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 γ. 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 Oleft( K/γ2 right). 2. Under the more challenging weak linear separability condition, we design an efficient algorithm with a mistake bound of min (2^widetildeO(K log2 (1/γ)), 2^widetildeO(√1/γ log K)). Our algorithm is based on kernel Perceptron, which is inspired by the work of (Klivans and Servedio, 2008) on improperly learning intersection of halfspaces.