class07 (1) - Stability of Tikhonov Regularization 9.520...

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Stability of Tikhonov Regularization 9.520 Class 07, March 2003 Alex Rakhlin
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Plan Review of Stability Bounds Stability of Tikhonov Regularization Algorithms
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Uniform Stability Review notation : S = { z 1 , ..., z } ; S i,z = { z 1 , ..., z i - 1 , z, z i +1 , ..., z } c ( f, z ) = V ( f ( x ) , y ), where z = ( x , y ). An algorithm A has uniform stability β if ( S, z ) ∈ Z +1 , i, sup u ∈Z | c ( f S , u ) - c ( f S i,z , u ) | ≤ β. Last class : Uniform stability of β = O 1 implies good generalization bounds. This class : Tikhonov Regularization has uniform stability of β = O 1 . Reminder : The Tikhonov Regularization algorithm: f S = arg min f ∈H 1 X i =1 V ( f ( x i ) , y i ) + λ k f k 2 K
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Generalization Bounds Via Uniform Stability If β = k for some k , we have the following bounds from the last lecture: P | I [ f S ] - I S [ f S ] | ≥ k + 2 exp - 2 2( k + M ) 2 ! . Equivalently, with probability 1 - δ , I [ f S ] I S [ f S ] + k + (2 k + M ) s 2 ln(2 ) .
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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 0 ∈ Y , | V ( y 1 , y 0 ) - V ( y 2 , y 0 ) | ≤ L | y 1 - y 2 | . Put differently, if we have two functions f 1 and f 2 , under an L -Lipschitz loss function, sup ( x ,y ) | V ( f 1 ( x ) , y ) - V ( f 2 ( x ) , y ) | ≤ L | f 1 - f 2 | . Yet another way to write it: | c ( f 1 , · ) - c ( f 2 , · ) | L | f 1 ( · ) - f 2 ( · ) |
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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. The converse is false — it is possible for the difference in loss of two functions to be small, yet the functions to be far apart (in L ). Example: constant loss. The hinge loss and the -insensitive loss are both L -Lipschitz with L = 1. The square loss function is L Lipschitz if we can bound the y values and the f ( x ) values generated. The 0 - 1 loss function is not L
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