Abstract
We study the problem of deterministically predicting boolean valuesby combining the boolean predictions of several experts.Previous on-line algorithms for this problem predict with the weightedmajority of the experts' predictions.These algorithms give each expert an exponential weight βmwhere β is a constant in [0,1) and m is the number of mistakesmade by the expert in the past. We show that it is better to usesums of binomials as weights.In particular, we present a deterministic algorithmusing binomial weights that has a better worst case mistake bound than thebest deterministic algorithm using exponential weights.The binomial weights naturally arise from a version space argument.We also show how both exponential and binomial weighting schemes can beused to make prediction algorithms robust against noise.
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References
Aarts, E. & Korst, J. (1989). Simulated Annealing and Boltzmann Machines. John Wiley and Sons.
Alon, N., Spencer, J.H. & Erdös, P. (1992). The Probabilistic Method. John Wiley and Sons.
Angluin, D. (1988). Queries and concept learning. Machine Learning, 2:319–342.
Aslam, J.A. & Dhagat, A. (1991). Searching in the presence of linearly bounded errors. In Proceedings of the 23rd ACM Symposium on the Theory of Computation, pages 486–493. ACM Press.
Auer, P. & Long, P.M. (to appear). Structural results about on-line learning models with and without queries. Machine Learning.
Auer, P. & Long, P.M. (1994). Simulating access to hidden information while learning. In Proceedings of the 26th ACM Symposium on the Theory of Computation, pages 263–272. ACM Press.
Bardzin, J.M. & Freivalds, R.V. (1972). On the prediction of general recursive functions. Soviet Math.Dokl., 13:1224–1228.
Berlekamp, E.R. (1968). Error-Correcting Codes. John Wiley and Sons.
Cesa-Bianchi, N., Freund, Y., Helmbold, D.P., Haussler, D., Schapire, R. & Warmuth, M.K. (1995). How to use expert advice. To appear in Journal of the ACM.
Cesa-Bianchi, N., Long, P.M. & Warmuth, M.K. (1996). Worst-case quadratic loss bounds for a generalization of the Widrow-Hoff rule. IEEE Transactions on Neural Networks, 7(2): 604-619.
Chernoff, H. (1952). Ameasure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics, 23:493–507.
Graham, R.L., Knuth, D.E. & Patashnik, O. (1989). Concrete Mathematics. Addison Wesley.
Kivinen, J. & Warmuth, M.K. (1994). Using experts for predicting continuous outcomes. In Computational Learning Theory: Eurocolt’ 93.The Institute of Mathematics and its Applications Conference Series, number 53, pages 109-120, Oxford: Oxford University Press.
Littlestone, N. (1988). Learning quickly when irrelevant attributes abound: a new linear-threshold algorithm. Machine Learning, 2(4):285–318.
Littlestone, N. (1989). Mistake Bounds and Logarithmic Linear-threshold Learning Algorithms. PhD thesis, University of California at Santa Cruz.
Littlestone, N., Long, P.M. & Warmuth, M.K. (1995). On-line learning of linear functions. Computational Complexity, 5(1):1–23.
Littlestone, N. & Warmuth, M.K. (1994). The weighted majority algorithm. Information and Computation, 108:212–261.
Mitchell, T.M. (1977). Version spaces: A candidate elimination approach to rule learning. In Proceedings International Joint Conference on Artificial Intelligence, pages 305–310, Cambridge, Mass.
Spencer, J. (1992). Ulam's searching game with a fixed number of lies. Theoretical Computer Science, 95:307–321.
Ulam, S. (1977). Adventures of a Mathematician. Scribners.
Vovk, V.G. (1990). Aggregating strategies. In Proceedings of the 3rd Annual Workshop on Computational Learning Theory, pages 372–383.
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Cesa-Bianchi, N., Freund, Y., Helmbold, D.P. et al. On-line Prediction and Conversion Strategies. Machine Learning 25, 71–110 (1996). https://doi.org/10.1023/A:1018348209754
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DOI: https://doi.org/10.1023/A:1018348209754