Support Vector Machines

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Support Vector Machines

Chih-Jen Lin

Department of Computer Science National Taiwan University

Talk at Machine Learning Summer School 2006, Taipei

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Outline

Basic concepts

SVM primal/dual problems

Training linear and nonlinear SVMs

Parameter/kernel selection and practical issues Multi-class classification

Discussion and conclusions

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Basic concepts

Outline

Basic concepts

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Basic concepts

Why SVM and Kernel Methods

SVM: in many cases competitive with existing classification methods

Relatively easy to use

Kernel techniques: many extensions

Regression, density estimation, kernel PCA, etc.

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Basic concepts

Support Vector Classification

Training vectors : xi, i = 1, . . . , l Feature vectors. For example, A patient = [height, weight, . . .]

Consider a simple case with two classes:

Define an indicator vector y y_i =

1 if xi in class 1

−1 if xⁱ in class 2, A hyperplane which separates all data

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Basic concepts

w^Tx+ b = h₊₁

−10

i A separating hyperplane: w^Tx+ b = 0

(w^Txi) + b > 0 if yi = 1 (w^Tx_i) + b < 0 if yi = −1

Decision function f (x) = sgn(w^Tx+ b), x: test data Many possible choices of w and b

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Basic concepts

Maximal Margin

Distance between w^Tx+ b = 1 and −1:

2/kwk = 2/√ w^Tw A quadratic programming problem [Boser et al., 1992]

minw,b

1 2w^Tw

subject to yi(w^Txi + b) ≥ 1, i = 1, . . . , l.

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Basic concepts

Data May Not Be Linearly Separable

An example:

Allow training errors

Higher dimensional ( maybe infinite ) feature space φ(x) = (φ1(x), φ2(x), . . .).

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Basic concepts

Standard SVM [Cortes and Vapnik, 1995]

w,b,ξmin 1

2w^Tw+C

l

X

i=1

ξi

subject to yi(w^Tφ(xi)+ b) ≥ 1 −ξi, ξi ≥ 0, i = 1, . . . , l.

Example: x ∈ R³, φ(x) ∈ R¹⁰ φ(x) = (1,√

2x₁,√

2x₂,√

2x₃, x₁², x₂², x₃²,√

2x1x₂,√

2x1x₃,√

2x2x₃)

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Basic concepts

Finding the Decision Function

w: maybe infinite variables The dual problem

minα

1

2α^TQα− e^Tα

subject to 0 ≤ αⁱ ≤ C , i = 1, . . . , l y^Tα = 0,

where Qij = yiyjφ(xi)^Tφ(xj) and e = [1, . . . , 1]^T At optimum

w = Pl

i=1αiy_iφ(xi)

A finite problem: #variables = #training data

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Basic concepts

Kernel Tricks

Qij = yiyjφ(xi)^Tφ(xj) needs a closed form Example: xi ∈ R³, φ(xi) ∈ R¹⁰

φ(xi) = (1,√

2(xi)1,√

2(xi)2,√

2(xi)3, (xi)²₁, (xi)²₂, (xi)²₃,√

2(xi)1(xi)2,√

2(xi)1(xi)3,√

2(xi)2(xi)3) Then φ(xi)^Tφ(xj) = (1 + x^T_i xj)².

Kernel: K (x, y) = φ(x)^Tφ(y); common kernels:

e^−γkxⁱ^−x^j^k², (Radial Basis Function) (x^T_i x_j/a + b)^d (Polynomial kernel)

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Basic concepts

Can be inner product in infinite dimensional space Assume x ∈ R¹ and γ > 0.

e^−γkxⁱ^−x^j^k² = e^−γ(xⁱ^−x^j⁾² = e^−γxⁱ²^+2γxⁱ^x^j^−γx^j²

=e^−γxⁱ²^−γx^j² 1 + 2γxix_j

1! + (2γxix_j)²

2! + (2γxix_j)³

3! + · · ·

=e^−γxⁱ²^−γx^j² 1 · 1+r 2γ

1!x_i ·r 2γ 1!x_j+

r(2γ)² 2! x_i² ·

r(2γ)² 2! x_j² +

r(2γ)³ 3! x_i³ ·

r(2γ)³

3! x_j³ + · · · = φ(xi)^Tφ(xj), where

φ(x) = e^−γx²

1,r 2γ 1!x,

r(2γ)² 2! x²,

r(2γ)³

3! x³, · · ·

T

.

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Basic concepts

More about Kernels

How do we know kernels help to separate data?

In R^l, any l independent vectors

⇒ linearly separable



 (x¹)^T

...

(x^l)^T



w = +e

−e

If K positive definite ⇒ data linearly separable K = LL^T.

Transforming training points to independent vectors in R^l

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Basic concepts

So what kind of kernel should I use?

What kind of functions are valid kernels?

How to decide kernel parameters?

