Introduction to SVMs. SVMs Geometric –Maximizing Margin Kernel Methods –Making nonlinear decision boundaries linear –Efficiently! Capacity –Structural.

Introduction to SVMsIntroduction to SVMs

• Geometric – Maximizing Margin

• Kernel Methods– Making nonlinear decision boundaries linear– Efficiently!

• Capacity– Structural Risk Minimization

Linear Classifiersf x

denotes +1

denotes -1

f(x,w,b) = sign(w. x - b)

How would you classify this data?

denotes +1

denotes -1

denotes +1

denotes -1

denotes +1

denotes -1

denotes +1

denotes -1

Any of these would be fine..

..but which is best?

Classifier Marginf x

denotes +1

denotes -1

Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.

Maximum Marginf x

denotes +1

denotes -1

The maximum margin linear classifier is the linear classifier with the maximum margin.

This is the simplest kind of SVM (Called an LSVM)Linear SVM

Maximum Marginf x

denotes +1

denotes -1

The maximum margin linear classifier is the linear classifier with the, um, maximum margin.

This is the simplest kind of SVM (Called an LSVM)

Support Vectors are those datapoints that the margin pushes up against

Linear SVM

Why Maximum Margin?

denotes +1

denotes -1

The maximum margin linear classifier is the linear classifier with the, um, maximum margin.

This is the simplest kind of SVM (Called an LSVM)

Support Vectors are those datapoints that the margin pushes up against

1. Intuitively this feels safest.

2. If we’ve made a small error in the location of the boundary (it’s been jolted in its perpendicular direction) this gives us least chance of causing a misclassification.

3. There’s some theory (using VC dimension) that is related to (but not the same as) the proposition that this is a good thing.

4. Empirically it works very very well.

A “Good” Separator

Noise in the Observations

Ruling Out Some Separators

Lots of Noise

Maximizing the Margin

Specifying a line and margin

• How do we represent this mathematically?

• …in m input dimensions?

Plus-Plane

Minus-Plane

Classifier Boundary

“ Predict Class

= +1”

“ Predict Class

= -1”

Specifying a line and margin

• Plus-plane = { x : w . x + b = +1 }• Minus-plane = { x : w . x + b = -1 }

Plus-Plane

Minus-Plane

Classifier Boundary

“ Predict Class

= +1”

“ Predict Class

= -1”

Classify as..

wx+b=1

wx+b=0

wx+b=-

+1 if w . x + b >= 1

-1 if w . x + b <= -1

Universe explodes

if -1 < w . x + b < 1

Computing the margin width

Claim: The vector w is perpendicular to the Plus Plane. Why?

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

How do we compute M in terms of w and b?

Claim: The vector w is perpendicular to the Plus Plane. Why?

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

Let u and v be two vectors on the Plus Plane. What is w . ( u – v ) ?

And so of course the vector w is also perpendicular to the Minus Plane

• Plus-plane = { x : w . x + b = +1 }• Minus-plane = { x : w . x + b = -1 }• The vector w is perpendicular to the Plus Plane• Let x- be any point on the minus plane• Let x+ be the closest plus-plane-point to x-.

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

Any location in m: not necessarily a datapoint

Any location in Rm: not necessarily a datapoint

• Plus-plane = { x : w . x + b = +1 }• Minus-plane = { x : w . x + b = -1 }• The vector w is perpendicular to the Plus Plane• Let x- be any point on the minus plane• Let x+ be the closest plus-plane-point to x-.• Claim: x+ = x- + w for some value of . Why?

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

• Plus-plane = { x : w . x + b = +1 }• Minus-plane = { x : w . x + b = -1 }• The vector w is perpendicular to the Plus Plane• Let x- be any point on the minus plane• Let x+ be the closest plus-plane-point to x-.• Claim: x+ = x- + w for some value of . Why?

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

The line from x- to x+ is perpendicular to the planes.

So to get from x- to x+ travel some distance in direction w.

What we know:• w . x+ + b = +1 • w . x- + b = -1 • x+ = x- + w• |x+ - x- | = M

It’s now easy to get M in terms of w and b

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

What we know:• w . x+ + b = +1 • w . x- + b = -1 • x+ = x- + w• |x+ - x- | = M

It’s now easy to get M in terms of w and b

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width

w . (x - + w) + b = 1

w . x - + b + w .w = 1

-1 + w .w = 1

What we know:• w . x+ + b = +1 • w . x- + b = -1 • x+ = x- + w• |x+ - x- | = M•

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width =

M = |x+ - x- | =| w |=

www .|| λλ

Learning the Maximum Margin Classifier

Given a guess of w and b we can• Compute whether all data points in the correct half-planes• Compute the width of the margin

So now we just need to write a program to search the space of w’s and b’s to find the widest margin that matches all the datapoints. How?

