$3\mathrm{.}1(20$ points$)$ Perceptron Algorithm $-$ Deterministic Iteration

In this section, we will apply the perceptron learning algorithm to solve the same binary classification problem as logistic regression above: We need to predict a binary label y in $\{1,1\}$ for a feature vector x $=[$x$0,$x$1].$ The decision rule of the perceptron model is defined as:

$(1,$ if wT x $+$ b $>=0,$

f $($x; w$,$ b$)=1,$ otherwise. $(2)$

where w $=[$w$0,$ w$1]$ is the weight vector, and b is the bias scalar. Given a training dataset Straining $=\{($xi$,$ yi$)\},$ i $=1,...,$ n$\},$ we define the training error etraining as:

$1$ Xn

etraining $=$ n

and the test error etest on the test set Stest can be defined in the same way. In the perceptron algorithm, we aim to directly minimize the training error etraining to obtain the optimal parameters w$,$b$.$ If we represent data points in training set Straining as matrices X $=[$x$1,$x$2,...,$xn$]$T and Y $=[$y$1,$y$2,...,$yn$]$T$,$ the perceptron algorithm is shown as below: