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\lecture{CS5350: Machine Learning}{HW4: Loss and Optimization}{Due 21 Oct 2008}

\section{Written Exercises}

Answer the following questions in 25-100 words each:

\bee

\item Consider the degree-two polynomial kernel defined by $K(\vec
  x,\vec z) = (1 + \vec x \T \vec z)^2$.  Expand this out completely
  for the three-dimensional case (i.e., $\vec x = \langle x_1, x_2,
  x_3 \rangle$ and $\vec z = \langle z_1, z_2, z_3 \rangle$.  Verify
  that this has the same form as the quadratic expansion, although
  with different coefficients on the terms.

\item Continuing from the previous question, what is the form of
  $\Phi$ so that $K(\vec x, \vec z) = \Phi(\vec x)\T\Phi(\vec z)$?
  (You need only consider the three-dimensional data case.)  How does
  this differ from the expansion $\Phi(\vec x) = \langle x_1, x_2,
  x_3, x_1^2, x_2^2, x_3^2, x_1 x_2, x_1 x_3, x_2, x_3 \rangle$ that
  we discussed in class?

\item Consider optimizing an SVM with \emph{squared} loss on the $\xi$
  variables.  That is, an optimization problem of the form:

  \begin{align*}
    \min_{\vec w,b} & \frac 1 2 \norm{\vec w}^2 + \la \sum_n \xi_n^2 \\
    \text{s.t.} & y_n (\vec w\T\vec x_n + b) \geq 1 - \xi_n \quad\quad (\forall n)\\
                & \xi_n \geq 0 \quad\quad (\forall n)
  \end{align*}

  Construct the dual formulation for this problem.  In particular,
  construct the Lagrangian, optimize it with respect to $\vec w$ and
  $b$, plug these solutions back in and get an optimization problem
  just in terms of the dual (Lagrange) variables $\vec \al$.  How does
  this compare to the dual formulation for the standard SVM?

\item (6350 only) For $D$ dimensional data, consider using the degree
  $d$ polynomial kernel defined by $K(\vec x, \vec z) = (1 + \vec x\T
  \vec z)^d$.  What is the general form of the expansion?  What are
  the coefficients on all the different forms in the expansion?
\ene


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