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class15 - Stability of Tikhonov Regularization 9.520 April...

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Unformatted text preview: Stability of Tikhonov Regularization 9.520 Class 15, 05 April, 2006 Sasha Rakhlin Plan Review of Stability Bounds • Stability of Tikhonov Regularization Algorithms • Uniform Stability Review notation : S = { z 1 , ..., z n } ; S i,z = { z 1 , ..., z i − 1 , z, z i +1 , ..., z n } An algorithm A has uniform stability β if ∀ ( S, z ) ∈ Z n +1 , ∀ i, sup V ( f S , u ) − V ( f S i,z , u ) ≤ β. | | u ∈Z Last class : Uniform stability of β = O 1 implies good n generalization bounds. This class : Tikhonov Regularization has uniform stability = O 1 of β . n Reminder : The Tikhonov Regularization algorithm: n 1 f S = arg min V ( f ( x i ) , y i ) + λ f 2 K n i =1 f ∈H Generalization Bounds Via Uniform Stability If β = k for some k , we have the following bounds from n the last lecture: k n 2 P I [ f S ] − I S [ f S ] + ≤ 2 exp . | | ≥ n − 2( k + M ) 2 Equivalently, with probability 1 − δ , k 2 ln(2 /δ ) I [ f S ] ≤ I S [ f S ] + + (2 k + M ) . n n Lipschitz Loss Functions, I We say that a loss function (over a possibly bounded do- main X ) is Lipschitz with Lipschitz constant L if ∀ y 1 ,y 2 ,y , V ( y 1 ,y ) − V ( y 2 ,y ) . ≤ L y 1 − y 2 ∈ Y | | | | Put differently, if we have two functions f 1 and f 2 , under an L-Lipschitz loss function, sup V ( f 1 ( x ) ,y ) − V ( f 2 ( x ) ,y ) . ≤ L f 1 − f 2 ( x ,y ) | | | | ∞ Yet another way to write it: V ( f 1 , · ) − V ( f 2 , · ) f 1 ( · ) − f 2 ( · ) | | ∞ ≤ L | | ∞ Lipschitz Loss Functions, II If a loss function is L-Lipschitz, then closeness of two func- tions (in L ∞ norm) implies that they are close in loss....
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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class15 - Stability of Tikhonov Regularization 9.520 April...

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