Non-parametric Methods - Machine...

ML-Overview Curve Fitting Decision Theory Probability Theory Kernel Density Estimation Nearest-neighbour Conclusion

Non-parametric MethodsMachine Learning

Alireza Ghane

Non-Parametric Methods Alireza Ghane / Torsten Möller 1


Outline

Machine Learning: What, Why, and How?

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

Decision Theory: ML, Loss Function, MAP

Probability Theory: (e.g.) Probabilities and Parameter Estimation

Kernel Density Estimation

Nearest-neighbour

Conclusion



Outline






Nearest-neighbour

Conclusion



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 classification

problem• Given a training set {(x1, t1), . . . , (xN , tN )}, learning problem

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



Stock Price Prediction

• Problems in which ti is continuous are called regression• E.g. ti is stock price, xi contains company profit, debt, cash

flow, 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






Nearest-neighbour

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 data



Polynomial Curve Fitting

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

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

M

• 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, choose thatwhich minimizes this error

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t

x

y(xn,w)

tn

xn



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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• 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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• Note the ERMS for the training set. Perfect match of training setwith 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) on training

set• Choose model (e.g. order of polynomial) with minimum error on

validation set



Cross-validation

run 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 is

number of training data points• Cross-validation is an effective method for model selection, but

can be slow• Models with multiple complexity parameters: exponential

number 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 on newdata



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 with

lowest test error• Performance on test set is no longer good estimate of performance

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 a separatevalidation 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 penalize

being “too complex” (e.g. large weight magnitudes)• Need to balance error vs. regularization (λ)

• Choose λ using cross-validation

• Get more data



Outline






Nearest-neighbour

Conclusion



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 34


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 = arg maxk{ p (x|Ck)︸︷︷︸ }

Likelihood







Likelihood



Decision: Minimum Misclassification Rate

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

∫R1p (x, C2) dx +

∫R2p (x, C1) dx

q(mistake) =∑k

∑j

∫Rjp (x, Ck) dx

R1 R2

x0 x

p(x, C1)

p(x, C2)

x

x: decision boundary.x0: optimal decision boundary

x0 : arg minR1

{p (mistake)}



Decision: Minimum Loss/Cost

• Misclassification rate:

R : arg min{Ri|i∈{1,··· ,K}}

∑k

∑j

L (Rj , Ck)

• Weighted loss/cost function:

R : arg min{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 than asingle point estimate.

• Can utilize Apriori Information in the decision.



Outline






Nearest-neighbour

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, thelikelihood 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 which

maximizes the likelihood of the data

µML = arg maxµ

p(D|µ)

• Since ln(·) is monotone increasing:

µML = arg maxµ

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+ hNon-Parametric Methods Alireza Ghane / Torsten Möller 49



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? Do Ibelieve 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 prior knowledge

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 = arg maxµ

P (µ|D)

• Known as point estimation• However, correct Bayesian thing to do is to use the full

distribution over µ• i.e. Compute

Eµ[f ] =

∫p(µ|D)f(µ)dµ

• This integral is usually hard to compute





µMAP = arg maxµ

P (µ|D)



Eµ[f ] =

∫p(µ|D)f(µ)dµ




Polynomial Curve Fitting: What We Did

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

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

M

• 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, choose thatwhich minimizes this error

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t

x

y(xn,w)

tn

xn



Curve Fitting: Probabilistic Approach

t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

p(t|x,w, β) =N∏n=1

N(tn|y(xn,w), β−1

)Intro. to Machine Learning Alireza Ghane 65



t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

p(t|x,w, β) =N∏n=1


)

ln (p(t|x,w, β)) = − β2

N∑n=1

{y(xn,w)− tn}2︸︷︷︸βE(w)

+N

2lnβ︸︷︷︸

const.

− N2

ln (2π)︸︷︷︸const.




t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

p(t|x,w, β) =

N∏n=1


)

ln (p(t|x,w, β)) = − β2

N∑n=1

{y(xn,w)− tn}2︸︷︷︸βE(w)

+N

2lnβ︸︷︷︸

const.

− N2

ln (2π)︸︷︷︸const.

Maximize log-likelihood⇔Minimize E(w).Can optimize for β as well.



Curve Fitting: Bayesian Approach

t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

p(t|x,w, β) =

N∏n=1


)




t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

p(t|x,w, β) =

N∏n=1


)

Posterior Dist.:p (w|x, t, α, β) ∝ p (t|x,w, β) p (w|α)

Minimize:β

2

N∑n=1

{y(xn,w)− tn}2︸︷︷︸βE(w)

+α

2wTw︸︷︷︸

regularization.



