- Model Representation
Recall that in *regression problems*, we are taking input variables and trying to fit the output onto a *continuous* expected result function.
Linear regression with one variable is also known as "univariate linear regression."
Univariate linear regression is used when you want to predict a **single output** value from a **single input** value. We're doing **supervised learning** here, so that means we already have an idea about what the input/output cause and effect should be.
- The Hypothesis Function
Our hypothesis function has the general form: $$h_\theta(x) = \theta_0 + \theta_1 x$$
Note that this is like the equation of a straight line. We give to $$h_\theta(x)$$ values for $$\theta_0$$ and $$\theta_1$$ to get our output 'y'. In other words, we are trying to create a function called $$h_\theta$$ that is trying to map our input data (the x's) to our output data (the y's).
Example:
| x (input) | y (output) |
|---|---|
| 0 | 4 |
| 1 | 7 |
| 2 | 7 |
| 3 | 8 |
Now we can make a random guess about our $$h_\theta$$ function: $$\theta_0 = 2$$ and $$\theta_1 = 2$$. The hypothesis function becomes $$h_\theta(x) = 2 + 2 x$$.
So for input of 1 to our hypothesis, y will be 4. This is off by 3. Note that we will be trying out various values of $$\theta_0$$ and $$\theta_1$$ to try to find values which provide the best possible "fit" or the most representative "straight line" through the data points mapped on the x-y plane.
- Cost Function
We can measure the accuracy of our hypothesis function by using a **cost function**. This takes an average (actually a fancier version of an average) of all the results of the hypothesis with inputs from x's compared to the actual output y's.
$$J(\theta_0, \theta_1) = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left (h_\theta (x_{i}) - y_{i} \right)^2$$
To break it apart, it is $$\frac{1}{2} \bar{x}$$ where $$\bar{x}$$ is the mean of the squares of $$h_\theta (x_{i}) - y_{i}$$, or the difference between the predicted value and the actual value.
This function is otherwise called the "Squared error function", or "Mean squared error". The mean is halved ($$\frac{1}{2m}$$) as a convenience for the computation of the gradient descent, as the derivative term of the square function will cancel out the $$\frac{1}{2}$$ term.
Now we are able to concretely measure the accuracy of our predictor function against the correct results we have so that we can predict new results we don't have.
If we try to think of it in visual terms, our training data set is scattered on the x-y plane. We are trying to make straight line (defined by $$h_\theta(x)$$) which passes through this scattered set of data. Our objective is to get the best possible line. The best possible line will be such so that the average distances of the scattered points from the line will be the least. In the best case, the line should pass through all the points of our training data set. In such a case the value of $$J(\theta_0, \theta_1)$$ will be 0.
- Gradient Descent
So we have our hypothesis function and we have a way of measuring how accurate it is. Now what we need is a way to automatically improve our hypothesis function. That's where gradient descent comes in.
Imagine that we graph our hypothesis function based on its fields $$\theta_0$$ and $$\theta_1$$ (actually we are graphing the cost function for the combinations of parameters). This can be kind of confusing; we are moving up to a higher level of abstraction. We are not graphing x and y itself, but the parameter range of our hypothesis function and the cost resulting from selecting particular set of parameters.
We put $$\theta_0$$ on the x axis and $$\theta_1$$ on the y axis, with the cost function on the vertical z axis. The points on our graph will be the result of the **cost function** using our hypothesis with those specific theta parameters.
We will know that we have succeeded when our cost function is at the very bottom of the pits in our graph, i.e. when its value is the minimum.
The way we do this is by taking the **derivative** (the line tangent to a function) of our cost function. The slope of the tangent is the derivative at that point and it will give us a direction to move towards. We make steps down that derivative by the parameter $$\alpha$$, called the **learning rate**.
The gradient descent algorithm is:
$$ \text{repeat until convergence:} $$
$$ \theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1)$$
- for j=0 and j=1**
Intuitively, this could be thought of as:
$$\text{repeat until convergence:}$$
$$ \theta_j := \theta_j - \alpha \[\text{Slope of tangent aka derivative in j dimension}\]$$
- Gradient Descent for Linear Regression
When specifically applied to the case of linear regression, a new form of the gradient descent equation can be derived. We can substitute our actual cost function and our actual hypothesis function and modify the equation to (the derivation of the formulas are out of the scope of this course, but a really great one can be [found here](https://math.stackexchange.com/questions/70728/partial-derivative-in-gradient-descent-for-two-variables/189792#189792)):
$$\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*}$$
where $$m$$ is the size of the training set, $$\theta_0$$ a constant that will be changing simultaneously with $$\theta_1$$ and $$x_{i}, y_{i}$$ are values of the given training set (data).
Note that we have separated out the two cases for $$\theta_j$$ into separate equations for $$\theta_0$$ and $$\theta_1$$; and that for $$\theta_1$$ we are multiplying $$x_{i}$$ at the end due to the derivative.
The point of all this is that if we start with a guess for our hypothesis and then repeatedly
apply these gradient descent equations, our hypothesis will become more and more accurate.
- Gradient Descent for Linear Regression: visual worked example
[See this video](https://www.youtube.com/watch?v=WnqQrPNYz5Q) that some may find useful as it visualizes the improvement of the hypothesis as the error function reduces.
- What's Next
Instead of using linear regression on just one input variable, we'll generalize and expand our concepts so that we can predict data with multiple input variables. Also, we'll solve for $$\theta_0$$ and $$\theta_1$$ exactly without needing an iterative function like gradient descent.
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