Stanford Machine Learning: (4). Advice for applying Machine Learning

Deciding what to try next

We now know many techniques

But, there is a big difference between someone who knows an algorithm vs. someone less familiar and doesn't understand how to apply them
Make sure you know how to chose the best avenues to explore the various techniques
Here we focus deciding what avenues to try

Debugging a learning algorithm

So, say you've implemented regularized linear regression to predict housing prices





Trained it
But, when you test on new data you find it makes unacceptably large errors in its predictions
:-(

What should you try next?

There are many things you can do;

Get more training data

Sometimes more data doesn't help
Often it does though, although you should always do some preliminary testing to make sure more data will actually make a difference (discussed later)

Try a smaller set a features

Carefully select small subset
You can do this by hand, or use some dimensionality reduction technique (e.g. PCA - we'll get to this later)

Try getting additional features

Sometimes this isn't helpful
LOOK at the data
Can be very time consuming

Adding polynomial features

You're grasping at straws, aren't you...

Building your own, new, better features based on your knowledge of the problem

Can be risky if you accidentally over fit your data by creating new features which are inherently specific/relevant to your training data

Try decreasing or increasing λ

Change how important the regularization term is in your calculations

These changes can become MAJOR projects/headaches (6 months +)

Sadly, most common method for choosing one of these examples is to go by gut feeling (randomly)
Many times, see people spend huge amounts of time only to discover that the avenue is fruitless

No apples, pears, or any other fruit. Nada.

There are some simple techniques which can let you rule out half the things on the list

Save you a lot of time!

Machine learning diagnostics

Tests you can run to see what is/what isn't working for an algorithm
See what you can change to improve an algorithm's performance
These can take time to implement and understand (week)

But, they can also save you spending months going down an avenue which will never work

Evaluating a hypothesis

When we fit parameters to training data, try and minimize the error

We might think a low error is good - doesn't necessarily mean a good parameter set

Could, in fact, be indicative of overfitting
This means you model will fail to generalize

How do you tell if a hypothesis is overfitting?

Could plot hθ(x)
But with lots of features may be impossible to plot

Standard way to evaluate a hypothesis is

Split data into two portions

1st portion is training set
2nd portion is test set

Typical split might be 70:30 (training:test)





NB if data is ordered, send a random percentage

(Or randomly order, then send data)
Data is typically ordered in some way anyway

So a typical train and test scheme would be

1) Learn parameters θ from training data, minimizing J(θ) using 70% of the training data]
2) Compute the test error

Jtest(θ)
= average square error as measured on the test set





This is the definition of the test set error

What about if we were using logistic regression

The same, learn using 70% of the data, test with the remaining 30%





Sometimes there a better way - misclassification error (0/1 misclassification)

We define the error as follows





Then the test error is





i.e. its the fraction in the test set the hypothesis mislabels

These are the standard techniques for evaluating a learned hypothesis

Model selection and training validation test sets

How to chose regularization parameter or degree of polynomial (model selection problems)
We've already seen the problem of overfitting

More generally, this is why training set error is a poor predictor of hypothesis accuracy for new data (generalization)

Model selection problem

Try to chose the degree for a polynomial to fit data





d = what degree of polynomial do you want to pick

An additional parameter to try and determine your training set

d =1 (linear)
d=2 (quadratic)
...
d=10

Chose a model, fit that model and get an estimate of how well you hypothesis will generalize

You could

Take model 1, minimize with training data which generates a parameter vector θ1 (where
d =1)
Take mode 2, do the same, get a different θ2 (where
d = 2)
And so on
Take these parameters and look at the test set error for each using the previous formula

Jtest(θ1)
Jtest(θ2)
...
Jtest(θ10)

You could then

See which model has the lowest test set error

Say, for example, d=5 is the lowest

Now take the d=5 model and say, how well does it generalize?

You could use Jtest(θ5)
BUT, this is going to be an optimistic estimate of generalization error, because our parameter is fit to that test set (i.e. specifically chose it because the test set error is small)
So not a good way to evaluate if it will generalize

To address this problem, we do something a bit different for model selection

Improved model selection

Given a training set instead split into three pieces

1 - Training set (60%) - m values
2 - Cross validation (CV) set (20%)mcv
3 - Test set (20%) mtest

As before, we can calculate

Training error
Cross validation error
Test error





So

Minimize cost function for each of the models as before
Test these hypothesis on the cross validation set to generate the cross validation error
Pick the hypothesis with the lowest cross validation error

e.g. pick θ5

Finally

Estimate generalization error of model using the test set

Final note

In machine learning as practiced today - many people will select the model using the test set and then check the model is OK for generalization using the test error (which we've said is bad because it gives a bias analysis)

With a MASSIVE test set this is maybe OK

But considered much better practice to have separate training and validation sets

Diagnosis - bias vs. variance

If you get bad results usually because of one of

High bias - under fitting problem
High variance - over fitting problem

Important to work out which is the problem

Knowing which will help let you improve the algorithm

Bias/variance shown graphically below





The degree of a model will increase as you move towards overfitting
Lets define training and cross validation error as before
Now plot

x = degree of polynomial d
y = error for both training and cross validation (two lines)

CV error and test set error will be very similar





This plot helps us understand the error

We want to minimize both errors

Which is why that d=2 model is the sweet spot

How do we apply this for diagnostics

If cv error is high we're either at the high or the low end of d





if d is too small --> this probably corresponds to a high bias problem
if d is too large --> this probably corresponds to a high variance problem

For the high bias case, we find both cross validation and training error are high

Doesn't fit training data well
Doesn't generalize either

For high variance, we find the cross validation error is high but training error is low

So we suffer from overfitting (training is low, cross validation is high)

i.e. training set fits well
But generalizes poorly

Regularization and bias/variance

How is bias and variance effected by regularization?