Curve Fitting: Bayesian

p (t|x0,w, β, α) = N (t|Pw(x0), Qw,β,α(x0))

p (t|x,x, t) = N(t|m(x), s2(x)

)

m(x) = φ(x)TS

N∑n=1

φ(xn)tn

s2(x) = β−1(1 + φ(x)TSφ(x)

)S−1 =

α

βI +

N∑n=1

φ(xn)φ(xn)T

t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

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p (t|x,x, t) = N(t|m(x), s2(x)

)

m(x) = φ(x)TS

N∑n=1

φ(xn)tn

s2(x) = β−1(1 + φ(x)TSφ(x)

)S−1 =

α

βI +

N∑n=1

φ(xn)φ(xn)T

t

xx0

2σy(x0,w)

y(x,w)

p(t|x0,w, β)

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Outline






Nearest-neighbour

Conclusion



Histograms

• Consider the problem of modelling the distribution of brightnessvalues in pictures taken on sunny days versus cloudy days

• We could build histograms of pixel values for each classIntro. to Machine Learning Alireza Ghane 75


Histograms

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• E.g. for sunny days• Count ni number of datapoints (pixels) with

brightness value falling into each bin:pi = ni

N∆i

• Sensitive to bin width ∆i

• Discontinuous due to bin edges• In D-dim space with M bins per dimension,MD bins



Histograms

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N∆i





Local Density Estimation

• In a histogram we use nearby points to estimate density• For a small region around x, estimate density as:

p(x) =K

NV

• K is number of points in region, V is volume of region, N istotal number of datapoints




• Try to keep idea of using nearby points to estimate density, butobtain smoother estimate

• Estimate density by placing a small bump at each datapoint• Kernel function k(·) determines shape of these bumps

• Density estimate is

p(x) ∝ 1

N

N∑n=1

k

(x− xnh

)




��

0 0.5 10

5

��

0 0.5 10

5

��

0 0.5 10

5

• Example using Gaussian kernel:

p(x) =1

N

N∑n=1

1

(2πh2)1/2exp

{−||x− xn||

2

2h2

}




!3 !2 !1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

!5 0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

• Other kernels: Rectangle, Triangle, Epanechnikov

• Fast at training time, slow at test time – keep all datapoints• Sensitive to kernel bandwidth h




!3 !2 !1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

!5 0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

• Other kernels: Rectangle, Triangle, Epanechnikov• Fast at training time, slow at test time – keep all datapoints

• Sensitive to kernel bandwidth h




!3 !2 !1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

!5 0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

• Other kernels: Rectangle, Triangle, Epanechnikov• Fast at training time, slow at test time – keep all datapoints• Sensitive to kernel bandwidth h



Outline






Nearest-neighbour

Conclusion



Nearest-neighbour��

0 0.5 10

5

��

0 0.5 10

5

��

0 0.5 10

5

• Instead of relying on kernel bandwidth to get proper densityestimate, fix number of nearby points K:

p(x) =K

NV

• Note: diverges, not proper density estimateIntro. to Machine Learning Alireza Ghane 88


Nearest-neighbour for Classification

x1

x2

(a)x1

x2

(b)

• K Nearest neighbour is often used for classification• Classification: predict labels ti from xi

• e.g. xi ∈ R2 and ti ∈ {0, 1}, 3-nearest neighbour

• K = 1 referred to as nearest-neighbour




x1

x2

(a)

x1

x2

(b)

• K Nearest neighbour is often used for classification• Classification: predict labels ti from xi• e.g. xi ∈ R2 and ti ∈ {0, 1}, 3-nearest neighbour

• K = 1 referred to as nearest-neighbour




x1

x2

(a)x1

x2

(b)

• K Nearest neighbour is often used for classification• Classification: predict labels ti from xi• e.g. xi ∈ R2 and ti ∈ {0, 1}, 3-nearest neighbour

• K = 1 referred to as nearest-neighbourIntro. to Machine Learning Alireza Ghane 91



• Good baseline method• Slow, but can use fancy data structures for efficiency (KD-trees,

Locality Sensitive Hashing)• Nice theoretical properties

• As we obtain more training data points, space becomes morefilled with labelled data

• As N →∞ error no more than twice Bayes error



Bayes Error

R1 R2

x0 x

p(x, C1)

p(x, C2)

x

• Best classification possible given features• Two classes, PDFs shown• Decision rule: C1 if x ≤ x; makes errors on red, green, and blue

regions• Optimal decision rule: C1 if x ≤ x0, Bayes error is area of green

and blue regions



Bayes Error

R1 R2

x0 x

p(x, C1)

p(x, C2)

x



and blue regions



Outline






Nearest-neighbour

Conclusion



Conclusion

• Readings: Chapter 1.1, 1.3, 1.5, 2.1• 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



Conclusion

• Readings: Ch. 2.5• Kernel density estimation

• Model density p(x) using kernels around training datapoints• Nearest neighbour

• Model density or perform classification using nearest trainingdatapoints

• Multivariate Gaussian• Needed for next week’s lectures, if you need a refresher read

pp. 78-81


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