Learning binary relations using weighted majority voting

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270 GOLDMAN AND WARMUTH Furthermore, for y >_ 2 @2 _ 7y + 2 4 1 > - > - 6y 2-10y+3 - 7 2 and thus it suffices to have x _< ym/2 which is the case. Finally, it can be verified that fxxfyy - (fxy)2 is for y_> 2. y2 4y(y - 1) + 1 +bm In 2 + 2bray In 2(y - 2) + 2 in ~(-N-sTy_~ n(r~--l) ) ,_y (Q. 2 - 4y + 1) ) 16b 2 (In 2) 2 (y - 1)2re(f (x, y))4 This completes the proof that f(x, y) is concave over the desired interval. Notes 1. The adversary, who tries to maximize the learner's mistakes, knows the learner's algorithm and has unlimited computing power. 2. The halving algorithm is described in more detail in Section 2. 3, Throughout this paper we let lg denote the base 2 logarithm and In the natural logarithm. Euler's constant is denoted by e. 4. The same result holds if one performs the updates after each trial. 5. A version of this second algorithm is described in detail in Figure 3. For the sake of simplicity it is parameterized by an update the factor 3' = 2~/(1 + ~) instead of/3. See Section 5 for a discussion of this algorithm called Learn-Relation(7). 6. We can obtain a bound of half of (1) by either letting the algorithm predict probabilistically in {0, 1} or deterministically in the interval [0, 1] (Cesa-Bianchi, Freund, Helmbold, Haussler & Schapire 1993; Kivinen & Warmuth, 1994). In the case of probabilistic predictions this quantity is an upper bound for the expected number of mistakes. And in the case of deterministic predictions in [0, 1], this quantity is an upper bound on the loss of the algorithm (where the loss is just the sum over all trials of the absolute difference between the prediction and the correct value). 7. Recall that 9(z) = 1 1+2z+2~ 2 and 9(ee) = 0. 8. Again, we can obtain improvements of a factor of 2 by either letting the algorithm predict probabilistically in {0, 1} or deterministically in the interval [0, 1] (Cesa-Bianchi, Freund, Helmbold, Haussler, Schapire & Warmuth, 1993; Kivinen & Warmuth, 1994). References Angluin, D. (1988). Queries and concept learning. Machine Learning, 2(4): pp. 319-342. Barzdin, J. & Freivald, R. (1972). On the prediction of general recursive functions. Soviet Mathematics Doklady, 13: pp. 1224-1228. Cesa-Bianchi, N., Freund, Y., Helmbold, D., Haussler, D., Schapire, R. & Warmuth, M. (1993). How to use expert advice. Proceedings of the Twenty Fifth Annual ACM Symposium on Theory of Computing, pp. 382-391. Chert, W. (1992). Personal communication. Goldman, S., Rivest, R. & Schapire, R. (1993). Learning binary relations and total orders. SIAM Journal of Computing, 22(5): pp. 1006-1034. >_0
LEARNING BINARY RELATIONS 271 Kivinen, J. & and Warmuth, M. (1994) Using Experts for Predicting Continuous Outcomes. Computational Learning Theory: Eurocolt "93 pp. 109-120, Oxford University Press, Oxford, ISBN 0-19-853492-2. Littlestone, N. (1988) Learning when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2(4): pp. 285-318. Littlestone, N. (1989) Mistake Bounds and Logarithmic Linear-threshold Learning algorithms. PhD thesis, U. C. Santa Cruz. Littlestone, N., Long, P. & Warmuth, M. (1991) On-line learning of linear functions. Proceedings of the Twenty Third Annual ACM Symposium on Theory of Computing, pp. 465-475. To appear in Journal oJ Computational Complexiß'. Littlestone, N. & Warmuth, M. (1994) The weighted majority algorithm. Information and Computation, 108(2): pp. 212-261. Vovk, V. (1990) Aggregating strategies. Proceedings of the Third Annual Workshop on Computational Learning Theory, pp. 371-383. Morgan Kaufmann. Received January 17, 1994 Accepted November 15, 1994 Final Manuscript January 25, 1995
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