Bandit Multiclass Linear Classification: Efficient Algorithms for the Separable Case

We study the problem of efficient online multiclass linear classification with bandit feedback, where all examples belong to one of K classes and lie in the d-dimensional Euclidean space. Previous works have left open the challenge of designing efficient algorithms with finite mistake bounds when the data is linearly separable by a margin 'y. In this work, we take a first step towards this problem. We consider two notions of linear separability, strong and weak. 1. Under the strong linear separability condition, we design an efficient algorithm that achieves a near-optimal mistake bound of O (K/gamma(2)). 2. Under the more challenging weak linear separability condition, we design an efficient algorithm with a mistake bound of min (2((O) over tilde (K log2(1/gamma))), 2((O) over bar(root 1/gamma log K)))(1). Our algorithm is based on kernel Perceptron and is inspired by the work of Klivans & Servedio (2008) on improperly learning intersection of halfspaces.