Classification Yan Pan. Under and Over Fitting Probability Theory Non-negativity and unit measure 0 ≤ p(y), p( ) = 1, p( ) = 0 Conditional probability

Classification

Yan Pan

Under and Over Fitting

Probability Theory• Non-negativity and unit measure

• 0 ≤ p(y) , p() = 1, p() = 0• Conditional probability – p(y|x)

• p(x, y) = p(y|x) p(x) = p(x|y) p(y)• Bayes’ Theorem

• p(y|x) = p(x|y) p(y) / p(x) • Marginalization

• p(x) = y p(x, y) dy • Independence

• p(x1, x2) = p(x1) p(x2) p(x1|x2) = p(x1)

Chris Bishop, “Pattern Recognition & Machine Learning”

The Univariate Gaussian Density

• p(x|,) = exp( -(x – )2/22) / (22)½

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The Multivariate Gaussian Density

• p(x|,) = exp( -½ (x – )t -1 (x – ) )/ (2)D/2||½

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• p(|a,b) = a-1(1 – )b-1 (a+b) / (a)(b)

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Probability Distribution Functions• Bernoulli: Single trial with probability of success =

• n {0, 1}, [0, 1]• p(n|) = n(1 – )1-n

• Binomial: N iid Bernoulli trials with n successes• n {0, 1, …, N}, [0, 1], • p(n|N,) = NCn n(1 – )N-n

A Toy Example• We don’t know whether a coin is fair or not. We are told that heads occurred n times in N coin flips.

• We are asked to predict whether the next coin flip will result in a head or a tail.

• Let y be a binary random variable such that y = 1 represents the event that the next coin flip will be a head and y = 0 that it will be a tail

• We should predict heads if p(y=1|n,N) > p(y=0|n,N)

The Maximum Likelihood Approach• Let p(y=1|n,N) = and p(y=0|n,N) = 1 - so that we should predict heads if > ½

• How should we estimate ?

• Assuming that the observed coin flips followed a Binomial distribution, we could choose the value of that maximizes the likelihood of observing the data

• ML = argmax p(n|) = argmax NCn n(1 – )N-n

= argmax n log() + (N – n) log(1 – ) = n / N

• We should predict heads if n > ½ N

The Maximum A Posteriori Approach• We should choose the value of maximizing the posterior probability of conditioned on the data

• We assume a•Binomial likelihood : p(n|) = NCn n(1 – )N-n •Beta prior : p(|a,b)=a-1(1–)b-1(a+b)/(a)(b)

• MAP = argmax p(|n,a,b) = argmax p(n|) p(|a,b) = argmax n (1 – )N-n a-1 (1–)b-1

= (n+a-1) / (N+a+b-2) as if we saw an extra a – 1 heads & b – 1 tails

• We should predict heads if n > ½ (N + b – a)

The Bayesian Approach

Classification

Binary Classification

Approaches to Classification• Memorization

• Can not deal with previously unseen data• Large scale annotated data acquisition cost might be very high

• Rule based expert system• Dependent on the competence of the expert.• Complex problems lead to a proliferation of rules, exceptions, exceptions to exceptions, etc.• Rules might not transfer to similar problems

• Learning from training data and prior knowledge• Focuses on generalization to novel data

Notation• Training Data

• Set of N labeled examples of the form (xi, yi)• Feature vector – x D. X = [x1 x2 … xN]• Label – y {1}. y = [y1, y2 … yN]t. Y=diag(y)

• Example – Gender Identification

(x1 = , y1 = +1) (x2 = , y2 = +1)

(x3 = , y3 = +1) (x4 = , y4 = -1)


wtx + b = 0

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= [w ; b]

Machine Learning from the Optimization View

• Before we go into the details of classification and regression methods, we should take a close look at the objective functions of machine learning

• Machine Learning：根据数据找规律（从多个候选规律里面选最好的），选择的标准是什么？

• 把候选规律放到训练数据上预测一下，看看预测的错误率是多少，预测错误最少的规律就是我们要找的。

Supervised Learning

Common Form of Supervised Learning Problems

• Minimize the following objective function– Regularization term + Loss function

• Regularization term: control the model complexity, avoid over fitting

• Loss function: measure the quality of the learned function, i.e. predict error on the training data.

