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class14 - Generalization Bounds and Stability 9.520 April...

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Unformatted text preview: Generalization Bounds and Stability 9.520 Class 14, 03 April 2006 Sasha Rakhlin Plan Generalization Bounds • Stability • Generalization Bounds Using Stability • Algorithms We define an algorithm A to be a mapping from a training set S = { z 1 , . . . , z n } to a function f S . Here, z i ≡ ( x i , y i ). Throughout the next several lectures, we assume that A is deterministic, and that A does not depend on the ordering of the points in the training set. How can we measure quality of f S ? Risks Recall that in Lecture 2 we’ve defined the true (expected) risk: I [ f S ] = IE ( x ,y ) [ V ( f S ( x ) , y )] = V ( f S ( x ) , y ) dµ ( x , y ) and the empirical risk: 1 n I S [ f S ] = V ( f S ( x i ) , y i ) . n i =1 Note : the true and empirical risks are denoted in Bous- ˆ quet & Elisseeff as R ( A , S ) and R ( A , S ), respectively, to emphasize the algorithm that produced f S . Note : we will denote the loss function as V ( f, z ) or as V ( f ( x ) , y ), where z = ( x , y ). Generalization Bounds Our goal is to choose an algorithm A so that I [ f S ] will be small. This is difficult because we can’t measure I [ f S ]. We can, however, measure I S [ f S ]. A generalization bound is a (probabilistic) bound on how big the defect D [ f S ] = I [ f S ] − I S [ f S ] can be. If we can bound the defect and we can observe that I S [ f S ] is small, then I [ f S ] must be small. Properties of Generalization Bounds, I What will a generalization bound depend on? A gener- alization bound is a way of saying that the performance of a function on the training set has to be similar to its performance on future examples. For this reason, gener- alization bounds are always probabilistic : they hold with some (high) probability, to take into account the (low) chance that you’ll see a very unrepresentative training set. Properties of Generalization Bounds, II Generalization bounds depend on some measure of the size of the hypothesis space we allow ourselves to choose from. As the hypothesis space gets smaller, the generalization bound will get tighter (but the empirical performance will often go down). As the hypothesis space gets bigger, the generalization bound will get looser....
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class14 - Generalization Bounds and Stability 9.520 April...

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