  In this paper, we study a special bandit setting of online stochastic linear
optimization, where only one-bit of information is revealed to the learner at
each round. This problem has found many applications including online
advertisement and online recommendation. We assume the binary feedback is a
random variable generated from the logit model, and aim to minimize the regret
defined by the unknown linear function. Although the existing method for
generalized linear bandit can be applied to our problem, the high computational
cost makes it impractical for real-world problems. To address this challenge,
we develop an efficient online learning algorithm by exploiting particular
structures of the observation model. Specifically, we adopt online Newton step
to estimate the unknown parameter and derive a tight confidence region based on
the exponential concavity of the logistic loss. Our analysis shows that the
proposed algorithm achieves a regret bound of $O(d\sqrt{T})$, which matches the
optimal result of stochastic linear bandits.