Ex.1 Linear Regression

• E(w) = ½ n (yn - wtxn)^2 + ½wtw

Ex.2 Logistic Regression (classification method)

• (w, b) = ½wtw + I log(1+exp(-yi(b+wtxi)))

Ex.3 SVM

• E(w) = ½wtw + I max(0,1-yiwtxi)

• Or• E(w) = ½wtw + I max(0,1-yiwtxi)^2

How to measure error?

• True: yi

• Predicted: wtxi

• 越像越好。相等？– I (yi ！ = wtxi ）– （ yi - wtxi ） ^2

• 假设取值范围为 [-1,1]: 乘积尽量大– yi wtxi

Approximate the Zero-One Loss

• Squared Error• Exponential Loss• Logistic Loss• Hinge Loss• Sigmoid Loss

Regularized Logistic Regression

Zhu & Hastie, “KLR and the Import Vector Machine”, NIPS 01

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Convex Functions

• Convex f : f(x1 + (1- )x2) f(x1) + (1- )f(x2)• The Hessian 2f is always positive semi-definite• The tangent is always a lower bound to f

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Gradient Descent

• Iteration : xn+1 = xn - nf(xn)• Step size selection : Armijo rule• Stopping criterion : Change in f is “miniscule”

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Gradient Descent – Logistic Regression(w, b) = ½wtw + I log(1+exp(-yi(b+wtxi)))

w(w, b) = w – I p(-yi|xi,w) yi xi

b(w, b) = – I p(-yi|xi,w) yi

Beware of numerical issues while coding!

Gradient Decent Algorithm• Input: x0, objective f(x), e, T• Output: x_star that minimize f(x)• t=0• While (t==0 || (f(x_{t-1}) – f(x_{t})>e && T<100000 )) {• g_t = gradient of f(x) at x_t• for( i=10; i>=-6; i--)• {• s=2^i• x_{t+1}=x_t – s*g_t• if (f(x_{t+1} < f(x_t))• break;• }• t++;• }• Output x_t

Newton Methods

• Iteration : xn+1 = xn - nH-1f(xn)• Approximate f by a 2nd order Taylor expansion• The error can now decrease quadratically

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Newton Decent Algorithm• Input: x0, objective f(x), e, T• Output: x_star that minimize f(x)• t=0• While (t==0 || (f(x_{t-1}) – f(x_{t})>e && T<10)) {• g_t = gradient of f(x) at x_t• h_t = hessian matrix of f(x) at x_t• s = inverse matrix of h_t• • x_{t+1}=x_t – s*g_t• • t++;• }• Output x_t

Quasi-Newton Methods• Computing and inverting the Hessian is expensive• Quasi-Newton methods can approximate H-1 directly (LBFGS)• Iteration : xn+1 = xn - nBn

-1f(xn)• Secant equation : f(xn+1) – f(xn) = Bn+1(xn+1 – xn)• The secant equation does not fully determine B • LBFGS updates Bn+1

-1 using two rank one matrices

Machine Learning Problems from the

Probability View

Bayes’ Decision Rule• Bayes’ decision rule

p(y=+1|x) > p(y=-1|x) ? y = +1 : y = -1 p(y=+1|x) > ½ ? y = +1 : y = -1

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Bayesian Approachp(y|x,X,Y) = f p(y,f|x,X,Y) df

= f p(y|f,x,X,Y) p(f|x,X,Y) df= f p(y|f,x) p(f|X,Y) df

• This integral is often intractable. • To solve it we can

• Choose the distributions so that the solution is analytic (conjugate priors)• Approximate the true distribution of p(f|X,Y) by a simpler distribution (variational methods)• Sample from p(f|X,Y) (MCMC)

Maximum A Posteriori (MAP)p(y|x,X,Y) = f p(y|f,x) p(f|X,Y) df

= p(y|fMAP,x) when p(f|X,Y) = (f – fMAP)

• The more training data there is the better p(f|X,Y) approximates a delta function• We can make predictions using a single function, fMAP, and our focus shifts to estimating fMAP.

