We are interested in performing ordinary least squares regression given data $x,Y$ to find parameters $,{}_0$ that minimize the mean squared error objective:

$J(,{}_0)=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{L}_s(x^{(i)},y^{(i)}$;${}_s{}_0)$

where the squared loss is

$L_s(x^{(i)},y^{(i)}$;$,{}_0)=(hat(y{)}^{(i)}-y^{(i)}{)}^2=({}^T{x}^{(i)}+{}_0-y^{(i)}{)}^2$

Note that the $L_s(x^{(i)},y^{(i)}$;$,{}_0)$ notation here is used to emphasize that the loss depends on both the data sample $(x^{(i)},y^{(i)}),$ and the parameters, $$ and ${}_0.$ Compared to the $L_s$ notation as used in some part of the notes and the lab, we see that $L_s(x^{(i)},y^{(i)}$;$,{}_0)=L_s(h(x^{(i)}$;$,{}_0),y^{(i)}),$ where $h(x^{(i)}$;$,{}_0)={}^T{x}^{(i)}+{}_0.$ Now implement ${}^{**}$ as found in the previous problem, using symbols $x$ and $Y$ for the data matrix and outputs, and np$.$dot$($or @ shorthand$),$ np$.$transpose $($or $.$ T shorthand$),$ np$.$linalg.inv.

# Enter an expression to compute and set th to the optimal theta th $=$ None