Gradient descent for multiple variables

Continue from the preceding example

[Ex]
$$\begin{array}{llll} \hfill\mathrm{Size~in~feet^2 (x_1)}\hfill & \hfill\mathrm{\#~bedrooms(x_2)}\hfill & \hfill\mathrm{\#~ floors(x_3)}\hfill & \hfill\mathrm{Age(x_4)}\hfill & \hfill\mathrm{Price~$1000~(y)}\hfill \\ \hline \\ 2104 & 5 & 1 & 45& 460 \\ 1416 & 3&2&40&232 \\ 1534 & 3&2&30&315 \\ 852 & 2&1&36&178 \\ ... & ...& ...& ...& ... \\ \end{array}$$

Notation:

• n = number of variables
• m = number of examples
• $x^{(i)}$ = input variables of  $i^{th}$ training example.
• $x^{(i)}_j$ = value of input variable j  in $i^{th}$ training example.

Hypothesis: $h_\theta(x) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4$

For convenience of notation, define $x_0 = 1$, it means $x^{(i)}_0 = 1$, so the hypothesis can transfer as:

$h_\theta(x) = \theta_0 x_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4$

         $= \theta^T x$

So, the definition is as below:

[Def]
$$\begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0...n}\newline \rbrace\end{align*}$$