Last lecture summary. Basic terminology tasks – classification – regression learner, algorithm – each has one or several parameters influencing its behavior

Last lecture summary

Basic terminology• tasks

– classification– regression

• learner, algorithm– each has one or several parameters influencing its

behavior• model

– one concrete combination of learner and parameters– tune the parameters using the training set– the generalization is assessed using test set

(previously unseen data)

• learning (training)– supervised

• a target vector t is known, parameters are tuned to achieve the best match between prediction and the target vector

– unsupervised• training data consists of a set of input vectors x without

any corresponding target value• clustering, vizualization

• for most applications, the original input variables must be preprocessed– feature selection– feature extraction

x784x6x5x4x3x2x1 . . .

x456x103x5x1

x784x6x5x4x3x2x1 . . .

x*666x*

309x*152x*

18

x*784x*

6x*5x*

4x*3x*

2x*1 . . .

selection extraction

• feature selection/extraction = dimensionality reduction– generally good thing– curse of dimensionality

• example:– learner: regression (polynomial, y = w0 + w1x + w2x2 + w3x3 + …)– parameters: weights (coeffiients) w, order of polynomial

• weights– adjusted so the the sum of squared errors SSE (error function)

is as small as possible

predicted known target

𝑺𝑺𝑬=12∑𝑛=1

𝑁

(𝑦 (𝑥𝑛 ,𝒘 )− 𝑡𝑛 )2suma čtverců chyb

New stuff

Model selection

overfitting

𝑦 (𝑥 ,𝑤 )=∑𝑗=0

𝑀

𝑤 𝑗𝑥𝑗=w0=const 𝑦 (𝑥 ,𝑤 )=∑

𝑗=0

𝑀

𝑤 𝑗𝑥𝑗=𝑤0+𝑤1𝑥


𝑁

(𝑦 (𝑥𝑛 ,𝒘 )− 𝑡𝑛 )2

𝑹𝑴𝑺=√ 2×𝑆𝑆𝐸𝑁

=√ 1𝑁 ∑

𝑛=1

𝑁

(𝑦 (𝑥𝑛 ,𝒘 )−𝑡𝑛 )2

𝐌𝑺𝑬= 1𝑁 ∑

𝑛=1

𝑁

(𝑦 (𝑥𝑛 ,𝒘 )−𝑡𝑛 )2

RMS – root mean squared errorodmocnina střední kvadratické chyby

MSE – mean squared errorstřední kvadratická chyba

comparing error for data sets of different size – root mean squared error RMS

Summary of errors

sum of squared errors

mean squared error

root mean squared error


𝑁

(𝑦 (𝑥𝑛 ,𝒘 )− 𝑡𝑛 )2

𝑴𝑺𝑬= 1𝑁 ∑

𝑛=1

𝑁

(𝑦 (𝑥𝑛 ,𝒘 )−𝑡𝑛 )2

𝑹𝑴𝑺=√ 2×𝑆𝑆𝐸𝑁

=√ 1𝑁 ∑

𝑛=1

𝑁

(𝑦 (𝑥𝑛 ,𝒘 )−𝑡𝑛 )2=√𝑀𝑆𝐸

Training set

Test set

• the bad result for M = 9 may seem paradoxical because– polynomial of given order contains all lower order

polynomials as special cases (M=9 polynomial should be at least as good as M=3 polynomial)

• OK, let’s examine the values of the coefficients w* for polynomials of various orders

M = 0 M = 1 M = 3 M = 9w0

* 0.19 0.82 0.31 0.35

w1* -1.27 7.99 232.37

w2* -25.43 -5321.83

w3* 17.37 48568.31

w4* -231639.30

w5* 640042.26

w6* -1061800.52

w7* 1042400.18

w8* -557682.99

w9* 125201.43

for a given model complexity the overfitting problem becomes lesssevere as the size of the data setincreases

M = 9N = 15

M = 9N = 100

or in other words, the larger the data set is, the more complex (flexible) model can be fitted

Overfitting in classification

Bias-variance tradeoff

• low flexibility (low degree of polynomial) models have large bias and low variance– bias means large quadratic error of the model– variance means that the predictions of the model

will depend only little on the particular sample that was used for building the model

• i.e. there is little change in the model if training data set is changed

• thus there is little change between predictions for given x for different models

• high flexibility models have low bias and large variance– Large degree will make the polynomial

very sensitive to the details of the sample.– Thus the polynomial changes dramatically upon

the change of the data set.– However, bias is low, as the quadratic error is low.

• A polynomial with too few parameters (too low degree) will make large errors because of a large bias.

• A polynomial with too many parameters (too high degree) will make large errors because of a large variance.

• The degree of the ”best” polynomial must be somewhere ”in-between” - bias-variance tradeoff

MSE = variance + bias2

• This phenomenon is not specific to polynomial regression!

• In fact, it shows-up in any kind of model.• Generally, the bias-variance tradeoff principle can

be stated as:– Models with too few parameters are inaccurate

because they are not flexible enough (large bias, large error of the model).

