Math Of Intelligence : Linear Regression

Let $x$ be the input feature and $y$ is the output that we are interested in.

For linear regression, we need a hypothesis function that predicts y, given the input feature x.

Let us assume that y is linearly dependent on x, so our hypothesis function is:

$$\begin{equation} h_\theta(x) = \theta_0 + \theta_1x \end{equation}$$

Here $\theta_i$’s are the parameters(or weights). To simplify the notation, we will drop the $\theta$ in the subscript of $h_\theta(x)$ and mention it simply as $h(x)$.

Now, we need to find a way to measure the error between our predicted output $h(x)$ and the actual value y for all our training examples.

One way to measure this error is the ordinary least squared method. TODO: Explore other cost functions

So, the cost function(or loss function)* $J(\theta)$ according to the ordinary least square method will be as follows:

*there’s some debate about whether they are the same or not but for now we’ll assume they are the same

$$\begin{equation} J(\theta) = \frac{1}{2}(h(x)-y)^2 \end{equation}$$

On expanding $h(x)$, we get

$$\begin{equation} J(\theta) = \frac{1}{2}(\theta_0 + \theta_1 x-y)^2 \end{equation}$$

Our objective is to find the values of $\theta_0$ and $\theta_1$ that minimize the loss function.

One way to do this is by using the Gradient descent method. TODO:Explore other methods to find the global minima of a function

$$\begin{equation} \theta_j = \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta) \end{equation}$$

In this method, we first initialize $\theta_j$ randomly and then update it according to the above rule to come closer the minima with each update.

Here, $\alpha$ is the learning rate.

Hence, in order to update $\theta_j$, we need to find out the partial derivative of $J(\theta)$ w.r.t. $\theta_j$. In our case j = 0 and 1

w.r.t. $\theta_0$

$$\begin{equation} \frac{\partial}{\partial \theta_0} = 2*\frac{1}{2}(\theta_0 + \theta_1 x-y).(1) \end{equation}$$ $$\begin{equation} \frac{\partial}{\partial \theta_0} = \theta_0 + \theta_1 x-y \end{equation}$$

w.r.t. $\theta_1$

$$\begin{equation} \frac{\partial}{\partial \theta_1} = 2*\frac{1}{2}(\theta_0 + \theta_1 x-y).(x) \end{equation}$$ $$\begin{equation} \frac{\partial}{\partial \theta_1} = (\theta_0 + \theta_1 x-y)x \end{equation}$$

Combining equations (4) and (6) as well as (4) and (8) we get:

$$\begin{equation} \theta_0 = \theta_0 - \alpha(\theta_0 + \theta_1 x-y) \end{equation}$$ $$\begin{equation} \theta_1 = \theta_1 - \alpha(\theta_0 + \theta_1 x-y)(x) \end{equation}$$

The above equations can be used to update the weights and hence improve the hypothesis function with every training example.

References:

1. https://see.stanford.edu/materials/aimlcs229/cs229-notes1.pdf