Uniform Convergence (Combinatorics)
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. For a class of predicates H,! defined on a set X,! and a set of samples x=(x_{1},x_{2},dots,x_{m}),!, where x_{i}in X,!, the empirical frequency of hin H,! on x,! is widehat{Q_{x}}(h)=frac{1}{m} {i:1leq ileq m,h(x_{i})=1} ,!. The Uniform Convergence Theorem states, roughly,that if H,! is "simple" and we draw samples independently (with replacement) from X,! according to a distribution P,!, then with high probability all the empirical frequency will be close to its exp...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. For a class of predicates H,! defined on a set X,! and a set of samples x=(x_{1},x_{2},dots,x_{m}),!, where x_{i}in X,!, the empirical frequency of hin H,! on x,! is widehat{Q_{x}}(h)=frac{1}{m} {i:1leq ileq m,h(x_{i})=1} ,!. The Uniform Convergence Theorem states, roughly,that if H,! is "simple" and we draw samples independently (with replacement) from X,! according to a distribution P,!, then with high probability all the empirical frequency will be close to its expectation, where the expectation is given by Q_{P}(h)=P{yin X:h(y)=1},!. Here "simple" means that the Vapnik-Chernovenkis dimension of the class H,! is small relative to the size of the sample.In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole.
