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lecture10 notes - Lecture 10 Symmetrization Pessimistic VC...

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± ± ± ± ± ± ± ± ± ± ² ³ ± ± ± ± ± ´ ± ± ± ± ± ² ± ± ± ± ± ´ ± ± ± ± ± ² ± ± ± ± ± ± ± ± ± ± ± ± µ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ² ´ µ ´ ´ ³ ± ± ± ± ± ´ ± ± ± ± ± ± ± ± ± ± ² · ± ± ± ± ± ´ ± ± ± ± ± ± ± ± ± ± ´ ± ± ± ± ± ± ± ± ± ± ´ ´ ± ± ± ± ± Lecture 10 Symmetrization. Pessimistic VC inequality. 18.465 We are interested in bounding 1 P sup C ∈C n n ´ i =1 I ( X i C ) P ( C ) t In Lecture 7 we hinted at Symmetrization as a way to deal with the unknown P ( C ). Lemma 10.1. [Symmetrization] If t 2 , then n n sup C ∈C 1 n n n i =1 ´ i =1 1 1 I ( X i C ) P ( C ) 2 P I ( X i C ) I ( X i C ) t/ 2 P sup C ∈C t . n n i =1 Proof. Suppose the event sup C ∈C n ´ i =1 1 n I ( X i C ) P ( C ) t 1 n occurs. Let X = ( X 1 , . . . , X n ) ∈ { sup C ∈C I ( X i C ) P ( C ) Then t } . I ( X i C X ) P ( C X ) n i =1 n n ´ i =1 1 such that C X t. For a fxed C , P X 1 n ² 2 n n i =1 ´ i =1 1 t 2 / 4 I ( X i C ) P ( C ) t/ 2 I ( X i C ) P ( C ) = P n ¸ 2 1
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