– Models with too many parameters are inaccurate because they overfit data (large variance, too much sensitivity to the data)

– Identifying the best model requires identifying the proper “model complexity” (number of parameters).

Test-data and Cross Validation

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8 No Single 85K Yes

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10 No Single 90K Yes 10

attributes, input/independent variables, features

object

instance

sampleclass

Attribute types• discrete

– Has only a finite or countably infinite set of values.– nominal (also categorical)

• the values are just different labels (e.g. ID number, eye color)• central tendency given by mode (median, mean not defined)

– ordinal• their values reflect the order (e.g. ranking, height in {tall,

medium, short})• central tendency given by median, mode (mean not defined)

– binary attributes - special case of discrete attributes• continuous (also quantitative)

– Has real numbers as attribute values.– central tendency given by mean, + stdev, …

A regression problem

y = f(x) + noiseCan we learn from this data?

Consider three methods

x

y

taken from Cross Validation tutorialby Andrew Moorehttp://www.autonlab.org/tutorials/overfit.html

Linear regression

What will the regression model will look like?

y = ax + b

Univariate linear regression with a constant term.

x

y

taken from Cross Validation tutorial by Andrew Moore http://www.autonlab.org/tutorials/overfit.html

Quadratic regression

What will the regression model will look like?

y = ax2 + bx + c

x

y


Join-the-dots

Also known as piecewise linearnonparametric regression if thatmakes you feel better.

x

y


Which is best?

Why not to choose the method with the best fit to data?


What do we really want ?

Why not to choose the method with the best fit to data?

How well are you going to predict future data?


The test set method1. Randomly choose 30%

of data to be in test set.

2. The remainder is training set.

3. Perform regression on thetraining set.

4. Estimate future performancewith the test set.x

y

linear regressionMSE = 2.4







y

quadratic regressionMSE = 0.9







y

join-the-dotsMSE = 2.2


Test set method• good news

– very simple– Model selection: choose method with the best score.

• bad news– wastes data (we got an estimate of the best method by

using 30% less data)

– if you don’t have enough data, test set may be just lucky/unlucky

test set estimator of performance has high variance

Train Test


training error

testing error

model complexity

the above examples were for different algorithms, this one is about the model complexity (for the given algorithm)

• stratified division– same proportion of data in the training and test

sets

LOOCV (Leave-one-out Cross Validation)

y

x

1. choose one data point2. remove it from the set3. fit the remaining data points4. note your error

Repeat these steps for all points. When you are done report the mean square error.


MSELOOCV = 2.12

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MSELOOCV = 0.962

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MSELOOCV = 3.33

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Which kind of Cross Validation?

Good BadTest set Cheap Variance

Wastes dataLOOCV Doesn’t waste data Expensive

Can we get best of both worlds?


k-fold Cross Validation

x

y

Randomly break data set into k partitions.In our case k = 3.

Red partition: Train on all points not in the red partition. Find the test set sum of errors on the red points.

Blue partition: Train on all points not in the blue partition. Find the test set sum of errors on the blue points.

Green partition: Train on all points not in the green partition. Find the test set sum of errors on the green points.

Then report the mean error. linear regressionMSE3fold = 2.05


Results of 3-fold Cross Validation

MSE3fold

linear 2.05quadratic 1.11join-the-dots 2.93


Which kind of Cross Validation?Good Bad

Test set Cheap. VarianceWastes data.

LOOCV Doesn’t waste data. Expensive.3-fold Slightly better than test-

set.Wastier than LOOCV.More expensive than test-set.

10-fold Only wastes 10%.Only 10 times more expensive instead of R times.

Wastes 10%.10 times more expensive instead of R times (as LOOCV is).

R-fold is identical to LOOCV

Model selection via CV• We are trying to decide which model to use. For the

polynomial regression decide about the degree of polynom.

• Train each machine and make a table.

• Whichever model gave best CV score: train it with all the data. That’s the predictive model you’ll use.

degree MSEtrain MSE10-fold Choice

1

2

3

4

5

6

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Selection and testing• Complete procedure to algorithm selection and

estimation of its quality1. Divide data to train/test

2. By Cross Validation on the Train choose the algorithm

3. Use this algorithm to construct a classifier using Train

4. Estimate its quality on the Test

Train Test

Train

Test

Train Val

• Training error can not be used as an indicator of model’s performance due to overfitting.

• Training data set - train a range of models, or a given model with a range of values for its parameters.

• Compare them on independent data – Validation set.– If the model design is iterated many times, then

some overfitting to the validation data can occur and so it may be necessary to keep aside a third

• Test set on which the performance of the selected model is finally evaluated.

Fnally comes our first machine learning algorithm

• Which class (Blue or Orange) would you predict for this point?

• And why?• classification boundary

x

y?

x

y?

• And now?• Classification boundary is quadratic

x

y?

• And now?• And why?

Nearest Neighbors Classification

instances

• But, what does it mean similar?

A B C D

source: Kardi Teknomo’s Tutorials, http://people.revoledu.com/kardi/tutorial/index.html

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Last lecture summary. Basic terminology tasks – classification – regression learner, algorithm – each has one or several parameters influencing its behavior