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No DataModerate DataLots of Data

MAP & Maximum Likelihood (ML)fMAP = argmaxf p(f|X,Y)

= argmaxf p(X,Y|f) p(f) / p(X,Y) = argmaxf p(X,Y|f) p(f)

fML argmaxf p(X,Y|f) (Maximum Likelihood)

• Maximum Likelihood holds if• There is a lot of training data so that

p(X,Y|f) >> p(f)• Or if there is no prior knowledge so that p(f) is uniform (improper)

IID Data fML = argmaxf p(X,Y|f)

= argmaxf I p(xi,yi|f)

• The independent and identically distributed assumption holds only if we know everything about the joint distribution of the features and labels. In particular, p(X,Y) I p(xi,yi)

Discriminative Methods Logistic Regression

Discriminative Methods MAP = argmax p() I p(xi,yi| )• We assume that

• p() = p(w) p(w)• p(xi,yi| ) = p(yi| xi, ) p(xi| )

= p(yi| xi, w) p(xi| w) MAP = [argmaxw p(w) I p(yi| xi, w)] * [argmaxw

p(w) I p(xi|w)]

• It turns out that only w plays a role in determining the posterior distribution p(y|x,X,Y) = p(y|x, MAP) = p(y|x, wMAP) where wMAP = argmaxw p(w) I p(yi| xi, w)

Disc. Methods – Logistic RegressionMAP = argmaxw,b p(w) I p(yi| xi, w)

• Regularized Logistic Regression• Gaussian prior – p(w) = exp( -½ wtw)• Logistic likelihood–

p(yi| xi, w) = 1 / (1 + exp(-yi(b + wtxi)))

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Regularized Logistic RegressionMAP = argmaxw,b p(w) I p(yi| xi, w)

= argminw,b ½wtw + I log(1+exp(-yi(b+wtxi)))

• Bad news: No closed form solution for w and b• Good news: We have to minimize a convex function

• We can obtain the global optimum• The function is smooth

Tom Minka, “A comparison of numerical optimizers for LR” (Matlab code)Keerthi et al., “A Fast Dual Algorithm for Kernel Logistic Regression”, ML 05Andrew and Gao, “OWL-QN” ICML 07Krishnapuram et al., “SMLR” PAMI 05



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Convex Functions

• Convex f : f(x1 + (1- )x2) f(x1) + (1- )f(x2)• The Hessian 2f is always positive semi-definite• The tangent is always a lower bound to f

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Gradient Descent

• Iteration : xn+1 = xn - nf(xn)• Step size selection : Armijo rule• Stopping criterion : Change in f is “miniscule”

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Gradient Descent – Logistic Regression(w, b) = ½wtw + I log(1+exp(-yi(b+wtxi)))

w(w, b) = w – I p(-yi|xi,w) yi xi

b(w, b) = – I p(-yi|xi,w) yi

Beware of numerical issues while coding!

Newton Methods

• Iteration : xn+1 = xn - nH-1f(xn)• Approximate f by a 2nd order Taylor expansion• The error can now decrease quadratically

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Quasi-Newton Methods• Computing and inverting the Hessian is expensive• Quasi-Newton methods can approximate H-1 directly (LBFGS)• Iteration : xn+1 = xn - nBn

-1f(xn)• Secant equation : f(xn+1) – f(xn) = Bn+1(xn+1 – xn)• The secant equation does not fully determine B • LBFGS updates Bn+1

-1 using two rank one matrices

Multi-class Logistic Regression• Multinomial Logistic Regression• 1-vs-All

• Learn L binary classifiers for an L class problem• For the lth classifier, examples from class l are +ve while examples from all other classes are –ve• Classify new points according to max probability

• 1-vs-1• Learn L(L-1)/2 binary classifiers for an L class problem by considering every class pair• Classify novel points by majority vote• Classify novel points by building a DAG

