machine learning – some basics - bishop prml ch....

Machine Learning Model Selection Goal Function Coin Tossing Conclusion

Machine Learning – some basicsBishop PRML Ch. 1

Torsten Möller


Outline

Machine Learning: What, Why, and How?

Curve Fitting: (e.g.) Regression and Model Selection

Goal Function: Decision Theory—ML, Loss Function, MAP

Probability Theory: (e.g.) Coin Tossing and Parameter Estimation

Conclusion

Machine Learning–The basics ©Möller / Mori 1


References

• Christopher M. Bishop: Pattern Recognition and MachineLearning; Springer, 2007

• Han, Kamber: Data Mining: Concepts and Techniques;Elsevier, 2012

• Hastie, Tibshirani, Friedman: The Elements of StatisticalLearning; Springer, 2009

• James, Witten, Hastie, Tibshirani: An Introduction toStatistical Learning with Applications in R; Springer, 2015

• Kevin Murphy: Machine Learning, A ProbabilisticPerspective, The MIT Press, 2012

• Tan, Steinbach, Kumar: Introduction to Data Mining;Pearson, 2005



Outline





Conclusion



What is Machine Learning (ML)?

• Algorithms that automatically improve performance throughexperience

• Often this means define a model by hand, and use data to fitits parameters



Why ML?

• The real world is complex – difficult to hand-craft solutions.• ML is the preferred framework for applications in many fields:

• Computer Vision• Natural Language Processing, Speech Recognition• Robotics• …



Hand-written Digit Recognition

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Belongie et al. PAMI 2002

• Difficult to hand-craft rules about digits



Hand-written Digit Recognition

xi =

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ti = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0)

• Represent input image as a vector xi ∈ R784.• Suppose we have a target vector ti

• This is supervised learning• Discrete, finite label set: perhaps ti ∈ {0, 1}10, a classificationproblem

• Given a training set {(x1, t1), . . . , (xN , tN )}, learningproblem is to construct a “good” function y(x) from these.

• y : R784 → R10



Face Detection

Schneiderman and Kanade, IJCV 2002

• Classification problem• ti ∈ {0, 1, 2}, non-face, frontal face, profile face.



Spam Detection

• Classification problem• ti ∈ {0, 1}, non-spam, spam• xi counts of words, e.g. Viagra, stock, outperform,

multi-bagger



Caveat - Horses (source?)• Once upon a time there were two neighboring farmers, Jedand Ned. Each owned a horse, and the horses both liked tojump the fence between the two farms. Clearly the farmersneeded some means to tell whose horse was whose.

• So Jed and Ned got together and agreed on a scheme fordiscriminating between horses. Jed would cut a small notchin one ear of his horse. Not a big, painful notch, but just bigenough to be seen. Well, wouldn’t you know it, the day afterJed cut the notch in horse’s ear, Ned’s horse caught on thebarbed wire fence and tore his ear the exact same way!

• Something else had to be devised, so Jed tied a big bluebow on the tail of his horse. But the next day, Jed’s horsejumped the fence, ran into the field where Ned’s horse wasgrazing, and chewed the bow right off the other horse’s tail.Ate the whole bow!



Caveat - Horses (source?)

• Finally, Jed suggested, and Ned concurred, that they shouldpick a feature that was less apt to change. Height seemedlike a good feature to use. But were the heights different?Well, each farmer went and measured his horse, and do youknow what? The brown horse was a full inch taller than thewhite one!

Moral of the story: ML provides theory and tools for settingparameters. Make sure you have the right model and features.Think about your “feature vector x.”



Stock Price Prediction

• Problems in which ti is continuous are called regression• E.g. ti is stock price, xi contains company profit, debt, cashflow, gross sales, number of spam emails sent, …



Clustering Images

Wang et al., CVPR 2006

• Only xi is defined: unsupervised learning• E.g. xi describes image, find groups of similar images



Types of Learning Problems

• Supervised Learning• Classification• Regression

• Unsupervised Learning• Density estimation• Clustering: k-means, mixture models, hierarchical clustering• Hidden Markov models

• Reinforcement Learning



Outline





Conclusion



An Example - Polynomial Curve Fitting

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• Suppose we are given training set of N observations(x1, . . . , xN ) and (t1, . . . , tN ), xi, ti ∈ R

• Regression problem, estimate y(x) from these dataMachine Learning–The basics ©Möller / Mori 19


Polynomial Curve Fitting

• What form is y(x)?• Let’s try polynomials of degree M :

y(x,w) = w0+w1x+w2x2+. . .+wMxM

• This is the hypothesis space.

• How do we measure success?• Sum of squared errors:

E(w) =1

2

N∑n=1

{y(xn,w)− tn}2

• Among functions in the class, choosethat which minimizes this error

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t

x

y(xn,w)

tn

xn





y(x,w) = w0+w1x+w2x2+. . .+wMxM



E(w) =1

2

N∑n=1

{y(xn,w)− tn}2


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−1

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1

t

x

y(xn,w)

tn

xn




• Error function

E(w) =1

2

N∑n=1

{y(xn,w)− tn}2

• Best coefficients

w∗ = argminw

E(w)

• Found using pseudo-inverse



Which Degree of Polynomial?

