A second order regret bound for NormalHedge

TL;DR

This paper introduces a variant of NormalHedge that achieves a second-order regret bound for easy sequences, leveraging continuous-time analysis and self-concordance techniques.

Contribution

It presents a novel second-order regret bound for NormalHedge, extending its theoretical guarantees for prediction with expert advice on easy sequences.

Findings

Achieves a second-order $oldsymbol{ extit{ extbf{O}}}(\sqrt{V_T ext{log}(V_T/ ext{ extbf{epsilon}})})$ regret bound.

Uses continuous-time limit and stochastic differential equations for analysis.

Employs self-concordance techniques in the discrete-time setting.

Abstract

We consider the problem of prediction with expert advice for ``easy'' sequences. We show that a variant of NormalHedge enjoys a second-order $\epsilon$-quantile regret bound of $O\big(\sqrt{V_T \log(V_T/\epsilon)}\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.