Gradient descent? Simulated Annealing? Matrix Inversion? EM? Newton’s Method?

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

M = Margin Width =

• Linear Programming

find w

argmax cw

subject to

wai bi, for i = 1, …, m

wj 0 for j = 1, …, n

Don’t worry…

it’s good for you…

Don’t worry…

it’s good for you…

There are fast algorithms for solving linear programs including the

simplex algorithm and Karmarkar’s algorithm

Learning via Quadratic Programming

• QP is a well-studied class of optimization algorithms to maximize a quadratic function of some real-valued variables subject to linear constraints.

Quadratic Programming

2maxarg

TT Find

nmnmnn

buauaua

22222121

11212111

)()(22)(11)(

)2()2(22)2(11)2(

)1()1(22)1(11)1(

enmmenenen

nmmnnn

buauaua

And subject to

n additional linear inequality constraints

linear

uality

Quadratic criterion

Subject to

Quadratic Programming

2maxarg

TT Find

Subject to

nmnmnn

buauaua

22222121

11212111

)()(22)(11)(

)2()2(22)2(11)2(

)1()1(22)1(11)1(

enmmenenen

nmmnnn

buauaua

And subject to

n additional linear inequality constraints

linear

uality

Quadratic criterion

There exist algorithms for

finding such constrained

quadratic optima much

more efficiently and

reliably than gradient

ascent.

(But they are very fiddly…you

probably don’t want to

write one yourself)

Learning the Maximum Margin ClassifierGiven guess of w , b we can• Compute whether all data

points are in the correct half-planes

• Compute the margin width

Assume R datapoints, each (xk,yk) where yk = +/- 1

“ Predict Class

= +1”

“ Predict Class

= -1”

zonewx+b=1

wx+b=0

wx+b=-

What should our quadratic optimization criterion be?

How many constraints will we have?

What should they be?

Minimize w.w

w . xk + b >= 1 if yk = 1

w . xk + b <= -1 if yk = -1

Uh-oh!

denotes +1

denotes -1

This is going to be a problem!

What should we do?

Idea 1:

Find minimum w.w, while minimizing number of training set errors.

Problem: Two things to minimize makes for an ill-defined optimization

Uh-oh!

denotes +1

denotes -1

What should we do?

Idea 1.1:

Minimize

w.w + C (#train errors)

There’s a serious practical problem that’s about to make us reject this approach. Can you guess what it is?

Tradeoff parameter

Uh-oh!

denotes +1

denotes -1

What should we do?

Idea 1.1:

Minimize

w.w + C (#train errors)

There’s a serious practical problem that’s about to make us reject this approach. Can you guess what it is?

Tradeoff parameterCan’t be expressed as a Quadratic

Programming problem.

Solving it may be too slow.

(Also, doesn’t distinguish between disastrous errors and near

misses)

So… any

ideas?

Uh-oh!

denotes +1

denotes -1

What should we do?

Idea 2.0:

Minimize w.w + C (distance of error points to their correct place)

Learning Maximum Margin with NoiseGiven guess of w , b we can

• Compute sum of distances of points to their correct zones

wx+b=1

wx+b=0

wx+b=-

How many constraints will we have?

04/21/23 38

Large-margin Decision Boundary• The decision boundary should be as far away from the data of both

classes as possible– We should maximize the margin, m– Distance between the origin and the line wtx=k is k/||w||

Class 1

Class 2

04/21/23 39

Finding the Decision Boundary

• Let {x1, ..., xn} be our data set and let yi {1,-1} be the class label of xi

• The decision boundary should classify all points correctly

• The decision boundary can be found by solving the following constrained optimization problem

• This is a constrained optimization problem. Solving it requires some new tools– Feel free to ignore the following several slides; what is important is the

constrained optimization problem above

04/21/23 40

Back to the Original Problem

• The Lagrangian is

– Note that ||w||2 = wTw

• Setting the gradient of w.r.t. w and b to zero, we have

• The Karush-Kuhn-Tucker conditions,

04/21/23 41

04/21/23 42

The Dual Problem• If we substitute to , we have

• Note that

• This is a function of i only

The Dual Problem

• The new objective function is in terms of i only

• It is known as the dual problem: if we know w, we know all i; if we know all i, we know w

• The original problem is known as the primal problem• The objective function of the dual problem needs to be maximized!• The dual problem is therefore:

