Bandit Multiclass Linear Classification: Efficient Algorithms for the Separable Case

Abstract

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 $\gamma$. 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 $O\left(K/{\gamma }^2\right)$. 2. Under the more challenging weak linear separability condition, we design an efficient algorithm with a mistake bound of $\min (2^{\tilde{O}(K{\log }^2(1/\gamma ))},2^{\tilde{O}(\sqrt{1/\gamma }\log K)})$. Our algorithm is based on kernel Perceptron, which is inspired by the work of (Klivans and Servedio, 2008) on improperly learning intersection of halfspaces.

41 pages, 8 figures

References in corpus (1)

• Efficient Optimal Learning for Contextual Bandits