Learning binary relations using weighted majority voting

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264 GOLDMAN AND WARMUTH Proof: Let p be any partition of size kp and noise O~p. We begin by noting that for any m cluster i and column j in partition p, (Si,j 2}. Clearly the total number of mistakes, #, can be expressed as #= ZPi+ ~#i. iES1 iES2 Recall that in Lemma 4 we showed that F.i > #i - m. Observe that if ni = 1 then Fi = 0, and thus it follows that ~iEs~ #i 2 for all i. As we have discussed, the base of our proof is provided by Lemma 2. We then apply Lemmas 4, 5 and 7 to obtain an upper bound on the number of mistakes made by Learn-Relation. We now proceed independently with the two lower bounds for Fi given in Lemma 4. Applying Lemma 2 with the first lower bound for Fi given in Lemma 4, summing over the clusters in p, and solving for # yields, kp b kp # = ~ .i
LEARNING BINARY RELATIONS 265 Finally by combining Inequalities (2), (3), and (4) we obtain that: #
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Page 9 and 10: LEARNING BINARY RELATIONS 253 Learn
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Page 15 and 16: LEARNING BINARY RELATIONS 259 In th
Page 17 and 18: LEARNING BINARY RELATIONS 261 depen
Page 19: LEARNING BINARY RELATIONS 263 m V S
Page 23 and 24: LEARNING BINARY RELATIONS 267 32500
Page 25 and 26: LEARNING BINARY RELATIONS 269 mance
Page 27: LEARNING BINARY RELATIONS 271 Kivin

Learning binary relations using weighted majority voting

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