Properties of i when we introduce the Lagrange multipliers

The result when we differentiate the original Lagrangian w.r.t. b

04/21/23 44

The Dual Problem

• This is a quadratic programming (QP) problem

– A global maximum of i can always be found

• w can be recovered by

04/21/23 45

Characteristics of the Solution• Many of the i are zero

– w is a linear combination of a small number of data points– This “sparse” representation can be viewed as data compression

as in the construction of knn classifier

• xi with non-zero i are called support vectors (SV)

– The decision boundary is determined only by the SV

– Let tj (j=1, ..., s) be the indices of the s support vectors. We can write

• For testing with a new data z

– Compute and classify z

as class 1 if the sum is positive, and class 2 otherwise

– Note: w need not be formed explicitly

04/21/23 46

A Geometrical Interpretation

Class 1

Class 2

04/21/23 47

Non-linearly Separable Problems

• We allow “error” i in classification; it is based on the output of the discriminant function wTx+b

• i approximates the number of misclassified samples

Class 1

Class 2

wx+b=1

wx+b=0

wx+b=-

Minimize

How many constraints will we have? R

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

wx+b=1

wx+b=0

wx+b=-

Minimize

Our original (noiseless data) QP had m+1 variables: w1, w2, … wm, and b.

Our new (noisy data) QP has m+1+R variables: w1, w2, … wm, b, k , 1 ,… R

m = # input dimension

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

R= # records

Learning Maximum Margin with NoiseGiven guess of w , b we

can• Compute sum of

distances of points to their correct zones

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

wx+b=1

wx+b=0

wx+b=-

Minimize

There’s a bug in this QP. Can you spot it?

wx+b=1

wx+b=0

wx+b=-

Minimize

How many constraints will we have? 2R

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

k >= 0 for all k

wx+b=1

wx+b=0

wx+b=-

Minimize

How many constraints will we have? 2R

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

k >= 0 for all k

An Equivalent Dual QP

Maximize

kk Qααα

1 11 2

1where ).( lklkkl yyQ xx

Subject to these constraints:

kCαk 0 01

kkk yα

Minimize

w . xk + b >= 1-k if yk = 1

w . xk + b <= -1+k if yk = -1

k >= 0 ， for all k

An Equivalent Dual QP

Maximize

kk Qααα

1 11 2

1where ).( lklkkl yyQ xx

kCαk 0

Then define:

kkkk yα

maxarg where

Then classify with:

kkk yα

Example XOR problem revisited:

Let the nonlinear mapping be :

(x) = (1,x12, 21/2 x1x2, x2

2, 21/2 x1 , 21/2 x2)T

And: (xi)=(1,xi12, 21/2 xi1xi2, xi2

2, 21/2 xi1 , 21/2 xi2)T

Therefore the feature space is in 6D with input data in 2D

x1 = (-1,-1), d1= - 1 x2 = (-1, 1), d2= 1 x3 = ( 1,-1), d3= 1 x4 = (-1,-1), d4= -1

Q(a)= ai – ½ ai aj di dj xi) Txj)

=a1 +a2 +a3 +a4 – ½(9 a1 a1 - 2a1 a2 -2 a1 a3 +2a1 a4

+9a2 a2 + 2a2 a3 -2a2 a4 +9a3 a3 -2a3 a4 +9 a4 a4 )

To minimize Q, we only need to calculate

(due to optimality conditions) which gives

1 = 9 a1 - a2 - a3 + a4

1 = -a1 + 9 a2 + a3 - a4

1 = -a1 + a2 + 9 a3 - a4

1 = a1 - a2 - a3 + 9 a4

4,...,1,0)( i

The solution of which gives the optimal values: a0,1 =a0,2 =a0,3 =a0,4 =1/8

w0 = a0,i di xi) = 1/8[x1)- x2)- x3)+ x4)]

Where the first element of w0 gives the bias b

From earlier we have that the optimal hyperplane is defined by:

w0T x) = 0

That is:

100 21

w0T x)

which is the optimal decision boundary for the XOR problem. Furthermore we note that the solution is unique since the optimal decision boundary is unique

Output for polynomial RBF

Harder 1-dimensional datasetRemember how

permitting non-linear basis functions made linear regression so much nicer?

Let’s permit them here too

x=0 ),( 2kkk xxz

For a non-linearly separable problem we have to first map data onto feature space so that they are linear separable

Given the training data sample {(xi,yi), i=1, …,N}, find the optimum values of the weight vector w and bias b

w = a0,i yi xi)

where a0,i are the optimal Lagrange multipliers determined by maximizing the following objective function

subject to the constraints

ai yi =0 ; ai >0

ii xxddaaaaQ

SVM building procedure:

1. Pick a nonlinear mapping 2. Solve for the optimal weight vector

However: how do we pick the function

• In practical applications, if it is not totally impossible to find it is very hard

• In the previous example, the function is quite complex: How would we find it?

