Answers.Body: <p>SVM looks to maximize the margin <span class="math-container">$ρ$</span> of support vectors to reduce the model's effective complexity for overfitting and improves its generalization ability. The bound <span class="math-container">$d_{VC}(ρ)≤d_{VC}(0)$</span> shows that while both SVM and PLA may have the same VC dimension for linear separators in worst case scenarios, the margin maximization in SVM effectively reduces its VC-dimension <em>on average</em> compared to PLA, making it a better choice for generalization.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-12-15T08:49:05.067Z
Answers.LastActivityDate: 2024-12-15T08:49:05.067Z
Answers.UserDisplayName: cinch
Body: <p>For linear separators in <span class="math-container">$d$</span> dimensions, we have
  <span class="math-container">$d_{VC}(\rho) \le d_{VC}(0) =d+1$</span> where <span class="math-container">$\rho$</span> is the margin.</p>
  <p>Based on the above result, does that mean SVM consistently produces a classifier
  with superior generalization performance compared to any linear separator derived from the Perceptron Learning Algorithm (PLA), as PLA does not maximize the separation margin between classes?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-12-15T22:18:31.927Z
Comments.Text: Can you clarify what $d_{VC}(p)$ means? VC probably stands for the VC dimension, but can you describe function and number? I suppose that's the VC dimension of some function of an SVM, but please put that into words. Also, where did you take this formula from?
Comments.UserDisplayName: nbro
Tags: <support-vector-machine><vc-dimension><vc-theory>
Title: Confusion about the use of VC in SVMs
UserDisplayName: DSPinfinity
ThreadText: # Confusion about the use of VC in SVMs
  
  (Asked by: username_0 on 2024-12-14)
  
  For linear separators in $d$ dimensions, we have
  $d\_{VC}(\rho) \le d\_{VC}(0) =d+1$ where $\rho$ is the margin.
  
  Based on the above result, does that mean SVM consistently produces a classifier
  with superior generalization performance compared to any linear separator derived from the Perceptron Learning Algorithm (PLA), as PLA does not maximize the separation margin between classes?
  
  **Tags:** support-vector-machine, vc-dimension, vc-theory
  
  > **Comments:**
  > **username_1**: Can you clarify what $d_{VC}(p)$ means? VC probably stands for the VC dimension, but can you describe function and number? I suppose that's the VC dimension of some function of an SVM, but please put that into words. Also, where did you take this formula from?  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2024-12-15. Score: 1
  
  SVM looks to maximize the margin $ρ$ of support vectors to reduce the model's effective complexity for overfitting and improves its generalization ability. The bound $d\_{VC}(ρ)≤d\_{VC}(0)$ shows that while both SVM and PLA may have the same VC dimension for linear separators in worst case scenarios, the margin maximization in SVM effectively reduces its VC-dimension *on average* compared to PLA, making it a better choice for generalization.
  
  
  ---
  
  