class15

# class15 - Stability of Tikhonov Regularization 9.520 April...

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 24

class15 - Stability of Tikhonov Regularization 9.520 April...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online