Answer: the Kernel Trick

Notice that in the dual problem the image of input vectors only involved as an inner product meaning that the optimization can be performed in the (lower dimensional) input space and that the inner product can be replaced by an inner-product kernel

How do we relate the output of the SVM to the kernel K?

Look at the equation of the boundary in the feature space and use the optimality conditions derived from the Lagrangian formulations

1( ) ( ) ( )2

1 ( , )2

ii i j i j j ji i j

i i j i j i ji i j

Q a a a a d d x x

a a a d d K x x

( ) 0; ( ) 1

: ( ) [ ( ), ( ),..., ( )]

: ( ) 0

Hyperplane is defined by

w x where x

writing x x x x

we get w x

from optimality conditions w a d x

: ( ) ( ) 0

: ( , ) 0

( ) ( , )

: ( , ) ( ) ( )

i i ii

ii j jj

Thus a d x x

and so boundary is a d K x x

and Output w x a d K x x

where K x x x x

In the XOR problem, we chose to use the kernel function:K(x, xi) = (x T

xi+1)2

= 1+ x12 xi1

2 + 2 x1x2 xi1xi2 + x22 xi2

2 + 2x1xi1 ,+ 2x2xi2

Which implied the form of our nonlinear functions:(x) = (1,x1

2, 21/2 x1x2, x22, 21/2 x1 , 21/2 x2)T

And: (xi)=(1,xi12, 21/2 xi1xi2, xi2

2, 21/2 xi1 , 21/2 xi2)T

However, we did not need to calculate at all and could simply have used the kernel to calculate:

Q(a) = ai – ½ ai aj di dj xixj

Maximized and solved for ai and derived the hyperplane via:

ii xxKda

We therefore only need a suitable choice of kernel function cf:Mercer’s Theorem:

Let K(x,y) be a continuous symmetric kernel that defined in the closed interval [a,b]. The kernel K can be expanded in the form

(x,y) = x) T y)

provided it is positive definite. Some of the usual choices for K are:

Polynomial SVM (x T xi+1)p p specified by user

RBF SVM exp(-1/(2) || x – xi||2) specified by user

MLP SVM tanh(s0 x T xi + s1)

Maximize

where ).( lklkkl yyQ xx

kCαk 0

Then define:

kkkk yα

maxarg where

Then classify with:

kkk yα

Datapoints with k > 0 will be the support vectors

..so this sum only needs to be over the support vectors.

kk Qααα

1 11 2

1An Equivalent Dual QP

Quadratic Basis

Functions

Constant Term

Linear Terms

Pure Quadratic

Quadratic Cross-Terms

Number of terms (assuming m input dimensions) = (m+2)-choose-2

= (m+2)(m+1)/2

= (as near as makes no difference) m2/2

You may be wondering what those

’s are doing.

•You should be happy that they do no harm

•You’ll find out why they’re there soon.

QP with basis functionswhere ))().(( lklkkl yyQ xΦxΦ

kCαk 0

Then define:

maxarg where

Then classify with:

f(x,w,b) = sign(w. (x) - b)

kkk yα

0 s.t.

)(kαk

kkk yα xΦw

Maximize

kk Qααα

1 11 2

QP with basis functionswhere ))().(( lklkkl yyQ xΦxΦ

kCαk 0

Then define:

maxarg where

Then classify with:

kkk yα

We must do R2/2 dot products to get this matrix ready.

Each dot product requires m2/2 additions and multiplications

The whole thing costs R2 m2 /4. Yeeks!

……or does it?or does it?

0 s.t.

)(kαk

kkk yα xΦw

Maximize

kk Qααα

1 11 2

)()( bΦaΦ

iii ba

ijjiji bbaa

Quadratic Dot Products

)()( bΦaΦ

ijjiji

iii bbaababa

Just out of casual, innocent, interest, let’s look at another function of a and b:

2)1.( ba

1.2).( 2 baba

iii baba

1211 1

jjjii bababa

122)(11 11

ijjjii

iii babababa

)()( bΦaΦ

Just out of casual, innocent, interest, let’s look at another function of a and b:

2)1.( ba

1.2).( 2 baba

iii baba

1211 1

jjjii bababa

122)(11 11

ijjjii

iii babababa

They’re the same!