Multi-class Logistic Regression• Assume

• Non-linear multi-class classifier• Number of classes = L• Number of training points per class = N• Algorithm training time for M points = O(M3)• Classification time given M training points=O(M)

Multi-class Logistic Regression• Multinomial Logistic Regression

• Training time = O(L6N3)• Classification time for a new point = O(L2N)

• 1-vs-All• Training time = O(L4N3)• Classification time for a new point = O(L2N)

• 1-vs-1• Training time = O(L2N3)• Majority vote classification time = O(L2N)• DAG classification time = O(LN)

Multinomial Logistic RegressionMAP = argmaxw,b p(w) I p(yi| xi, w)

• Regularized Multinomial Logistic Regression• Gaussian prior

p(w) = exp( -½ lwltwl)

• Multinomial logistic posteriorp(yi = l | xi, w) = efl(xi) / k efk(xi) where fk(xi) = wk

txi + bk

Note that we have to learn an extra classifier by not explicitly enforcing l p(yi = l | xi, w) = 1

Multinomial Logistic Regression(w, b) = ½ kwk

twk + I [log(k fk(xi)) - kkyi fk(xi)]

wk(w, b) = wk + I [ p(yi = k | xi,w) - kyi

] xi

bk(w, b) = I [ p(yi = k | xi,w) - kyi

]

Multi-class Logistic Regression

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From Probabilities to Loss FunctionsMAP = argminw,b ½wtw + I log(1+exp(1-yi(b+wtxi)))

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0/1HingeSquare HingeLogRobust

Support Vector Machines


A Separating Hyperplane

Maximum Margin Hyperplane

Geometric Intuition: Choose the perpendicular bisector of the shortest line segment joining the convex hulls of the two classes

wtx + b = 0

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SVM Notation

wtx + b = +1

wtx + b = -1

Support Vector

Support Vector

Support Vector

Support Vector

Margin = 2 / wtw

Calculating the Margin • Let x+ be any point on the +ve supporting plane and x- the closest point on the –ve supporting plane

Margin = |x+ – x-|= |w| (since x+ = x- + w)= 2 |w|/|w|2 (assuming = 2/|w|2)= 2/|w|

wtx+ + b = +1wtx- + b = -1 wt(x+ – x-)= 2 wtw= 2 = 2/|w|2

Hard Margin SVM Primal• Maximize 2/|w| such that wtxi + b +1 if yi = +1

wtxi + b -1 if yi = -1

• Difficult to optimize directly

• Convex Quadratic Program (QP) reformulation• Minimize ½wtw such that yi(wtxi + b) 1

• Convex QPs can be easy to optimize

Linearly Inseparable Data• Minimize ½wtw + C #(Misclassified points) such that yi(wtxi + b) 1 (for “good” points)

• The optimization problem is NP Hard in general• Disastrous errors are penalized the same as near misses

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wtx + b = +1

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Misclassified point

Support Vector

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The C-SVM Primal Formulation• Minimize ½wtw + C i i

such that yi(wtxi + b) 1 – i i 0

• The optimization is a convex QP• The globally optimal solution will be obtained• Number of variables = D + N + 1• Number of constraints = 2N • Solvers can train on 800K points in 47K (sparse) dimensions in less than 2 minutes on a standard PC

Fan et al., “LIBLINEAR” JMLR 08Bordes et al., “LaRank” ICML 07

The C-SVM Dual Formulation• Maximize 1t – ½tYKY

such that 1tY = 0 0 C

• K is a kernel matrix such that Kij = K(xi, xj) = xitxj

• are the dual variables (Lagrange multipliers)• Knowing gives us w and b • The dual is also a convex QP

• Number of variables = N• Number of constraints = 2N + 1

Fan et al., “LIBSVM” JMLR 05 Joachims, “SVMLight”

SVMs versus Regularized LR

Most of the SVM s are zero!

