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

Batch gradient descent

[Ex]
Continue from the preceding article about cost function, here we will use this method to minimize the cost function, and using just two parameters \( \theta_0 \) and  \(\theta_1\) . It's a more general algorithm, and not only in linear regression.

outline:

• start with some  \( \theta_0 \) and  \(\theta_1\) 
• keep changing the  \( \theta_0 \) and  \(\theta_1\)  to reduce the \( J(\theta_0, \theta_1)\) and end up at a minimum

[Def]
repeat until converge {
\( \theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1)  \)        (for j = 0 and j = 1)
}

\( \alpha\) : learning rate, it means how big step we take with creating descent.
Each step of gradient descent uses ALL the training examples.

Notice: we need updating these parameters "simultaneously" !

[Ex]
Back to our example, we take one parameter \( \theta_1\) now to get this algorithm more simpler.
[Plot 1]

Assuming this is our cost function, and if our  \( \theta_1\) starting from the blue point, since the slope is positive, so the new \( \theta_1\) from the gradient descent will decrease and  move to the green point (minimum), the moving speed depend on the \(\alpha\). In contrast, if our  \( \theta_1\) starting from the red point, it'll increase and move to the green point, too.

Although the moving speed is depend on the \(\alpha\), when \(\alpha\) is bigger the converging speed is more faster, but if the \(\alpha\) is too big, the result will be diverging.
[Plot 2]

On the contrary, if the \(\alpha\) is too small, it'll take too much time to converge.
[Plot 3]

Here is the other property of gradient descent, if our initial  \( \theta_1\) is at the local optimum, then it leaves \( \theta_1\) unchanged, because the slope will nearly equal to zero.
[Plot 4]

So, as we approach a local minimum, the gradient descent will automatically take smaller steps, this is why we don't need to decrease \(\alpha\) over time.

[For linear regression]
[Def]
$$\frac{\partial}{\partial \theta_j}J(\theta_0,\theta_1) = \frac{\partial}{\partial \theta_j}\frac{1}{2m} \sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})^2$$

\( \begin{align*} \text{repeat until convergence: } \lbrace & \newline \theta_0 := & \theta_0 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m}(h_\theta(x_{i}) - y_{i}) \newline \theta_1 := & \theta_1 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m}\left((h_\theta(x_{i}) - y_{i}) x_{i}\right) \newline \rbrace& \end{align*}\)

Cost Function

[Ex]
Here is our training set and hypothesis as below:
\begin{array}{ll}
\hfill\mathrm{Size~in~feet^2 (x)}\hfill & \hfill\mathrm{Price($)~in~1000's(y)}\hfill
\\ \hline
\\ 2104 & 460
\\ 1416 & 232
\\ 1534 & 315
\\ 852 & 178
\\ ... & ...
\\ \end{array}

In this example, we want to fit a straight line to predict the house price, then we set a simple model.
Hypothesis: \( h_\theta(x) = \theta_0 + \theta_1 x\)

It's a simple regression problem, you can choose any number for \( \theta_0\) and \( \theta_1\), so that we can get the \( h_\theta(x)\) and it's meaning the value which the model predict to the input x. As a prediction model, we want the difference between \( h_\theta(x) \) and y to be small, in other words
is this hypothesis good fit to the data?

We can measure the accuracy of our hypothesis function by using a cost function or called squared error function, in other words it's almost the same as MSE in statistics.

[Def]
$$J(\theta_0, \theta_1) = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left ( \hat{y}_{i}- y_{i} \right)^2 = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left (h_\theta (x_{i}) - y_{i} \right)^2$$
m: the number of training example
\( h_\theta(x) \): form of hypothesis