Machine Learning

, Volume 72, Issue 1–2, pp 21–37 | Cite as

Regret to the best vs. regret to the average

  • Eyal Even-Dar
  • Michael Kearns
  • Yishay Mansour
  • Jennifer WortmanEmail author


We study online regret minimization algorithms in an experts setting. In this setting, the algorithm chooses a distribution over experts at each time step and receives a gain that is a weighted average of the experts’ instantaneous gains. We consider a bicriteria setting, examining not only the standard notion of regret to the best expert, but also the regret to the average of all experts, the regret to any given fixed mixture of experts, or the regret to the worst expert. This study leads both to new understanding of the limitations of existing no-regret algorithms, and to new algorithms with novel performance guarantees. More specifically, we show that any algorithm that achieves only \(O(\sqrt{T})\) cumulative regret to the best expert on a sequence of T trials must, in the worst case, suffer regret \(\varOmega(\sqrt{T})\) to the average, and that for a wide class of update rules that includes many existing no-regret algorithms (such as Exponential Weights and Follow the Perturbed Leader), the product of the regret to the best and the regret to the average is, in the worst case, Ω(T). We then describe and analyze two alternate new algorithms that both achieve cumulative regret only \(O(\sqrt{T}\log T)\) to the best expert and have only constant regret to any given fixed distribution over experts (that is, with no dependence on either T or the number of experts N). The key to the first algorithm is the gradual increase in the “aggressiveness” of updates in response to observed divergences in expert performances. The second algorithm is a simple twist on standard exponential-update algorithms.


Online learning Regret minimization Prediction with expert advice Lower bounds 


  1. Auer, P., Cesa-Bianchi, N., & Gentile, C. (2002). Adaptive and self-confident on-line learning algorithms. Journal of Computer and System Sciences, 64, 48–75. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge University Press. zbMATHGoogle Scholar
  3. Cesa-Bianchi, N., Mansour, Y., & Stoltz, G. (2007). Improved second-order bounds for prediction with expert advice. Machine Learning, 66(2/3), 321–352. CrossRefGoogle Scholar
  4. Cover, T. (1991). Universal portfolios. Mathematical Finance, 1(1), 1–19. zbMATHCrossRefMathSciNetGoogle Scholar
  5. Even-Dar, E., Kearns, M., Mansour, Y., & Wortman, J. (2007). Regret to the best versus regret to the average. In Twentieth annual conference on learning theory (pp. 233–247). Google Scholar
  6. Freund, Y. (2003). Predicting a binary sequence almost as well as the optimal biased coin. Information and Computation, 182(2), 73–94. zbMATHCrossRefMathSciNetGoogle Scholar
  7. Helmbold, D., Schapire, R., Singer, Y., & Warmuth, M. (1998). On-line portfolio selection using multiplicative updates. Mathematical Finance, 8(4), 325–347. zbMATHCrossRefGoogle Scholar
  8. Kalai, A., & Vempala, S. (2005). Efficient algorithms for on-line optimization. Journal of Computer and System Sciences, 71(3), 291–307. zbMATHCrossRefMathSciNetGoogle Scholar
  9. Littlestone, N., & Warmuth, M. K. (1994). The weighted majority algorithm. Information and Computation, 108(2), 212–261. zbMATHCrossRefMathSciNetGoogle Scholar
  10. Vovk, V. (1998). A game of prediction with expert advice. Journal of Computer and System Sciences, 56(2), 153–173. zbMATHCrossRefMathSciNetGoogle Scholar
  11. Zhang, T. (2007). Personal communication. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Eyal Even-Dar
    • 1
  • Michael Kearns
    • 2
  • Yishay Mansour
    • 1
    • 3
  • Jennifer Wortman
    • 2
    Email author
  1. 1.Google ResearchNew YorkUSA
  2. 2.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael

Personalised recommendations