Skip to main content
SpringerLink
Log in

Asymptotically Optimal Strategies for Online Prediction with History-Dependent Experts

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the d dimensional de Bruijn graph, where d is the number of days of history used by the experts. Previous work Drenska and Kohn (arXiv:2007.12732, 2020) established \(O(\varepsilon )\) optimal strategies for \(n=2\) experts and \(d\le 4\) days of history, while Drenska and Kohn (J Nonlinear Sci 30. 30(1), 137–173, 2020) established \(O(\varepsilon ^{1/3})\) optimal strategies for all \(n\ge 2\) and all \(d\ge 1\), where the game is played for N steps and \(\varepsilon =N^{-1/2}\). In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish \(O(\varepsilon )\) optimal strategies for all values of n and d.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Amin, K., Kale, S., Tesauro, G., Turaga, D.: Budgeted prediction with expert advice. In: Twenty-Ninth AAAI Conference on Artificial Intelligence (2015)

  2. Antunovic, T., Peres, Y., Sheffield, S., Somersille, S.: Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition. Commun. Partial Differ. Equ. 37(10), 1839–1869 (2012)

    Article  MathSciNet  Google Scholar 

  3. Armstrong, S.N., Smart, C.K.: A finite difference approach to the infinity Laplace equation and tug-of-war games. Trans. Am. Math. Soc. 364(2), 595–636 (2012)

    Article  MathSciNet  Google Scholar 

  4. Bayraktar, E., Ekren, I., Zhang, Y.: On the asymptotic optimality of the comb strategy for prediction with expert advice. Ann. Appl. Probab. (2020)

  5. Calder, J., Smart, C.K.: The limit shape of convex hull peeling. Duke Math. J. 169(11), 2079–2124 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cesa-Bianchi, N., Freund, Y., Haussler, D., Helmbold, D.P., Schapire, R.E., Warmuth, M.K.: How to use expert advice. J. ACM 44(3), 427–485 (1997)

    Article  MathSciNet  Google Scholar 

  7. Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, New York (2006)

    Book  Google Scholar 

  8. Cover, T.M.: Behavior of sequential predictors of binary sequences. Technical report, Stanford University California Stanford Electronics Labs (1966)

  9. Drenska, N.: A PDE Approach to a Prediction Problem Involving Randomized Strategies. PhD thesis, New York University, New York (2017)

  10. Drenska, N., Calder, J.: Online prediction with history-dependent experts: the general case. arXiv:2008.00052 (2020)

  11. Drenska, N., Kohn, R.V.: A PDE approach to the prediction of a binary sequence with advice from two history-dependent experts. arXiv:2007.12732 (2020)

  12. Drenska, N., Kohn, R.V.: Prediction with expert advice: a PDE perspective. J. Nonlinear Sci. 30(1), 137–173 (2020)

    Article  MathSciNet  Google Scholar 

  13. Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci. 55(1), 119–139 (1997)

    Article  MathSciNet  Google Scholar 

  14. Gravin, N., Peres, Y., Sivan, B.: Towards optimal algorithms for prediction with expert advice. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’16, pp. 528–547, Philadelphia, PA, USA (2016). Society for Industrial and Applied Mathematics

  15. Hannan, J.: Approximation to Bayes risk in repeated play. Contrib. Theory Games 3, 97–139 (1957)

    MathSciNet  MATH  Google Scholar 

  16. Haussler, D., Kivinen, J., Warmuth, M.K.: Tight worst-case loss bounds for predicting with expert advice. In: European Conference on Computational Learning Theory, pp. 69–83. Springer, Berlin (1995)

  17. Kobzar, V.A., Kohn, R.V., Wang, Z.: New potential-based bounds for prediction with expert advice. In: Conference on Learning Theory, pp. 2370–2405. PMLR (2020)

  18. Kobzar, V.A., Kohn, R.V., Wang, Z.: New potential-based bounds for the geometric-stopping version of prediction with expert advice. In: Mathematical and Scientific Machine Learning, pp. 537–554. PMLR (2020)

  19. Kohn, R.V., Serfaty, S.: A deterministic-control-based approach motion by curvature. Commun. Pure Appl. Math. 59(3), 344–407 (2006)

    Article  MathSciNet  Google Scholar 

  20. Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Commun. Pure Appl. Math. 63(10), 1298–1350 (2010)

    Article  MathSciNet  Google Scholar 

  21. Lewicka, M., Manfredi, J.J.: The obstacle problem for the p-Laplacian via optimal stopping of tug-of-war games. In: Probability Theory and Related Fields, pp. 1–30 (2015)

  22. Littlestone, N., Warmuth, M.K.: The weighted majority algorithm. Inf. Comput. 108(2), 212–261 (1994)

    Article  MathSciNet  Google Scholar 

  23. Naor, A., Sheffield, S.: Absolutely minimal Lipschitz extension of tree-valued mappings. Math. Ann. 354(3), 1049–1078 (2012)

    Article  MathSciNet  Google Scholar 

  24. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)

    Article  MathSciNet  Google Scholar 

  25. Peres, Y., Sheffield, S.: Tug-of-war with noise: a game-theoretic view of the \(p\)-Laplacian. Duke Math. J. 145(1), 91–120 (2008)

    Article  MathSciNet  Google Scholar 

  26. Rokhlin, D.: PDE approach to the problem of online prediction with expert advice: a construction of potential-based strategies. Int. J. Pure Appl. Math. 114, 05 (2017)

    Article  Google Scholar 

  27. Yadkori, Y.A., Bartlett, P.L., Gabillon, : Near minimax optimal players for the finite-time 3-expert prediction problem. In: Advances in Neural Information Processing Systems, pp. 3033–3042 (2017)

  28. Zhu, K.: Two problems in applications of PDE. http://pqdtopen.proquest.com/pubnum/3635320.html (2014)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, USA

    Jeff Calder & Nadejda Drenska

Authors
  1. Jeff Calder
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nadejda Drenska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadejda Drenska.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Funding Jeff Calder was supported by NSF-DMS Grant 1944925 and the Alfred P. Sloan foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calder, J., Drenska, N. Asymptotically Optimal Strategies for Online Prediction with History-Dependent Experts. J Fourier Anal Appl 27, 20 (2021). https://doi.org/10.1007/s00041-021-09815-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-021-09815-4

Keywords

Search

Navigation