Ex. 4.7

Ex. 4.7

Consider the criterion

\[\begin{equation} D^\ast(\beta,\beta_0) = -\sum_{i=1}^Ny_i(x_i^T\beta + \beta_0),\non \end{equation}\]

a generalization of (4.41) in the textbook where we sum over all the observations. Consider minimizing \(D^\ast\) subject to \(\|\beta\|=1\). Describe this criterion in words. Does it solve the optimal separating hyperplane problem?

Soln. 4.7

When \(\|\beta\| = 1\), \(\beta^Tx_i + \beta_0\) is the signed distance of \(x_i\) to the hyperplane \(\beta^Tx + \beta_0 = 0\). This does not solve the optimal separating hyperplane problem. Optimal separating hyperplane is actually solving a max-min problem such that each point satisfies the distance requirement, however minimizing \(D^\ast\) does not have such pointwise constraint.