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• A model selection problem• M = 9 → E(w∗) = 0: This is over-fitting



Generalization

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1TrainingTest

• Generalization is the holy grail of ML• Want good performance for new data

• Measure generalization using a separate set• Use root-mean-squared (RMS) error: ERMS =

√2E(w∗)/N



Controlling Over-fitting: Regularization

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• As order of polynomial M increases, so do coefficientmagnitudes

• Penalize large coefficients in error function:

E(w) =1

2

N∑n=1

{y(xn,w)− tn}2 +λ

2||w||2




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• As order of polynomial M increases, so do coefficientmagnitudes

• Penalize large coefficients in error function:

E(w) =1

2

N∑n=1

{y(xn,w)− tn}2 +λ

2||w||2




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1TrainingTest

• Note the ERMS for the training set. Perfect match of trainingset with the model is a result of over-fitting

• Training and test error show similar trend



Over-fitting: Dataset size

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• With more data, more complex model (M = 9) can be fit• Rule of thumb: 10 datapoints for each parameter



Validation Set

• Split training data into training set and validation set• Train different models (e.g. diff. order polynomials) ontraining set

• Choose model (e.g. order of polynomial) with minimum erroron validation set



Cross-validationrun 1

run 2

run 3

run 4

• Data are often limited• Cross-validation creates S groups of data, use S − 1 to train,other to validate

• Extreme case leave-one-out cross-validation (LOO-CV): S isnumber of training data points

• Cross-validation is an effective method for model selection,but can be slow

• Models with multiple complexity parameters: exponentialnumber of runs



Summary

• Want models that generalize to new data• Train model on training set• Measure performance on held-out test set

• Performance on test set is good estimate of performance onnew data



Summary - Model Selection

• Which model to use? E.g. which degree polynomial?• Training set error is lower with more complex model

• Can’t just choose the model with lowest training error• Peeking at test error is unfair. E.g. picking polynomial withlowest test error

• Performance on test set is no longer good estimate ofperformance on new data



Summary - Solutions I

• Use a validation set• Train models on training set. E.g. different degree polynomials• Measure performance on held-out validation set• Measure performance of that model on held-out test set

• Can use cross-validation on training set instead of aseparate validation set if little data and lots of time

• Choose model with lowest error over all cross-validation folds(e.g. polynomial degree)

• Retrain that model using all training data (e.g. polynomialcoefficients)



Summary - Solutions II

• Use regularization• Train complex model (e.g high order polynomial) but penalizebeing “too complex” (e.g. large weight magnitudes)

• Need to balance error vs. regularization (λ)• Choose λ using cross-validation

• Get more data



Common Task Framework

• common to many competitions (like kaggle)1. A publicly available training dataset involving, for each

observation, a list of (possibly many) feature measurements,and a class label for that observation.

2. A set of enrolled competitors whose common task is to infera class prediction rule from the training data.

3. A scoring referee, to which competitors can submit theirprediction rule. The referee runs the prediction rule against atesting dataset which is sequestered behind a Chinese wall.The referee objectively and automatically reports the score(prediction accuracy) achieved by the submitted rule


Outline





Conclusion


Goal Function – Decision Theory

For a sample x, decide which class(Ck) it is from.

Ideas:• Maximum Likelihood• Minimum Loss/Cost (e.g. misclassification rate)• Maximum Aposteriori (MAP)

Intro. to Machine Learning Alireza Ghane 42


Decision: Maximum Likelihood

• Inference step: Determine statistics from training data.p(x, t) OR p(x|Ck)

• Decision step: Determine optimal t for test input x:

t = argmaxk

{ p (x|Ck)︸︷︷︸ }

Likelihood






t = argmaxk

{ p (x|Ck)︸︷︷︸ }

Likelihood



Decision: Minimum Misclassification Rate

p(mistake) = p (x ∈ R1, C2) + p (x ∈ R2, C1)=

∫R1

p (x, C2) dx +∫R2

p (x, C1) dx

p(mistake) =∑k

∑j

∫Rj

p (x, Ck) dx

R1 R2

x0 x

p(x, C1)

p(x, C2)

x

x: decision boundary.x0: optimal decision boundary

x0 : argminR1

{p (mistake)}



Decision: Minimum Loss/Cost

• Misclassification rate:

R : argmin{Ri|i∈{1,··· ,K}}

∑k

∑j

L (Rj , Ck)

• Weighted loss/cost function:

R : argmin{Ri|i∈{1,··· ,K}}

∑k

∑j

Wj,kL (Rj , Ck)

Is useful when:

• The population of the classes are different• The failure cost is non-symmetric• · · ·



Decision: Maximum Aposteriori (MAP)

Bayes’ Theorem: P{A|B} = P{B|A}P{A}P{B}

p(Ck|x)︸︷︷︸Posterior

∝ p(x|Ck)︸︷︷︸Likelihood

p(Ck)︸︷︷︸Prior

• Provides an Aposteriori Belief for the estimation, rather thana single point estimate.

