Consider the dual with no offset and not slack.

In the dual we have that for data points that are support vectors:

$$\alpha_t > 0$$

and that the constraint is satisfied with equality (since a support vector!):

$$y^{(t)}\sum^n_{t'=1} \alpha_{t'}y^{(t')}x^{(t')} \cdot x^{(t)} = 1$$

However, just because $\alpha_t > 0$ its not obvious to me if all the support vector have the same value of $\alpha_t$. Do all of the support vectors have the same value of $\alpha_t$? I was kind of looking for a mathematical justification or intuition for my question. My guess is that they would have the same $\alpha_t$ value in the absence of slack. My other guess was that, $\alpha_i$ intuitively, says the important of that point, right? (maybe thats wrong) If thats true then, not all support vector have the same value of $\alpha_i$ because not all support vectors are essential support vectors. What I mean is, that some support vectors will lie on the margin boundary but removing them will not necessarily change the margin (for example, consider the case where you have 3 SVs in a line, you can remove the middle one and keep the same boundary). So do essential support vectors get larger value of $\alpha_t$?

Also, I was wondering, what would happen in the case of slack and offset to the value of $\alpha_t$. I was guessing that closer points to the decision boundary, get larger value of $\alpha$? However, are points inside the margin even consider support vectors?

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    $\begingroup$ @user603 in the case of no offset, for support vectors we have $y^{(i)} \theta \cdot x^{(i)} = 1$ (I said no slack and then later asked whats going on with slack). If $\theta = \sum^n_{i=1} \alpha_i y^{i} x^{(i)}$ then the constrain becomes: $y^{(i)} \sum^n_{i=1} \alpha_i y^{i} x^{(i)} \cdot x^{(i)} = 1$. Which is what I wrote. For support vectors we can change the inequality to an equality because its a support vector. There is nothing wrong about what I wrote. $\endgroup$ – Pinocchio Mar 20 '14 at 19:36
  • $\begingroup$ thanks for the clarification. I see now: I had read the equations a bit too fast and clearly not paid attention to the surrounding text. $\endgroup$ – user603 Mar 20 '14 at 19:39
  • $\begingroup$ Don't worry, I think I also wrote the question a little weirdly, let me write it clearer, I agree its not the best wording I could have chosen. You should try look for a clarification before writing an angry comment and down voting though :P I was so sad for a second hehe But I understand the misunderstanding. I'll improve my question. $\endgroup$ – Pinocchio Mar 20 '14 at 19:41

The cost function of SVM is: $$\min_{\alpha,\xi,b} \frac{1}{2}\|\mathbf{w}\|^2 + C\sum_{i=1}^n \xi_i$$

where $\xi$ are the slack variables (0 for hard margin) and $$ ||\mathbf{w}||^2 =\mathbf{w}^T\mathbf{w} = \sum_{i\in \mathcal{S}}\sum_{j\in \mathcal{S}} \alpha_i \alpha_j y_i y_j \kappa(\mathbf{x}_i,\mathbf{x}_j) $$

A couple of important properties can be derived from the Langrangian (here with slack variables):

$$L_p = \frac{1}{2}||\mathbf{w}||^2+C\sum_{i=1}^n\xi_i -\sum_{i=1}^n\alpha_i\Big[y_i\big(<\mathbf{w},\varphi(\mathbf{x}_i)>+b\big)-(1-\xi_i)\Big]-\sum_{i=1}^n\mu_i\xi_i. $$

By setting the partial derivative to $\xi$ (the slack variables) to zero, we obtain the following equation in $\alpha$: $$\frac{\partial L_p}{\partial \xi_i}=0 \quad \rightarrow \quad \alpha_i=C-\mu_i, \quad \forall i$$ In the presence of slack variables, all support values ($\alpha$'s) are bound by $0$ and $C$. They are not all equal.

In the absence of slack variables (hard margin SVM), $\alpha$ values are not equal because $\alpha$ is a direct part of the objective function as shown in the first two equations. In hard margin SVM, the cost function is a weighted quadratic function in $\alpha$ where the kernel evaluations between corresponding points determine the weights. Since not all weights are equal, the $\alpha$ values aren't either.

This is more apparant in least-squares SVM (LS-SVM), where all training points become support vectors due to the problem formulation. One of the original methods proposed to obtain sparsity in LS-SVM involved pruning support vectors with small $\alpha$ values.

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  • $\begingroup$ In your Lagrangian, what is $\mu_i$? $\endgroup$ – Pinocchio Mar 20 '14 at 19:58
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    $\begingroup$ @Pinocchio $\mu$ is the vector of dual variables (Langrange multipliers) associated with the slack variables $\xi$. $\endgroup$ – Marc Claesen Mar 20 '14 at 20:00
  • $\begingroup$ Aahhh I see. We have two lagrange multipler variables, one for the two sets of constraints. One for the slack constraints and the other for the "normal" margin one. Thanks now it makes sense. I was put me off because $\mu$ is usually reserved for denoting the mean, so I was very confused. I would have used $\beta_i$, since we are using $\alpha$ to denote the first set of constraints. :p But its not that important. $\endgroup$ – Pinocchio Mar 20 '14 at 20:09

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