IDEAS home Printed from https://ideas.repec.org/a/spr/dyngam/v5y2015i1p136-153.html
   My bibliography  Save this article

Exponential Weight Approachability, Applications to Calibration and Regret Minimization

Author

Listed:
  • Vianney Perchet

Abstract

Basic ideas behind the “exponential weight algorithm” (designed for aggregation or minimization of regret) can be transposed into the theory of Blackwell approachability. Using them, we develop an algorithm—that we call “exponential weight approachability”—bounding the distance of average vector payoffs to some product set, with a logarithmic dependency in the dimension of the ambient space. The classic strategy of Blackwell would get instead a polynomial dependency. This result has important consequences, in several frameworks that emerged both in game theory and machine learning. The most striking application is the construction of algorithms that are calibrated with respect to the family of all balls (we treat in details the case of the uniform norm), with dimension independent and optimal, up to logarithmic factors, rates of convergence. Calibration can also be achieved with respect to all Borel sets, covering and improving the previously known results. Exponential weight approachability can also be used to design an optimal and natural algorithm that minimizes refined notions of regret. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Vianney Perchet, 2015. "Exponential Weight Approachability, Applications to Calibration and Regret Minimization," Dynamic Games and Applications, Springer, vol. 5(1), pages 136-153, March.
    • Handle: RePEc:spr:dyngam:v:5:y:2015:i:1:p:136-153
    DOI: 10.1007/s13235-014-0119-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s13235-014-0119-x
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s13235-014-0119-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Sergiu Hart & Andreu Mas-Colell, 2013. "A General Class Of Adaptive Strategies," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 3, pages 47-76, World Scientific Publishing Co. Pte. Ltd..
    2. Serge A. Plotkin & David B. Shmoys & Éva Tardos, 1995. "Fast Approximation Algorithms for Fractional Packing and Covering Problems," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 257-301, May.
    3. Shie Mannor & Gilles Stoltz, 2009. "A Geometric Proof of Calibration," Working Papers hal-00442042, HAL.
    4. Foster, Dean P., 1999. "A Proof of Calibration via Blackwell's Approachability Theorem," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 73-78, October.
    5. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, December.
    6. Foster, Dean P. & Vohra, Rakesh V., 1997. "Calibrated Learning and Correlated Equilibrium," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 40-55, October.
    7. Shie Mannor & Gilles Stoltz, 2010. "A Geometric Proof of Calibration," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 721-727, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Flesch, János & Laraki, Rida & Perchet, Vianney, 2018. "Approachability of convex sets in generalized quitting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 411-431.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ehud Lehrer & Eilon Solan, 2016. "A General Internal Regret-Free Strategy," Dynamic Games and Applications, Springer, vol. 6(1), pages 112-138, March.
    2. Andrey Bernstein & Shie Mannor & Nahum Shimkin, 2014. "Opportunistic Approachability and Generalized No-Regret Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1057-1083, November.
    3. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. Mannor, Shie & Shimkin, Nahum, 2008. "Regret minimization in repeated matrix games with variable stage duration," Games and Economic Behavior, Elsevier, vol. 63(1), pages 227-258, May.
    5. Foster, Dean P. & Hart, Sergiu, 2018. "Smooth calibration, leaky forecasts, finite recall, and Nash dynamics," Games and Economic Behavior, Elsevier, vol. 109(C), pages 271-293.
    6. Foster, Dean P. & Young, H. Peyton, 2003. "Learning, hypothesis testing, and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 45(1), pages 73-96, October.
    7. Olszewski, Wojciech, 2015. "Calibration and Expert Testing," Handbook of Game Theory with Economic Applications,, Elsevier.
    8. Flesch, János & Laraki, Rida & Perchet, Vianney, 2018. "Approachability of convex sets in generalized quitting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 411-431.
    9. Ehud Lehrer & Eilon Solan, 2007. "Learning to play partially-specified equilibrium," Levine's Working Paper Archive 122247000000001436, David K. Levine.
    10. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision Making in Uncertain and Changing Environments," Discussion Papers 19, Kyiv School of Economics.
    11. Vladimir V'yugin, 2014. "Log-Optimal Portfolio Selection Using the Blackwell Approachability Theorem," Papers 1410.5996, arXiv.org, revised Jun 2015.
    12. Giovanni Di Bartolomeo & Debora Di Gioacchino, 2008. "Fiscal-monetary policy coordination and debt management: a two-stage analysis," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 35(4), pages 433-448, September.
    13. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2006. "Stochastic Approximations and Differential Inclusions, Part II: Applications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 673-695, November.
    14. Dean P. Foster & Sergiu Hart, 2021. "Forecast Hedging and Calibration," Journal of Political Economy, University of Chicago Press, vol. 129(12), pages 3447-3490.
    15. Sergiu Hart & Yishay Mansour, 2013. "How Long To Equilibrium? The Communication Complexity Of Uncoupled Equilibrium Procedures," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 10, pages 215-249, World Scientific Publishing Co. Pte. Ltd..
    16. Burkhard C. Schipper, 2022. "Strategic Teaching and Learning in Games," American Economic Journal: Microeconomics, American Economic Association, vol. 14(3), pages 321-352, August.
    17. Chernov, G. & Susin, I., 2019. "Models of learning in games: An overview," Journal of the New Economic Association, New Economic Association, vol. 44(4), pages 77-125.
    18. Germano, Fabrizio & Lugosi, Gabor, 2007. "Global Nash convergence of Foster and Young's regret testing," Games and Economic Behavior, Elsevier, vol. 60(1), pages 135-154, July.
    19. Eddie Dekel & Yossi Feinberg, 2006. "Non-Bayesian Testing of a Stochastic Prediction," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 73(4), pages 893-906.
    20. Nicolò Cesa-Bianchi & Gábor Lugosi & Gilles Stoltz, 2006. "Regret Minimization Under Partial Monitoring," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 562-580, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:dyngam:v:5:y:2015:i:1:p:136-153. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.