lecture10 notes

Assume that the event n 1 i xi cx p cx t2 n i1 occurs

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Unformatted text preview: I (Xi ∈ C ) − P (C ) ≥ t2 /4⎠ �n � n i=1 i=1 ≤ (by Chebyshev’s Ineq) = = since we chose t ≥ � 4 n 2 t2 n � i=1 4� n 2 t2 4E � 1 �n n i=1 �2 � I (Xi ∈ C ) − P (C ) t2 � � E(I (Xi ∈ C ) − P (C ))(I (Xj ∈ C ) − P (C )) i,j � E(I (Xi ∈ C ) − P (C ))2 = 4nP (C ) (1 − P (C ) 1 1 ≤ 2≤ 2 t2 n 2 nt 2 n. So, � � �� n � �1 � � � � � � � PX � � I (Xi ∈ CX ) − P (CX )� ≤ t/2�∃CX ≥ 1/2 �n � � i=1 if t ≥ � 2/n. Assume that the event �n � �1 � � � � � I (Xi ∈ CX ) − P (CX )� ≤ t/2 � �n � i=1 occurs. Recall that �n � �1 � � � � I (Xi ∈ CX ) − P (CX )� ≥ t. � �n � i=1 Hence, it must be that �n � n �1 � � 1� � � � I (Xi ∈ CX ) − I (Xi ∈ CX )� ≥ t/2. � �n � n i=1 i=1 21 Lecture 10 Symmetrization. Pessimistic VC inequality. 18.465 We conclude 1 2 � � �� n � �1 � � � � � � � ≤ PX � � I (Xi ∈ CX ) − P (CX )� ≤ t/2�∃CX �n...
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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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