The equation above describes fitting a high order polynomial with regularization (used to keep parameter values small)

Consider three cases

λ = large

All θ values are heavily penalized
So most parameters end up being close to zero
So hypothesis ends up being close to 0
So high bias -> under fitting data

λ = intermediate

Only this values gives the fitting which is reasonable

λ = small

Lambda = 0
So we make the regularization term 0
So high variance -> Get overfitting (minimal regularization means it obviously doesn't do what it's meant to)





How can we automatically chose a good value for λ?

To do this we define another function Jtrain(θ)
which is the optimization function without the regularization term (average squared errors)





Define cross validation error and test set errors as before (i.e. without regularization term)

So they are 1/2 average squared error of various sets

Choosing λ

Have a set or range of values to use
Often increment by factors of 2 so

model(1)= λ = 0
model(2)= λ = 0.01
model(3)= λ = 0.02
model(4) = λ = 0.04
model(5) = λ = 0.08

.

.

.
model(p) = λ = 10

This gives a number of models which have different λ
With these models

Take each one (pth)
Minimize the cost function
This will generate some parameter vector

Call this θ(p)

So now we have a set of parameter vectors corresponding to models with different λ values

Take all of the hypothesis and use the cross validation set to validate them

Measure average squared error on cross validation set
Pick the model which gives the lowest error
Say we pick θ(5)

Finally, take the one we've selected (θ(5))
and test it with the test set

Bias/variance as a function of λ

Plot λ vs.

Jtrain

When λ is small you get a small value (regularization basically
goes to 0)
When λ is large you
get a large vale corresponding to high bias

Jcv

When λ is small we see high variance

Too small a value means we over fit the data

When λ is large we end up underfitting, so this is bias

So cross validation error is high

Such a plot can help show you you're picking a good value for λ

Learning curves

A learning curve is often useful to plot for algorithmic sanity checking or improving performance
What is a learning curve?

Plot Jtrain (average
squared error on training set) or Jcv (average squared
error on cross validation set)
Plot against m (number of training examples)

m is a constant
So artificially reduce m and recalculate errors with the smaller training set sizes

Jtrain

Error on smaller sample sizes is smaller (as less variance to accommodate)
So as m grows error grows

Jcv

Error on cross validation set
When you have a tiny training set your generalize badly
But as training set grows your hypothesis generalize better
So cv error will decrease as m increases





What do these curves look like if you have

High bias

e.g. setting straight line to data
Jtrain

Training error is small at first and grows
Training error becomes close to cross validation
So the performance of the cross validation and training set end up being similar (but very poor)

Jcv

Straight line fit is similar for a few vs. a lot of data
So it doesn't generalize any better with lots of data because the function just doesn't fit the data

No increase in data will help it fit

The problem with high bias is because cross validation and training error are both high
Also implies that if a learning algorithm as high bias as we get more examples the cross validation error doesn't decrease

So if an algorithm is already suffering from high bias, more data does not help
So knowing if you're suffering from high bias is good!
In other words, high bias is a problem with the underlying way you're modeling your data

So more data won't improve that model
It's too simplistic

High variance

e.g. high order polynomial
Jtrain

When set is small, training error is small too
As training set sizes increases, value is still small
But slowly increases (in a near linear fashion)
Error is still low

Jcv

Error remains high, even when you have a moderate number of examples
Because the problem with high variance (overfitting) is your model doesn't generalize

An indicative diagnostic that you have high variance is that there's a big gap between training error and cross validation error
If a learning algorithm is suffering from high variance, more data is probably going to help





So if an algorithm is already suffering from high variance, more data will probably help

Maybe

These are clean curves
In reality the curves you get are far dirtier
But, learning curve plotting can help diagnose the problems your algorithm will be suffering from

What to do next (revisited)

How do these ideas help us chose how we approach a problem?

Original example

Trained a learning algorithm (regularized linear regression)
But, when you test on new data you find it makes unacceptably large errors in its predictions
What should try next?

How do we decide what to do?

Get more examples --> helps to fix high variance

Not good if you have high bias

Smaller set of features --> fixes high variance (overfitting)

Not good if you have high bias

Try adding additional features --> fixes high bias (because hypothesis is too simple, make hypothesis more specific)

Add polynomial terms --> fixes high bias problem

Decreasing λ --> fixes high bias

Increases λ -->
fixes high variance

Relating it all back to neural networks - selecting a network architecture

One option is to use a small neural network

Few (maybe one) hidden layer and few hidden units
Such networks are prone to under fitting
But they are computationally cheaper

Larger network

More hidden layers

How do you decide that a larger network is good?

Using a single hidden layer is good default

Also try with 1, 2, 3, see which performs best on cross validation set
So like before, take three sets (training, cross validation)

More units

This is computational expensive
Prone to over-fitting

Use regularization to address over fitting