I've seen other posts about using gradient descent for the primal form, but not the dual form.

In this book, the author discusses using (projected) gradient descent for the dual form: http://ciml.info/ (Chapter 11, Kernel Methods)

The problem is, it is unclear how to find the bias term, as it does not appear at all in the dual objective.

The objective is:

$L(\alpha)=\alpha^T 1 - \frac{1}{2}\alpha^TG\alpha $


$ G_{nm} = y_n y_m K(x_n, x_m) $

The constraint on $\alpha$ is:

$ C \ge \alpha_n \ge 0$

So, how can one find the bias term using this method?


Answering myself since I figured it out with help from http://disp.ee.ntu.edu.tw/~pujols/Support%20Vector%20Machine.pdf.

Using the KKT conditions, we have the 3 outcomes:

$ \alpha_i = 0 $, meaning $ x^{(i)} $ lies beyond the margin

$ 0 < \alpha_i < C $, implying $ x^{(i)} $ lies on the margin

$ \alpha_i = C $, meaning $ x^{(i)} $ violates the margin

We can find the data points which correspond to the second case, since if $ x^{(i)} $ lies on the margin, then:

$ y^{(i)}(w^Tx^{(i)}+b) = 1 $

Rearranging for b, we get:

$ b = y^{(i)} - w^Tx^{(i)} $


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