Programming Project $4$

Due: $11$:$59$pm$,$ Friday, Dec $6$th$,2024.$

Write the scripts for the following problems in jupyter notebook and combine them with your answers as a single

PDF file. Note that failing to submit the PDF report $($with code outputs$)$ will result in a $0\mathrm{.}5-$point deduction.

$(2^($pt$))$ For a data set $\{($x$\_($i$),$y$\_($i$))\}\_($i$)=1^($n$)$ with x$\_($i$)$inR$^($d$)$ and y$\_($i$)$in$\{-1,+1\},$ consider the loss function L$($w$,$b$)=$

$\backslash$sum$\_($i$=1)^$n l$\_($i$)($z$\_($i$))$ where l$\_($i$)($z$\_($i$))=((1-$z$\_($i$))^(2))/(2)$ and z$\_($i$)=$y$\_($i$)*$g$($x$\_($i$)).$ Here the linear classifier is defined by g$($x$)=$w$^($T$)$x$+$b$.$

Derive the gradient of L and then design a gradient descent algorithm that finds the linear classifier coefficients

w$,$b by minimizing the loss function L$.$ Test your code on the iris data matrix of size $100\backslash$times $4$y$\_($i$)=-1$ y$\_($i$)=+1$ w$\_(1),$dots,w$\_(4)$ and b$.$