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Informational Complexity Notion of Reduction for Concept Classes Shai Ben-David Cornell University, and Technion Joint work with Ami Litman Technion

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Measures of the Informational Complexity of a class The VC-dimension of the class. The sample complexity for learning the class from random examples. The optimal mistake bound for learning the class online (or the query complexity of learning this class using membership and equivalence queries). The size of the minimal compression scheme for the class.

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Outline of the talk Defining our reductions, and the induced notion of complete concept classes. Introducing a specific family of classes that contains many natural concept classes. Prove that the class of half-spaces is complete w.r.t. that family. Demonstrate some non-reducability results. Corollaries concerning the existence of compression schemes.

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Defining Reductions We consider pair of sets (X,Y) where X is a domain and Y is a set of concepts. A concept class is a relations R over XxY (so each y Y can be viewed as the subset {x: (x,y) R} of X ). An embedding of C=(X,Y,R) into C’=(X’,Y’,R’) is a pair of functions :X X’, :Y Y’, so that (x,y) R iff ( (x), (y)) R’. C reduces to C’, denoted C C,’ if such an embedding exits.

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Relationship to Info Complexity If C C’ then, for each of the complexity parameters mentioned above, C’ is at least as complex as C. E.g., if C C,’ then, for every and the sample complexity of learning C is at most that needed for learning C’. (This is in the agnostic prediction model)

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Immediate observations If we take into account the computational complexity of the embedding functions, then we can also bound the computational complexity of learning C by that of learning C’ For every k, the class of all binary functions on a k-size domain is minimal w.r.t. the family of all classes having VC-dimension k.

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Universal Classes We say that a concept class C is universal for a family of classes F if every member of F reduces to C. Universal classes play a role analogous to that of, say, NP-hard decision problems – they are as complex as any member of the family F

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Some important classes For an integer k, let HS k denote the class of half spaces over R k. That is HS k =(R k, R k+1, H) where ((x 1,….x k ),(a 1,…a k+1 )) H iff a i x i +a k+1 0 Let PHS k denote the class of positive half spaces, that is, half spaces in which a 1 =1. Finally, let HS k 0 denote the class of homogenous half spaces (I.e., those having a k+1 =0), and PHS k 0 the class of poditive and homogenous half spaces.

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Half Spaces and Completeness The first family of classes that comes to mind is the family VC n - the family of all concept classes having VC-dimensions n. Theorem: For any n>2, no class HS k is universal for VC n (This holds even if we consider only finite classes)

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Dudley Classes (1) Next, we define a rich subfamily of VC n for which classes of half spaces are universal. Let F be a family of real valued functions over some domain set X. For any function g, let h be any real valued function over X and define a concept class D F,h = (X, F, R F,h ) where R F,h = {(x,f) : f(x)+h(x) 0}. (Note that all the PPD’s defined by Adam yesterday were of this form)

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Dudley Classes (2) Classes of the form D F,h = (X, F, R F,h ) are called Dudley Classes if the family of functions F is a vector space over the reals (with respect to point-wise addition and scalar multiplication). Examples of Dudley classes: HS k, PHS k, HS k 0, PHS k 0, and the class of all balls in any Euclidean space R k

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Dudley’s Theorem Theorem: If the a family of functions F is a vector space, then, for every h, the VC dimension of D F,h equals the (linear) dimension of the vector space F. Corollary: Easy calculations for the VC dimension of the classes HS k, PHS k, HS k 0, PHS k 0, k-dimensional balls.

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A Completeness Theorem Theorem: For every k, PHS k+1 0 is universal, (and therefore, complete) for the family of all k - dimensional Dudley classes. Proof: Let f 1, …f k be a basis for the vector space F, define :X R k+1, :F R k+1, be x) f 1 x), …. f k (x), h(x)) and for f= a i f i f)=(a 1, …a k, 1, 0)

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Corollaries k-size compression schemes for any k-dimensional Dudley class. Learning algorithms for all Dudley classes. An easy proof to Dudley’s theorem. (show that for any k –dimensional F, the class HS k 0 is embeddable into D F,h, for h=0)

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