz functions under adversarial binary signals with unknown corruption, introducing the agnostic checking technique to achieve low regret in various loss settings.

Contribution

It introduces the agnostic checking method and provides new algorithms with regret bounds for learning Lipschitz functions under corruption.

Findings

Achieves regret $L imes O(C ext{log} T)$ for symmetric loss in 1D.

Provides regret bounds for higher dimensions with unknown corruption.

Develops algorithms for pricing loss with near-optimal regret.

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 $L\cdot \widetilde{O} (T^{d/(d+1)} + C\cdot T^{1/(d+1)})$.