Learning binary relations using weighted majority voting
Learning binary relations using weighted majority voting
Learning binary relations using weighted majority voting
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264 GOLDMAN AND WARMUTH<br />
Proof: Let p be any partition of size kp and noise O~p. We begin by noting that for any<br />
m<br />
cluster i and column j in partition p, (Si,j 2}. Clearly the total number of mistakes, #,<br />
can be expressed as<br />
#= ZPi+ ~#i.<br />
iES1 iES2<br />
Recall that in Lemma 4 we showed that F.i > #i - m. Observe that if ni = 1 then<br />
Fi = 0, and thus it follows that ~iEs~ #i 2 for all i.<br />
As we have discussed, the base of our proof is provided by Lemma 2. We then apply<br />
Lemmas 4, 5 and 7 to obtain an upper bound on the number of mistakes made by<br />
Learn-Relation.<br />
We now proceed independently with the two lower bounds for Fi given in Lemma 4.<br />
Applying Lemma 2 with the first lower bound for Fi given in Lemma 4, summing over<br />
the clusters in p, and solving for # yields,<br />
kp b kp<br />
# = ~ .i
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