Affine-Invariant Online Optimization and the Low-rank Experts Problem

We present a new affine-invariant optimization algorithm called Online Lazy Newton. The regret of Online Lazy Newton is independent of conditioning: the algorithm's performance depends on the best possible preconditioning of the problem in retrospect and on its intrinsic dimensionality. As an application, we show how Online Lazy Newton can be used to achieve an optimal regret of order rT for the low-rank experts problem, improving by a r factor over the previously best known bound and resolving an open problem posed by Hazan et al. [15].