A second order regret bound for NormalHedge

We consider the problem of prediction with expert advice for ``easy'' sequences. We show that a variant of NormalHedge enjoys a second-order $ε$-quantile regret bound of $O\big(\sqrt{V_T \log(V_T/ε)}\big) $ when $V_T > \log N$, where $V_T$ is the cumulative second moment of instantaneous per-expert regret averaged with respect to a natural distribution determined by the algorithm. The algorithm is motivated by a continuous time limit using Stochastic Differential Equations. The discrete time analysis uses self-concordance techniques.