## Monday, July 20, 2015

### What is Machine Learning?

Two definitions of Machine Learning are offered. Arthur Samuel described it as: "the field of study that gives computers the ability to learn without being explicitly programmed." This is an older, informal definition.

Tom Mitchell provides a more modern definition: "A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E."

Example: playing checkers.

E = the experience of playing many games of checkers
T = the task of playing checkers.
P = the probability that the program will win the next game.

Supervised Learning

In supervised learning, we are given a data set and already know what our correct output should look like, having the idea that there is a relationship between the input and the output.

Supervised learning problems are categorized into "regression" and "classification" problems. In a regression problem, we are trying to predict results within a continuous output, meaning that we are trying to map input variables to some continuous function. In a classification problem, we are instead trying to predict results in a discrete output. In other words, we are trying to map input variables into discrete categories.

Example:

Given data about the size of houses on the real estate market, try to predict their price. Price as a function of size is a continuous output, so this is a regression problem.

We could turn this example into a classification problem by instead making our output about whether the house "sells for more or less than the asking price." Here we are classifying the houses based on price into two discrete categories.

Unsupervised Learning

Unsupervised learning, on the other hand, allows us to approach problems with little or no idea what our results should look like. We can derive structure from data where we don't necessarily know the effect of the variables.

We can derive this structure by clustering the data based on relationships among the variables in the data.

With unsupervised learning there is no feedback based on the prediction results, i.e., there is no teacher to correct you. It’s not just about clustering. For example, associative memory is unsupervised learning.

Example:

Clustering: Take a collection of 1000 essays written on the US Economy, and find a way to automatically group these essays into a small number that are somehow similar or related by different variables, such as word frequency, sentence length, page count, and so on.

Associative: Suppose a doctor over years of experience forms associations in his mind between patient characteristics and illnesses that they have. If a new patient shows up then based on this patient’s characteristics such as symptoms, family medical history, physical attributes, mental outlook, etc the doctor associates possible illness or illnesses based on what the doctor has seen before with similar patients. This is not the same as rule based reasoning as in expert systems. In this case we would like to estimate a mapping function from patient characteristics into illnesses.

Linear Regression with One Variable

Model Representation

Recall that in *regression problems*, we are taking input variables and trying to map 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 what the input/output cause and effect should be.

The Hypothesis Function

Our hypothesis function has the general form:

$h_{θ}(x)=θ_{0}+θ_{1}x$

We give to $h_{θ}$ values for $θ_{0}$ and $θ_{1}$ to get our output 'y'. In other words, we are trying to create a function called $h_{θ}$ that is able to reliably 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 hθ function: $θ_{0}=2$ and $θ_{1}=2$. The hypothesis function becomes $h_{θ}(x)=2+2x$.

So for input of 1 to our hypothesis, y will be 4. This is off by 3.

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.

$J(θ_{0},θ_{1})=\frac{1}{2m}\sum\limits_{i=1}^{m}(h_{θ}(x^{(i)})−y^{(i)})^{2}$

To break it apart, it is $\frac{1}{2m}\overline{x}$ where $\overline{x}$ is the mean of the squares of $h_{θ}(x^{(i)})−y^{(i)}$, 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 $\frac{1}{2m}$ 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.

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 $θ_{0}$ and $θ_{1}$ (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 guesses of our hypothesis function.

We put $θ_{0}$ on the x axis and $θ_{1}$ on the z axis, with the cost function on the vertical y 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 α, called the learning rate.

repeat until convergence:
$θ_{j} := θ_{j} − α\frac{∂}{∂θ_{j}}J(θ_{0}, θ_{1})$
for j = 0 and j = 1

Intuitively, this could be thought of as:

repeat until convergence:
$θ_{j}:=θ_{j} − α$ [Slope of tangent aka derivative]

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:

repeat until convergence: {
$θ_{0} := θ_{0} − α \frac{1}{m}\sum\limits_{i=1}^{m}(h_{θ}(x^{(i)})−y^{(i)})^{2}$
$θ_{1} := θ_{1} − α \frac{1}{m}\sum\limits_{i=1}^{m}((h_{θ}(x^{(i)})−y^{(i)})x^{(i)})^{2}$
}
where m is the size of the training set, $θ_{0}$ a constant that will be changing simultaneously with $θ_{1}$ and $x^{(i)}$, $y^{(i)}$ are values of the given training set (data).

Note that we have separated out the two cases for $θ_{j}$ and that for $θ_{1}$ 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.

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 $θ_{0}$ and $θ_{1}$ exactly without needing an iterative function like gradient descent.

SOURCE: Coursera