# Calculating Lagrange coefficients for SVM in Python

I'm trying to write a full SVM implementation in Python and I have a few issues computing the Lagrange coefficients.

First let me rephrase what I understand from the algorithm to make sure I'm on the right path.

If $x_1, x_2, ..., x_n$ is a dataset and $y_i \in \{-1, 1\}$ is the class label of $x_i$, then $$\forall i, y_i(w^Tx_i + b) \geq 1$$

So we just need to solve an optimization problem to

minimize $\|w\|^2$

subject to $y_i(w^Tx_i + b) \geq 1$

In terms of Lagrange coefficients, this translates into finding $w$, $b$ and $\alpha=(\alpha_1, \alpha_2, ... \alpha_n) \neq0$ and $\geq0$ minimizing: $$L(\alpha, w, b) = \frac12 \|w\|^2 - \sum \alpha_i(y_i(w^Tx + b)-1)$$

Now since $$\frac{\partial L}{\partial w}=0 \implies w=\sum \alpha_i y_i x_i$$ and $$\frac{\partial L}{\partial b}=0 \implies \sum y_i \alpha_i = 0$$ we can rewrite it as $$L(\alpha, w, b) = Q(\alpha)=\sum \alpha_i - \frac12\sum \sum \alpha_i \alpha_j y_i y_j x_i^T x_j$$ with constraints $$\alpha_i \geq 0 \ \text{and} \ \sum \alpha_i y_i = 0$$

So I'm trying to solve the optimization problem using Python, and the only free package I could find is called cvxopt.

I'd like some help to solve this, I couldn't find any good example about this, and while I understand the theory, I'm having a hard time translating it into code (I would have expected the opposite since I'm more from a programming background).

Note that at some point I'll want to solve it using Kernels $$L(\alpha, w, b) = Q(\alpha)=\sum \alpha_i - \frac12\sum \sum \alpha_i \alpha_j y_i y_j K(x_i,x_j)$$ but I'm not sure what the implications are regarding solving this in code.

Any help would be greatly appreciated, I'm really lost on how to implement this in Python. If you have a better module to solve the optimization problem I'd like to read about it as well.

I have used cvxopt to implement an SVM before, however in matlab not python. It will definitely serve your purpose, whether its efficient enough will depend on what you are using it for. The most efficient SVMs do not use a QP solver package, they take advantage of some optimizations unique to SVM. Many use an SMO style algorithm to solve it.

LibSVM is an SVM package which uses the algorithm in Working Set Selection Using Second Order Information for Training Support Vector Machines. The code is open source, if you are interested in looking at how its implemented. It also does have a python interface.

SVMLight is another package, they use a different algorithm (see their site for references). It is also open source and has a python interface.

• Thanks for the informative answer (which I think supercedes mine), and welcome to scicomp! – Aron Ahmadia May 2 '12 at 18:21
• +1 interesting answer and I've started to look at your great links which are helping me a lot ! – Charles Menguy May 4 '12 at 1:39

The general form of your optimization problem is a Quadratic Program, regardless of whether you are using the kernel trick or a linear kernel. It sounds like cvxopt will be sufficient for what you're trying to do, but other pythonauts on here have had luck with OpenOpt as well.

• Aron, do you know if the Ipopt Python wrapper ever got fixed? – Geoff Oxberry May 2 '12 at 10:35
• One of David Ketcheson's students got it working with OpenOpt (which can use it with a quasi-Newton algorithm), but has had some difficulties with getting the OpenOpt stack going on OS X. – Aron Ahmadia May 2 '12 at 11:16