# VC dimension

(45 minutes to learn)

## Summary

In statistical learning theory, VC dimension is a measure of the complexity of a continuous hypothesis class (such as linear classifiers). While it often corresponds to the number of parameters in a model, this isn't always the case. There is a general bound on the generalization error of a classifier in terms of its error on a training set and its VC dimension.

## Context

This concept has the prerequisites:

- binary linear classifiers (Binary linear classifiers are a canonical example for VC dimension.)

## Goals

- Know the definition of VC dimension.

- What is the VC dimension of a linear classifier in D dimensions?

- Give an example of a hypothesis class whose VC dimension is different from the number of parameters. (Hint: you can find one with a single parameter but infinite VC dimension.)

## Core resources (read/watch one of the following)

## -Free-

→ Coursera: Machine Learning

An online machine learning course aimed at advanced undergraduates.

Other notes:

- Click on "Preview" to see the videos.

→ Stanford's Machine Learning lecture notes

Lecture notes for Stanford's machine learning course, aimed at graduate and advanced undergraduate students.

→ The Elements of Statistical Learning

A graudate-level statistical learning textbook with a focus on frequentist methods.

## -Paid-

→ An Introduction to Computational Learning Theory (1994)

A graduate level textbook on computational learning theory.

- Section 3.2, "The Vapnik-Chervonenkis dimension," pages 50-51
- Section 3.3, "Examples of the VC dimension," pages 51-54

## See also

- Structural risk minimization is an idea based on VC dimension which justifies algorithms such as SVMs.
- Rademacher complexity is an alternative measure of model complexity.