Online Stochastic Linear Optimization under One-bit Feedback

TL;DR

This paper introduces an efficient online algorithm for stochastic linear optimization with one-bit feedback, achieving optimal regret bounds by leveraging the logistic model's structure, applicable to online advertising and recommendation systems.

Contribution

The paper proposes a novel online learning algorithm that reduces computational complexity while maintaining optimal regret bounds in one-bit feedback linear bandit problems.

Findings

Achieves $O(d\sqrt{T})$ regret bound matching the optimal linear bandit results.

Develops an efficient online Newton step-based algorithm for logistic feedback.

Provides theoretical analysis demonstrating the algorithm's effectiveness.

Abstract

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.