Efficient online bandit multiclass learning with Õ(√T) regret

We present an efficient second-order algorithm with Ō(1/η√T)1 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 (Abernethy, J. and Rakhlin, A. 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.