Material to “ On Multilabel Classification and Ranking with Partial Feedback ”
نویسندگان
چکیده
If there exists i ∈ Ys which is not among the s-top ranked ones, then we could replace class i in position ji within Ys with class k / ∈ Ys such that pk,t > pi,t obtaining a smaller loss. Next, we show that the optimal ordering within Y ∗ s,t is precisely ruled by the nonicreasing order of pi,t. By the sake of contradiction, assume there are i and k in Y ∗ s,t such that i preceeds k in Y ∗ s,t but pk,t > pi,t. Specifically, let i be in position j1 and k be in position j2 with j1 < j2 and such that c(j1, s) > c(j2, s). Then, disregarding the (1− a)-factor, switching the two classes within Y ∗ s,t yields an expected loss difference of
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On Multilabel Classification and Ranking with Partial Feedback
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