$[$Perceptron$]$ The Perceptron algorithm finds a decision boundary for binary classification below.

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w$($x$1,$ x$2)=+1$ w$1$x$1+$ w$2$x$2>0$

w$($x$1,$ x$2)=1$ w$1$x$1+$ w$2$x$20$

$(1)$

$2\mathrm{.}1$ Assume a data set consists only of a single data point $\{($x$1,$ x$2),+1)\}.$ How many iterations will

be required until it finds a decision rule when the initial w$0=(0,0)$ and step size $=1?$

$2\mathrm{.}2$ How many iterations will be required until it finds a decision rule if the initial weight vector

w$0$ was initialized randomly and not as the all$-$zero vector?

$2\mathrm{.}3$ Suppose you have the three data points below. Please complete the iterative updates for wi

in Perceptron algorithm. The initial w$0=(0,0)$ and step size $=1$; the point $(0,1)$ is detected as

misclassification at the first iteration so w is updated by the point: w$1=$ w$0+1(0,1)$

iteration $$ w

$0$ w$0=(0,0)$

$1$ w$1=(0,0)+(0,1)=(0,1)$

$2...$

$2$