Constrained versions of Sauer’s lemma

AbstractLet [n]={1,…,n}. For a function h:[n]→{0,1}, x∈[n] and y∈{0,1} define by the width ωh(x,y) of h at x the largest nonnegative integer a such that h(z)=y on x−a≤z≤x+a. We consider finite VC-dimension classes of functions h constrained to have a width ωh(xi,yi) which is larger than N for all points in a sample ζ={(xi,yi)}1ℓ or a width no larger than N over the whole domain [n]. Extending Sauer’s lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes

Complexity of hyperconcepts

AbstractIn machine-learning, maximizing the sample margin can reduce the learning generalization error. Samples on which the target function has a large margin (γ) convey more information since they yield more accurate hypotheses. Let X be a finite domain and S denote the set of all samples S⊆X of fixed cardinality m. Let H be a class of hypotheses h on X. A hyperconcept h′ is defined as an indicator function for a set A⊆S of all samples on which the corresponding hypothesis h has a margin of at least γ. An estimate on the complexity of the class H′ of hyperconcepts h′ is obtained with explicit dependence on γ, the pseudo-dimension of H and m