VC Dimension and the Fundamental Theorem of Statistical Learning: A Complete Mathematical Derivation
By
alok-g
If you only eat one bagel today, this is the bagel.
Summary
This article explains the theoretical foundations of statistical learning theory, specifically addressing when learning from data is guaranteed to work. It builds from first principles—starting with Markov's inequality and Hoeffding's lemma—to prove the Fundamental Theorem of Statistical Learning, which states that a hypothesis class is learnable if and only if it has finite VC dimension. The article is a comprehensive, from-scratch mathematical treatment of VC dimension and its relationship to learnability.
Key quotes
· 3 pulledThis post answers a single question: when does learning from data actually work?
You train a model on samples, it performs well on those samples, and you hope it performs well on new data. When is that hope justified?
The answer turns out to be a clean equivalence: a hypothesis class is learnable if and only if it has finite VC dimension.
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