# Reformulating the maximal margin classifier optimization problem

Ok, so I've been trying to read up on how SVM's work and started with maximal margin classifiers. At page $$132$$ in ESL (Elements of Statistical Learning) the authors "reformulates" the optimization problem but I can't seem to understand what they are doing from $$(4.47)$$ to $$(4.48)$$. Does anyone know?

Here is an excerpt: Edit: I guess, what I don't understand is why we can arbitrarly set the magnitude of beta to $$\frac1M$$. What does a positively scaled multiple mean in this case? Just a multiple larger than $$0$$?

If $$(\beta, \beta_0)$$ satisfies the inequality $$(4.47)$$, then for any positive $$k$$, $$k>0$$, $$(k\beta, k\beta_0)$$ would satisfies the inequality as well.

Also, $$(\hat{\beta}, \hat{\beta}_0)=\left( \frac{\beta}{M\|\beta\|}, \frac{\beta_0}{M\|\beta\|} \right)$$ satiesfies the inequlity as well since

$$y_i(x_i^T\hat{\beta}+\hat{\beta}_0)=\frac{y_i}{M\|\beta\|}(x_i^T\beta+\beta_0) \ge \frac{M\|\beta|}{M\|\beta\|}=1=M\|\hat{\beta}\|$$

Let's find $$\hat{\beta}$$ and $$\hat{\beta}_0$$ directly. Note that $$\hat{\beta}$$ satisfies the property that $$\|\hat{\beta}\|=\frac1M.$$

Hence we let $$\|\hat{\beta}\|$$ to be $$\frac1M$$, hence reducing the inequality to be

$$y_i(x_i^T\hat{\beta} + \hat{\beta}_0) \ge 1$$ $$\|\hat{\beta}\|=\frac1M$$

We want to maximize $$M$$, hence, we minimize $$\frac1M$$, which is equivalent to minimizing $$\frac12\|\hat{\beta}\|^2$$.

• Thank you for answering. Okay, I understand the scaling part now. However I still don't get how we can just set beta to equal one over M. Isn't M what we want to optimize which should change between iterations of optimization?
– E.K
Mar 24, 2020 at 21:52
• Yes, I shouldn't have used the word "fix". I have tried to improve the answer. Yes, $M$ changes if you use an iterative algorithm to find $M$. Mar 25, 2020 at 2:33