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lecture11 notes - Lecture 11 Optimistic VC inequality...

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Lecture 11 Optimistic VC inequality. 18.465 Last time we proved the Pessimistic VC inequality: V 1 2 en 2 nt n t 4 n i =1 I ( X i C ) P ( C ) e 8 P sup , V C which can be rewritten with 8 2 en log 4 + V log + u t = V n as 1 n n i =1 8 2 en I ( X i C ) P ( C ) log 4 + V log 1 e u P + u sup . V n C V log n n Hence, the rate is . In this lecture we will prove Optimistic VC inequality, which will improve on this rate when P ( C ) is small. As before, we have pairs ( X i , Y i ), Y i = ± 1. These examples are labeled according to some unknown C 0 such that Y = 1 if X = C 0 and Y = 0 if X / C 0 . Let C = { C : C X} , a set of classifiers. C makes a mistake if X C \ C 0 C 0 \ C = C C 0 . Similarly to last lecture, we can derive bounds on n 1 I ( X i C C 0 ) P ( C C 0 ) sup C , n i =1 where ( ) is the generalization error. P C C 0 Let C = { C C 0 : C C} . One can prove that V C ( C ) V C ( C ) and n ( C , X 1 , . . . , X n ) ≤ � n ( C, X 1 , . . . , X n ).
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