Instead of updating $m_j$ by computing the mean, we use stochastic gradient descent,

to remodel the mi such that :

$m_i=m_i+{\displaystyle \underset{?}{\overset{?}{}}}{z}_{kn}(x_n-m_i)$

with learning rate $($step size$).$

Recall that in the update step of the standard K$-$mean algorithm, we assign each cluster

centre to be the mean $($centroid$)$ of the data points closest to that centre. It turns out that a

particular choice of the learning rate $($which may be different for each cluster$)$ makes the

two algorithms $($batch gradient descent and the standard k$-$means algorithm$)$ have identical

update steps. Let's focus on the update for the first cluster, with center $m_1.$

d$)$ Implement the equation instead of mean in the standard algorithm and find the

learning rate value, $$ such that the value of both algorithms perform the same

update for $m_1.$

$(11$ marks$)$