And this is only O(m) to compute!

ijjiji

iii bbaababa

QP with Quadratic basis functionswhere ))().(( lklkkl yyQ xΦxΦ

kCαk 0

Then define:

maxarg where

Then classify with:

kkk yα

Each dot product now only requires m additions and multiplications

0 s.t.

)(kαk

kkk yα xΦw

Maximize

kk Qααα

1 11 2

Higher Order Polynomials

Polynomial (x) Cost to build Qkl matrix traditionally

Cost if 100 inputs

(a).(b) Cost to build Qkl matrix efficiently

Cost if 100 inputs

Quadratic All m2/2 terms up to degree 2

m2 R2 /4 2,500 R2 (a.b+1)2 m R2 / 2 50 R2

Cubic All m3/6 terms up to degree 3

m3 R2 /12 83,000 R2 (a.b+1)3 m R2 / 2 50 R2

Quartic All m4/24 terms up to degree 4

m4 R2 /48 1,960,000 R2

(a.b+1)4 m R2 / 2 50 R2

QP with Quintic basis functions

Maximize

kk Qααα

where ))().(( lklkkl yyQ xΦxΦ

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

Maximize

kk Qααα

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

In 100-d, each dot product now needs 103 operations instead of 75 million

But there are still worrying things lurking away. What are they?

•The fear of overfitting with this enormous number of terms

•The evaluation phase (doing a set of predictions on a test set) will be very expensive (why?)

Maximize

kk Qααα

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

Because each w. (x) (see below) needs 75 million operations. What can be done?

The use of Maximum Margin magically makes this not a problem

Maximize

kk Qααα

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

Only Sm operations (S=#support vectors)

0 s.t.

)()()(kαk

kkk yα xΦxΦxΦw

0 s.t.

5)1(kαk

kkk yα xx

Maximize

kk Qααα

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

0 s.t.

)()()(kαk

kkk yα xΦxΦxΦw

0 s.t.

5)1(kαk

kkk yα xx

QP with Quintic basis functionswhere ))().(( lklkkl yyQ xΦxΦ

kCαk 0

Then define:

0 s.t.

)(kαk

kkk yα xΦw

maxarg where

Then classify with:

kkk yα

Why SVMs don’t overfit as much as you’d think:

No matter what the basis function, there are really only up to R parameters: 1, 2 .. R, and usually most are set to zero by the Maximum Margin.

Asking for small w.w is like “weight decay” in Neural Nets and like Ridge Regression parameters in Linear regression and like the use of Priors in Bayesian Regression---all designed to smooth the function and reduce overfitting.

0 s.t.

)()()(kαk

kkk yα xΦxΦxΦw

0 s.t.

5)1(kαk

kkk yα xx

Maximize

kk Qααα

1 11 2

SVM Kernel Functions

• K(a,b)=(a . b +1)d is an example of an SVM Kernel Function

• Beyond polynomials there are other very high dimensional basis functions that can be made practical by finding the right Kernel Function– Radial-Basis-style Kernel Function:

– Neural-net-style Kernel Function:

)(exp),(

).tanh(),( babaK

, and are magic parameters that must be chosen by a model selection method such as CV or VCSRM

SVM Implementations

• Sequential Minimal Optimization, SMO, efficient implementation of SVMs, Platt – in Weka

• SVMlight

– http://svmlight.joachims.org/

References

• Tutorial on VC-dimension and Support Vector Machines:

C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html

• The VC/SRM/SVM Bible:Statistical Learning Theory by Vladimir Vapnik, Wiley-

Interscience; 1998

Introduction to SVMs. SVMs Geometric –Maximizing Margin Kernel Methods –Making nonlinear decision boundaries linear –Efficiently! Capacity –Structural.

zonewx b

x bhow

x bdefine

x bany

vector w

terms of w

margin widthplusplane

margin widthhow

Documents

Learning Structural SVMs with Latent Variables

Learning Ranking Functions with SVMs

PORTAFOLIOS DE INVERSIÓN DE MARKOVITZ CON SVMs …

SVMs for (x) Recognition (From Moghaddam / Yang’s...

Introduction to SVMs

Microeconomics Supply Producing Producing efficiently...

Spring Portraits SVMS Flyerjpg

OEMs SVMs (LDV/MDV/HDV)

Support Vector Machines (SVMs). Semi-Supervised...

Customer Use Case: Efficiently Maximizing Retail Value ........

Classification using intersection kernel SVMs is...

Cutting-Plane Training of Structural SVMs...Cutting-Plane...

APPLYING SVMS POST-PROCESSINGSTRATEGIES FOR … · APPLYING...

Enhancing Exemplar SVMs using Part Level Transfer ...

Using SVMs for Time Series Prediction

SVMs in R tutorial