As we explored above, the exact solution for linear regression can be computed using an analytic solution. The solution for l$2$ regularized linear regression can be given as $

w$=($XXT$+\backslash$lambda I$)1$XY

$ implement this exact solution for the parameters w$.$

$[2]$

$0$s

def analytic$\_$solution$($X$,$ y$,$ lam$)$:

$"""$Compute the analytic solution for regularized regression using the equation above.

Return the value of the updated parameters.

ARGS:

X $-($n x m $)$ matrix of features

y $-($n x $1)$ column vector of targets

lam $-$ real number denoting the regularization term

$"""$

m$,$n $=$ X$.$shape

return np$.$linalg.inv$($X$.$T @ X $+$ lam$*$np$.$eye$($n$))$@X$.$T@y

Compute the analytic solution for the problem of sensory scores.

$[3]$

$0$s

start $=$ time.time$()$

print$("$Computing analytic soln for sensory score"$)$

w $=$ analytic$\_$solution$($X$\_$train, y$\_$train, DEFAULT$\_$LAMBDA$)$

duration $=$ time.time$()-$ start

training$\_$loss $=$ loss$($X$\_$train, y$\_$train, w$,$ L$2\_$LOSS$)$

test$\_$loss $=$ loss$($X$\_$test, y$\_$test, w$,$ L$2\_$LOSS$)$

print$("$Training loss achieved $\{$loss$\}".$format$($loss$=$training$\_$loss$))$

print$("$Test loss achieved $\{$loss$\}".$format$($loss$=$test$\_$loss$))$

print$("$Duration $\{$time$\}".$format$($time$=$duration$))$