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

翻译：本文研究在带有失真二进制信号的情况下学习Lipschitz函数的问题。学习器试图学习一个由对手选择的$L$-Lipschitz函数$f: [0,1]^d \rightarrow [0, L]$。总共有$T$轮交互。在每一轮$t$中，对手在输入空间中选择一个上下文向量$x_t$，学习器对真实函数值$f(x_t)$进行猜测，并接收一个二进制信号指示猜测偏高或偏低。在总共$C$轮中，信号可能被破坏，尽管$C$的值对学习器而言是\emph{未知}的。学习器的目标是实现较小的累积损失。本文引入了新的算法技术\emph{不可知检验}以及新的分析技术。我所设计的算法能够实现：对于对称损失，当$d = 1$时学习器获得$L\cdot O(C\log T)$的遗憾，当$d > 1$时获得$L\cdot O_d(C\log T + T^{(d-1)/d})$的遗憾；对于定价损失，学习器获得$L\cdot \widetilde{O} (T^{d/(d+1)} + C\cdot T^{1/(d+1)})$的遗憾。