Efficient Online Bandit Multiclass Learning with $\tilde{O}(\sqrt{T})$ Regret

We present an efficient second-order algorithm with Õ( 1 η √ T ) regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by η, for a range of η restricted by the norm of the competitor. The family of loss functions ranges from hinge loss (η = 0) to squared hinge loss (η = 1). This provides a solution to the open problem of (J. Abernethy and A. Rakhlin. An efficient bandit algorithm for √ T -regret in online multiclass prediction? In COLT, 2009). We test our algorithm experimentally, showing that it also performs favorably against earlier algorithms.