Drifting Games (original) (raw)

Abstract

We introduce and study a general, abstract game played between two players called the shepherd and the adversary. The game is played in a series of rounds using a finite set of “chips” which are moved about in ℝ_n_. On each round, the shepherd assigns a desired direction of movement and an importance weight to each of the chips. The adversary then moves the chips in any way that need only be weakly correlated with the desired directions assigned by the shepherd. The shepherd's goal is to cause the chips to be moved to low-loss positions, where the loss of each chip at its final position is measured by a given loss function.

We present a shepherd algorithm for this game and prove an upper bound on its performance. We also prove a lower bound showing that the algorithm is essentially optimal for a large number of chips. We discuss computational methods for efficiently implementing our algorithm.

We show that our general drifting-game algorithm subsumes some well studied boosting and on-line learning algorithms whose analyses follow as easy corollaries of our general result.

Article PDF

Similar content being viewed by others

References

• Blackwell, D. (1956). An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1, 1–8.
  Google Scholar
• Cesa-Bianchi, N., Freund, Y., Haussler, D., Helmbold, D. P., Schapire, R. E., & Warmuth, M. K. (1997). How to use expert advice. Journal of the Association for Computing Machinery, 44:3, 427–485.
  Google Scholar
• Cesa-Bianchi, N., Freund, Y., Helmbold, D. P., & Warmuth, M. K. (1996). On-line prediction and conversion strategies. Machine Learning, 25, 71–110.
  Google Scholar
• Freund, Y. (1995). Boosting a weak learning algorithm by majority. Information and Computation, 121:2, 256–285.
  Google Scholar
• Freund, Y. (2001). An adaptive version of the boost by majority algorithm. Machine Learning, 43:3, 293–318.
  Google Scholar
• Freund, Y. & Schapire, R. E. (1997). A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55:1, 119–139.
  Google Scholar
• Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:301, 13–30.
  Google Scholar
• Littlestone, N. & Warmuth, M. K. (1994). The weighted majority algorithm. Information and Computation, 108, 212–261.
  Google Scholar
• Rockafellar, R. T. (1970). Convex Analysis. Princeton, NJ: Princeton University Press.
  Google Scholar
• Schapire, R. E. & Singer, Y. (1999). Improved boosting algorithms using confidence-rated predictions. Machine Learning, 37:3, 297–336.
  Google Scholar

Download references

Author information

Authors and Affiliations

1. AT&T Labs—Research, Shannon Laboratory, 180 Park Avenue, Room A279, Florham Park, NJ, 07932-0971, USA
   Robert E. Schapire

Authors

1. Robert E. Schapire

Rights and permissions

About this article

Cite this article

Schapire, R.E. Drifting Games.Machine Learning 43, 265–291 (2001). https://doi.org/10.1023/A:1010800213066

Download citation

• Issue date: June 2001
• DOI: https://doi.org/10.1023/A:1010800213066