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