VC Dimension

Date 03/09/2025

VC Dimension in Machine Learning

The Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity or expressive power) of a hypothesis class in machine learning. It tells us how well a model class can fit different patterns in the data.

• Definition: The VC dimension is the size of the largest set of points that can be shattered (i.e., correctly classified in all possible ways) by the hypothesis class.
  
  

• Example:
  
  

  • A linear classifier in 2D has VC dimension = 3 (it can shatter any 3 points in general position, but not 4).
    
    

  • A set of intervals on the real line has VC dimension = 2.
    
    

• Interpretation:
  
  

  • Higher VC dimension → more complex model → higher capacity to fit data.
    
    

  • Very high VC dimension → risk of overfitting.
    
    

• Relevance:
  
  

  • Used in statistical learning theory to analyze the trade-off between model complexity and generalization ability.
    
    

  • Helps in bounding the generalization error of classifiers.
    
    

In short: VC dimension quantifies the complexity of a model class and plays a key role in understanding the balance between underfitting and overfitting in machine learning.

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