lecture11 notes

Theorem 111 optimistic vc inequality n v 1 p c n

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Unformatted text preview: C ∈ C }. One can prove that V C (C � ) ≤ V C (C ) and �n (C � , X1 , . . . , Xn ) ≤ �n (C, X1 , . . . , Xn ). By Hoeffding-Chernoff, if P (C ) ≤ 1 , 2 � n 1� P P (C ) − I (Xi ∈ C ) ≤ n i=1 � 2P (C ) t n � ≥ 1 − e−t . Theorem 11.1. [Optimistic VC inequality] � � �n � �V 1 P (C ) − n i=1 I (Xi ∈ C ) nt2 2en � P sup ≥t ≤4 e− 4 . V P (C ) C Proof. Let C be fixed. Then � � n 1� 1 � � P(Xi ) I (Xi ∈ C ) ≥ P (C ) ≥ n i=1 4 �n �n 1 1 � � whe...
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This note was uploaded on 01/23/2014 for the course MATH 18.465 taught by Professor Panchenko during the Spring '07 term at MIT.

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