Improved second-order bounds for prediction with expert advice

This work studies external regret in sequential prediction games with both positive and negative payoffs. External regret measures the difference between the payoff obtained by the forecasting strategy and the payoff of the best action. In this setting, we derive new and sharper regret bounds for the well-known exponentially weighted average forecaster and for a second forecaster with a different multiplicative update rule. Our analysis has two main advantages: first, no preliminary knowledge about the payoff sequence is needed, not even its range; second, our bounds are expressed in terms of sums of squared payoffs, replacing larger first-order quantities appearing in previous bounds. In addition, our most refined bounds have the natural and desirable property of being stable under rescalings and general translations of the payoff sequence. Editor: Avrim Blum An extended abstract appeared in the Proceedings of the 18th Annual Conference on Learning Theory, Springer, 2005. The work of all authors was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. The work was done while Yishay Mansour was a fellow in the Institute of Advance studies, Hebrew University. His work was also supported by a grant no. 1079/04 from the Israel Science Foundation and an IBM faculty award.