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Duality• Primal P = Minx f0(x)

s. t. fi(x) 0 1 i N hi(x) = 0 1 i M

• Lagrangian L(x,,) = f0(x) + i ifi(x) + i ihi(x)

• Dual D = Max, Minx L(x,,) s. t. 0

Duality• The Lagrange dual is always concave (even if the primal is not convex) and might be an easier problem to optimize

• Weak duality : P D • Always holds

• Strong duality : P = D • Does not always hold• Usually holds for convex problems • Holds for the SVM QP

Karush-Kuhn-Tucker (KKT) Conditions• If strong duality holds, then for x*, * and * to be optimal the following KKT conditions must necessarily hold

• Primal feasibility : fi(x*) 0 & hi(x*) = 0 for 1 i • Dual feasibility : * 0• Stationarity : x L(x*, *,*) = 0• Complimentary slackness : i*fi(x*) = 0

• If x+, + and + satisfy the KKT conditions for a convex problem then they are optimal

SVM – Duality• Primal P = Minw,,b ½wtw + Ct

s. t. Y(Xtw + b1) 1 – 0

• Lagrangian L(,, w,,b) = ½wtw + Ct – t –t[Y(Xtw + b1) – 1 + ]

• Dual D = Max 1t – ½tYKY

s. t. 1tY = 0 0 C

SVM – KKT Conditions• Lagrangian L(,, w,,b) = ½wtw + Ct – t

–t[Y(Xtw + b1) – 1 + ]

• Stationarity conditions• w L= 0 w* = XY* (Representer Theorem)• L= 0 C = * + *• b L= 0 *tY1 = 0

• Complimentary Slackness conditions• i* [ yi (xi

tw* + b*) – 1 + i*] = 0• i*i* = 0

Hinge Loss and Sparseness in • From the Stationarity and Complimentary Slackness conditions it is easy to show that

• i = 0 xi has been classified correctly and lies beyond its supporting hyperplane

• 0 < i < C xi is a support vector and lies on its supporting hyperplane

• i = C xi has been misclassified or is a margin violator

Hinge Loss and Sparseness in

• SVM s are sparse but LR s are not

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Linearly Inseparable Data

• This 1D dataset can not be separated using a single hyperplane (threshold)• We need a non-linear decision boundary

x

Increasing Dimensionality Non-linearly

• The dataset is now linearly separable in space

(x) = (x, x2)x

The Kernel Trick• Let the “lifted” training set be { ((xi), yi) } • Define the kernel such that Kij = K(xi, xj) = (xi)t (xj)

• Primal P = Minw,,b ½wtw + Ct

s. t. Y((X)tw + b1) 1 – 0

• Dual D = Max 1t – ½tYKY

s. t. 1tY = 0 0 C

• Classifier: f(x) = sign((x)tw + b) = sign(tYK(:,x) + b)

The Kernel Trick• Let (x) = [1, 2x1, … , 2xD , x1

2, … , xD2, 2x1x2, …, 2x1xD, …, 2xD-1xD]t

• Define K(xi, xj) = (xi)t (xj) = (xitxj + 1)2

• Primal• Number of variables = D + N + 1• Number of constraints = 2N• Number of flops for calculating (x)tw = O(D2)• Number of flops for deg 20 polynomial = O(D20)

• Dual• Number of variables = N• Number of constraints = 2N + 1• Number of flops for calculating Kij = O(D)• Number of flops for deg 20 polynomial = O(D)

Some Popular Kernels• Linear : K(xi,xj) = xi

t-1xj

• Polynomial : K(xi,xj) = (xit-1xj + c)d

• Gaussian (RBF) : K(xi,xj) = exp( –k k(xik – xjk)2)

• Chi-Squared : K(xi,xj) = exp( –2(xi, xj) )

• Sigmoid : K(xi,xj) = tanh(xitxj – c)

should be positive definite, c 0, 0 and d should be a natural number

Valid Kernels – Mercer’s Theorem• Let Z be a compact subset of D and K a continuous symmetric function. Then K is a kernel if

Z Z f(x) K(x,z) f(z) dx dz 0

for all square integrable real valued function f on Z.

Valid Kernels – Mercer’s Theorem• Let Z be a compact subset of D and K a continuous symmetric function. Then K is a kernel if

Z Z f(x) K(x,z) f(z) dx dz 0

for all square integrable real valued function f on Z.

