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：本文研究在含噪二值信号下学习Lipschitz函数的问题。学习者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$x_t$，学习者对真实函数值$f(x_t)$做出猜测，并收到一个指示猜测偏高或偏低的二值信号。在总共$C$轮中，信号可能被破坏，但学习者$unknown$ $C$的具体数值。学习者的目标是使累积损失尽可能小。本文引入了新的算法技术——\emph{不可知检查}以及新的分析技术。我设计的算法实现：对于对称损失，当$d=1$时学习者获得$L\cdot O(C\log T)$的遗憾，当$d>1$时获得$L\cdot O_d(C\log T + T^{(d-1)/d})$；对于定价损失，学习者获得$L\cdot \widetilde{O} (T^{d/(d+1)} + C\cdot T^{1/(d+1)})$的遗憾。