A one-parameter class with infinite VC dimension

ProofAdvanced⏱ 1 hourLearning Theoryvc dimensioncounterexampleparameter counting

Theory: VC Dimension, Sauer's Lemma and Rademacher Complexity →

Problem

Show that $\Hcal = \{x\mapsto\sign\sin(\omega x) : \omega>0\}$ shatters arbitrarily large finite subsets of $\R$ , hence has infinite VC dimension despite having one real parameter. Conclude that capacity is not parameter count.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

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