Online Learning with Low Rank Experts

We consider the problem of prediction with expert advice when the losses of the experts have low-dimensional structure: they are restricted to an unknown $d$-dimensional subspace. We devise algorithms with regret bounds that are independent of the number of experts and depend only on the rank $d$. For the stochastic model we show a tight bound of $\mathrm{\Theta }(\sqrt{dT})$, and extend it to a setting of an approximate $d$ subspace. For the adversarial model we show an upper bound of $O(d\sqrt{T})$ and a lower bound of $\mathrm{\Omega }(\sqrt{dT})$.