• K is a kernel if every finite symmetric matrix formed by evaluating K on pairs of points from Z is positive semi-definite

Operations on Kernels• The following operations result in valid kernels

• K(xi,xj) = k k Kk(xi,xj) (k 0)• K(xi,xj) = k Kk(xi,xj)• K(xi,xj) = f(xi) f(xj) (f : D )• K(xi,xj) = p(K1(xi,xj)) (p : +ve coeff poly)• K(xi,xj) = exp(K1(xi,xj))

• Kernels can be defined over graphs, sets, strings and many other interesting data structures

Kernels• Kernels should encode all our prior knowledge about feature similarities.

• Kernel parameters can be chosen through cross validation or learnt (see Multiple Kernel Learning).

• Non-linear kernels can sometimes boost classification performance tremendously.

• Non-linear kernels are generally expensive (both during training and for prediction)

Polynomial Kernel of Degree 2

1 2 3 4 5

1

2

3

4

5Poly Deg 2

1 2 3 4 5

1

2

3

4

5Decision Boundaries

Abs(Distances)

2

4

6

8

Classification

Polynomial Kernel of Degree 5

1 2 3 4 5

1

2

3

4

5Poly Deg 5

1 2 3 4 5

1

2

3

4


Abs(Distances)

5

10

15

Classification

RBF Kernel

1 2 3 4 5

1

2

3

4

5RBF =1.000

1 2 3 4 5

1

2

3

4


Abs(Distances)

0.2

0.4

0.6

0.8

1

Classification

Exponential 2 Kernel

1 2 3 4 5

1

2

3

4

5EChi2 =1.000

1 2 3 4 5

1

2

3

4


Abs(Distances)

0.5

1

1.5

Classification

Kernel Parameter Setting - Underfitting

1 2 3 4 5

1

2

3

4

5RBF =0.001

1 2 3 4 5

1

2

3

4


Abs(Distances)

0.5

1

1.5

2

2.5

Classification

Kernel Parameter Setting

1 2 3 4 5

1

2

3

4

5RBF =1.000

1 2 3 4 5

1

2

3

4


Abs(Distances)

0.2

0.4

0.6

0.8

1

Classification

Kernel Parameter Setting – Overfitting

1 2 3 4 5

1

2

3

4

5RBF =100.000

1 2 3 4 5

1

2

3

4


Abs(Distances)

0.2

0.4

0.6

0.8

1

Classification

Structured Output Prediction • Minimize f ½|f|2 + C i i

such that f(xi,yi) f(xi,y) + (yi,y) – i y yi

i 0

• Prediction argmaxy f(x,y)

• This formulation minimizes the hinge on the loss on the training set subject to regularization on f Can be used to predict sets, graphs, etc. for suitable choices of Taskar et al., “Max-Margin Markov Networks” NIPS 03Tsochantaridis et al., “Large Margin Methods for Structured & Interdependent Output Variables” JMLR 05

Multi-Class SVM• Minimize f ½|f|2 + C i i

such that f(xi,yi) f(xi,y) + (yi,y) – i y yi

i 0

• Prediction argmaxy f(x,y)

• (yi,y) = yi,y

• f(x,y) = wt [ (x) (y) ] + bt(y) = wy

t(x) + by (assuming (y) = ey)

Weston and Watkin, “SVMs for Multi-Class Pattern Recognition” ESANN 99Bordes et al., “LaRank” ICML 07

Multi-Class SVM• Minw,b ½ kwk

twk + C i i

s. t. wyit(xi) + byi

wyt(xi) + by + 1 – I

yyi

i 0

• Prediction argmaxy wyt(x) + by

• For L classes, with N points per class, the number of constraints is NL2

• Finding the exact solution for real world non-linear problems is often infeasible• In practice, we can obtain an approximate solution or switch to the 1-vs-All or 1-vs-1 formulations

Documents

Classification Yan Pan. Under and Over Fitting Probability Theory Non-negativity and unit measure 0 ≤ p(y), p( ) = 1, p( ) = 0 Conditional probability