Machine Learning

, Volume 66, Issue 2–3, pp 321–352 | Cite as

Improved second-order bounds for prediction with expert advice

  • Nicolò Cesa-Bianchi
  • Yishay Mansour
  • Gilles StoltzEmail author


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.


Individual sequences Prediction with expert advice Exponentially weighted averages 


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Copyright information

© Springer Science + Business Media, LLC 2007

Authors and Affiliations

  • Nicolò Cesa-Bianchi
    • 1
  • Yishay Mansour
    • 2
  • Gilles Stoltz
    • 3
    Email author
  1. 1.DSI, Università di MilanoMilanoItaly
  2. 2.School of Computer ScienceTel-Aviv UniversityTel AvivIsrael
  3. 3.CNRS and Département de Mathématiques et ApplicationsEcole Normale SupérieureParisFrance

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