Stanford Machine Learning: (3). Regularization

The problem of overfitting

So far we've seen a few algorithms - work well for many applications, but can suffer from the problem of overfitting
What is overfitting?
What is regularization and how does it help

Overfitting with linear regression

Using our house pricing example again

Fit a linear function to the data - not a great model

This is underfitting - also known as high bias
Bias is a historic/technical one - if we're fitting a straight line to the data we have a strong preconception that there should be a linear fit

In this case, this is not correct, but a straight line can't help being straight!

Fit a quadratic function

Works well

Fit a 4th order polynomial

Now curve fit's through all five examples

Seems to do a good job fitting the training set
But, despite fitting the data we've provided very well, this is actually not such a good model

This is overfitting - also known as high variance

Algorithm has high variance

High variance - if fitting high order polynomial then the hypothesis can basically fit any data
Space of hypothesis is too large





To recap, if we have too many features then the learned hypothesis may give a cost function of exactly zero

But this tries too hard to fit the training set
Fails to provide a general solution - unable to generalize (apply to new examples)

Overfitting with logistic regression

Same thing can happen to logistic regression

Sigmoidal function is an underfit
But a high order polynomial gives and overfitting (high variance hypothesis)





Addressing overfitting

Later we'll look at identifying when overfitting and underfitting is occurring
Earlier we just plotted a higher order function - saw that it looks "too curvy"

Plotting hypothesis is one way to decide, but doesn't always work
Often have lots of a features - here it's not just a case of selecting a degree polynomial, but also harder to plot the data and visualize to decide what features to keep and which to drop
If you have lots of features and little data - overfitting can be a problem

How do we deal with this?

1) Reduce number of features

Manually select which features to keep

Model selection algorithms are discussed later (good for reducing number of features)
But, in reducing the number of features we lose some information

Ideally select those features which minimize data loss, but even so, some info is lost

2) Regularization

Keep all features, but reduce magnitude of parameters θ
Works well when we have a lot of features, each of which contributes a bit to predicting y

Cost function optimization for regularization

Penalize and make some of the θ parameters really small

e.g. here θ3 and θ4





The addition in blue is a modification of our cost function to help penalize θ3 and θ4

So here we end up with θ3 and θ4 being
close to zero (because the constants are massive)
So we're basically left with a quadratic function





In this example, we penalized two of the parameter values

More generally, regularization is as follows

Regularization

Small values for parameters corresponds to a simpler hypothesis (you effectively get rid of some of the terms)
A simpler hypothesis is less prone to overfitting

Another example

Have 100 features x1, x2,
..., x100
Unlike the polynomial example, we don't know what are the high order terms

How do we pick the ones to pick to shrink?

With regularization, take cost function and modify it to shrink all the parameters

Add a term at the end

This regularization term shrinks every parameter
By convention you don't penalize θ0 -
minimization is from θ1 onwards





In practice, if you include θ0 has
little impact
λ is the regularization parameter

Controls a trade off between our two goals

1) Want to fit the training set well
2) Want to keep parameters small

With our example, using the regularized objective (i.e. the cost function with the regularization term) you get a much smoother curve which fits the data and gives a much better hypothesis

If λ is very large we end up penalizing ALL the parameters (θ1, θ2 etc.)
so all the parameters end up being close to zero

If this happens, it's like we got rid of all the terms in the hypothesis

This results here is then underfitting

So this hypothesis is too biased because of the absence of any parameters (effectively)

So, λ should be chosen carefully - not too big...

We look at some automatic ways to select λ later in the course

Regularized linear regression

Previously, we looked at two algorithms for linear regression

Gradient descent
Normal equation

Our linear regression with regularization is shown below





Previously, gradient descent would repeatedly update the parameters θj,
where j = 0,1,2...n simultaneously

Shown below





We've got the θ0 update
here shown explicitly

This is because for regularization we don't penalize θ0 so
treat it slightly differently

How do we regularize these two rules?

Take the term and add λ/m
* θj

Sum for every θ (i.e. j = 0 to n)

This gives regularization for gradient descent

We can show using calculus that the equation given below is the partial derivative of the regularized J(θ)





The update for θj

θj gets
updated to

θj - α
* [a big term which also depends on θj]

So if you group the θj terms
together





The term






Is going to be a number less than 1 usually
Usually learning rate is small and m is large

So this typically evaluates to (1 - a small number)
So the term is often around 0.99 to 0.95

This in effect means θj gets
multiplied by 0.99

Means the squared norm of θj a
little smaller
The second term is exactly the same as the original gradient descent

Regularization with the normal equation

Normal equation is the other linear regression model

Minimize the J(θ) using the normal equation
To use regularization we add a term (+ λ [n+1 x n+1]) to the equation

[n+1 x n+1] is the n+1 identity matrix





Regularization for logistic regression

We saw earlier that logistic regression can be prone to overfitting with lots of features

Logistic regression cost function is as follows;





To modify it we have to add an extra term





This has the effect of penalizing the parameters θ1, θ2 up
to θn

Means, like with linear regression, we can get what appears to be a better fitting lower order hypothesis

How do we implement this?

Original logistic regression with gradient descent function was as follows





Again, to modify the algorithm we simply need to modify the update rule for θ1,
onwards

Looks cosmetically the same as linear regression, except obviously the hypothesis is very different






Advanced optimization of regularized linear regression

As before, define a costFunction which takes a θ parameter and gives jVal and gradient back






use fminunc

Pass it an @costfunction argument
Minimizes in an optimized manner using the cost function

jVal

Need code to compute J(θ)

Need to include regularization term

Gradient

Needs to be the partial derivative of J(θ)
with respect to θi
Adding the appropriate term here is also necessary






Ensure summation doesn't extend to to the lambda term!

It doesn't, but, you know, don't be daft!