Bandit Multiclass Linear Classification: Efficient Algorithms for the Separable Case
This addresses the challenge of designing efficient algorithms with finite mistake bounds in bandit multiclass classification, which is incremental as it builds on prior work on linear separability.
The paper tackles efficient online multiclass linear classification with bandit feedback for linearly separable data, achieving a near-optimal mistake bound of O(K/γ^2) under strong separability and an exponential bound under weak separability.
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 $O\left( K/γ^2 \right)$. 2. Under the more challenging weak linear separability condition, we design an efficient algorithm with a mistake bound of $\min (2^{\widetilde{O}(K \log^2 (1/γ))}, 2^{\widetilde{O}(\sqrt{1/γ} \log K)})$. Our algorithm is based on kernel Perceptron, which is inspired by the work of (Klivans and Servedio, 2008) on improperly learning intersection of halfspaces.