• Can utilize Apriori Information in the decision.


Outline





Conclusion


Coin Tossing

• Let’s say you’re given a coin, and you want to find outP (heads), the probability that if you flip it it lands as “heads”.

• Flip it a few times: H H T

• P (heads) = 2/3

• Hmm... is this rigorous? Does this make sense?



Coin Tossing



• P (heads) = 2/3




Coin Tossing - Model

• Bernoulli distribution P (heads) = µ, P (tails) = 1− µ

• Assume coin flips are independent and identically distributed(i.i.d.)

• i.e. All are separate samples from the Bernoulli distribution• Given data D = {x1, . . . , xN}, heads: xi = 1, tails: xi = 0,the likelihood of the data is:

p(D|µ) =N∏

n=1

p(xn|µ) =N∏

n=1

µxn(1− µ)1−xn



Maximum Likelihood Estimation

• Given D with h heads and t tails• What should µ be?• Maximum Likelihood Estimation (MLE): choose µ whichmaximizes the likelihood of the data

µML = argmaxµ

p(D|µ)

• Since ln(·) is monotone increasing:

µML = argmaxµ

ln p(D|µ)



Maximum Likelihood Estimation• Likelihood:

p(D|µ) =N∏

n=1

µxn(1− µ)1−xn

• Log-likelihood:

ln p(D|µ) =N∑

n=1

xn lnµ+ (1− xn) ln(1− µ)

• Take derivative, set to 0:

d

dµln p(D|µ) =

N∑n=1

xn1

µ− (1− xn)

1

1− µ=

1

µh− 1

1− µt

⇒ µ =h

t+ hMachine Learning–The basics ©Möller / Mori 56



p(D|µ) =N∏

n=1

µxn(1− µ)1−xn

• Log-likelihood:

ln p(D|µ) =N∑

n=1



d

dµln p(D|µ) =

N∑n=1

xn1

µ− (1− xn)

1

1− µ=

1

µh− 1

1− µt

⇒ µ =h



Bayesian Learning

• Wait, does this make sense? What if I flip 1 time, heads? DoI believe µ=1?

• Learn µ the Bayesian way:

P (µ|D) =P (D|µ)P (µ)

P (D)

P (µ|D)︸︷︷︸posterior

∝ P (D|µ)︸︷︷︸likelihood

P (µ)︸︷︷︸prior

• Prior encodes knowledge that most coins are 50-50• Conjugate prior makes math simpler, easy interpretation

• For Bernoulli, the beta distribution is its conjugate



Bayesian Learning



P (µ|D) =P (D|µ)P (µ)

P (D)








Beta Distribution• We will use the Beta distribution to express our priorknowledge about coins:

Beta(µ|a, b) = Γ(a+ b)

Γ(a)Γ(b)︸︷︷︸normalization

µa−1(1− µ)b−1

• Parameters a and b control the shape of this distribution



Posterior

P (µ|D) ∝ P (D|µ)P (µ)

∝N∏

n=1

µxn(1− µ)1−xn

︸︷︷︸likelihood

µa−1(1− µ)b−1︸︷︷︸prior

∝ µh(1− µ)tµa−1(1− µ)b−1

∝ µh+a−1(1− µ)t+b−1

• Simple form for posterior is due to use of conjugate prior• Parameters a and b act as extra observations• Note that as N = h+ t → ∞, prior is ignored



Posterior

P (µ|D) ∝ P (D|µ)P (µ)

∝N∏

n=1

µxn(1− µ)1−xn



∝ µh(1− µ)tµa−1(1− µ)b−1

∝ µh+a−1(1− µ)t+b−1




Maximum A Posteriori

• Given posterior P (µ|D) we could compute a single value,known as the Maximum a Posteriori (MAP) estimate for µ:

µMAP = argmaxµ

P (µ|D)

• Known as point estimation• However, correct Bayesian thing to do is to use the fulldistribution over µ

• i.e. ComputeEµ[f ] =

∫p(µ|D)f(µ)dµ

• This integral is usually hard to compute





µMAP = argmaxµ

P (µ|D)



∫p(µ|D)f(µ)dµ




Conclusion

• Types of learning problems• Supervised: regression, classification• Unsupervised

• Learning as optimization• Squared error loss function• Maximum likelihood (ML)• Maximum a posteriori (MAP)

• Want generalization, avoid over-fitting• Cross-validation• Regularization• Bayesian prior on model parameters

• see also Bishop: Chapters 1.1, 1.3, 1.5, 2.1


machine learning – some basics - bishop prml ch....

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