Will be discussed later

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Basic concepts

Decision function

At optimum

w = Pl

i=1αiy_iφ(xi) Decision function

w^Tφ(x) + b

=

l

X

i=1

αiy_iφ(xi)^Tφ(x) + b

=

l

X

i=1

αiy_iK(xi, x) + b

Only φ(xi) of αi > 0 used ⇒ support vectors

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Basic concepts

Support Vectors: More Important Data

−1.5 −1 −0.5 0 0.5 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

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Basic concepts

So we have roughly shown basic ideas of SVM A 3-D demonstration

www.csie.ntu.edu.tw/˜cjlin/libsvmtools/svmtoy3d Further references, for example,

[Cristianini and Shawe-Taylor, 2000, Sch¨olkopf and Smola, 2002]

Also see discussion on kernel machines blackboard www.kernel-machines.org/phpbb/

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Outline

Basic concepts

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Deriving the Dual

Consider the problem without ξi

minw,b

1 2w^Tw

subject to y_i(w^Tφ(xi) + b) ≥ 1, i = 1, . . . , l.

Its dual

minα

1

2α^TQα− e^Tα

subject to 0 ≤ αⁱ, i = 1, . . . , l, y^Tα = 0.

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Lagrangian Dual

maxα≥0 min

w,b L(w, b, α), where

L(w, b, α) = 1

2kwk² −

l

X

i=1

αi y_i(w^Tφ(xi) + b) − 1 Strong duality (be careful about this)

min Primal = max

α≥0 min

w,b L(w, b, α)

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Simplify the dual. When α is fixed, minw,b L(w, b, α) =

(−∞ if Pl

i=1αiy_i 6= 0, minw

1

2w^Tw−Pl

i=1αi[yi(w^Tφ(xi) − 1] if Pl

i=1αiy_i = 0.

If Pl

i=1αiy_i 6= 0, decrease

−b

l

X

i=1

αiy_i in L(w, b, α) to −∞

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If Pl

i=1αiy_i = 0, optimum of the strictly convex

1

2w^Tw−Pl

i=1αi[yi(w^Tφ(xi) − 1] happens when

∂

∂wL(w, b, α) = 0.

Thus,

w =

l

X

i=1

αiy_iφ(xi).

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Note that

w^Tw =

l

X

i=1

αiy_iφ(xi)

T l

X

j=1

αjy_jφ(xj)

= X

i,j

αiαjy_iy_jφ(xi)^Tφ(xj)

The dual is maxα≥0







l

P

i=1

αi − ¹₂ P

i,j

αiαjy_iy_jφ(xi)^Tφ(xj) if Pl

i=1αiy_i = 0,

−∞ if Pl

i=1αiy_i 6= 0.

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Lagrangian dual: maxα≥0 minw,bL(w, b, α)

−∞ definitely not maximum of the dual Dual optimal solution not happen when

l

X

i=1

αiy_i 6= 0 .

Dual simplified to maxα∈R^l

l

X

i=1

αi − 1 2

l

X

i=1 l

X

j=1

αiαjy_iy_jφ(xi)^Tφ(xj) subject to y^Tα = 0,

αi ≥ 0, i = 1, . . . , l.

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Outline

Basic concepts

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Training Nonlinear SVMs

If using kernels, we solve the dual minα

1

2α^TQα− e^Tα

subject to 0 ≤ αⁱ ≤ C , i = 1, . . . , l y^Tα = 0

Large dense quadratic programming Q_ij 6= 0, Q : an l by l fully dense matrix 30,000 training points: 30,000 variables:

(30, 000² × 8/2) bytes = 3GB RAM to store Q:

Traditional methods:

Newton, Quasi Newton cannot be directly applied

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Decomposition Methods

Working on some variables each time (e.g.,

[Osuna et al., 1997, Joachims, 1998, Platt, 1998]) Similar to coordinate-wise minimization

Working set B , N = {1, . . . , l}\B fixed Sub-problem at each iteration:

minαB

1

2α^T_B (α^k_N)^TQ_BB Q_BN QNB QNN

α_B α^k_N

−

e^T_B (e^k_N)^Tα_B α^k_N

subject to 0 ≤ α^t ≤ C , t ∈ B, yB^Tα_B = −y^TNα^k_N

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Avoid Memory Problems

The new objective function 1

2α^T_BQ_BBα_B + (−e^B + QBNα^k_N)^Tα_B + constant B columns of Q needed

Calculated when used Trade time for space

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Does it Really Work?

Compared to Newton, Quasi-Newton Slow convergence

However, no need to have very accurate α sgn

l

X

i=1

αiy_iK(xi, x) + b

!

Prediction not affected much

In some situations, # support vectors ≪ # training points

Initial α¹ = 0, some elements never used

Machine learning knowledge affects optimization

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An example of training 50,000 instances using LIBSVM

$ ./svm-train -m 200 -c 16 -g 4 22features optimization finished, #iter = 24981

Total nSV = 3370 time 5m1.456s

On a Pentium M 1.4 GHz Laptop

Calculating Q may have taken more than 5 minutes

#SVs = 3,370 ≪ 50,000

A good case where some remain at zero all the time

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Issues of Decomposition Methods

Working set size/selection Asymptotic convergence

Finite termination & stopping conditions Convergence rate

Numerical issues

Optimization researchers are now also interested in these issues

If interested in them, check my talk to optimization researchers in Rome last year:

http://www.csie.ntu.edu.tw/^∼cjlin/talks/rome.pdf

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Caching and Shrinking

Speed up decomposition methods Caching [Joachims, 1998]

Store recently used kernel columns in computer memory

100K Cache

$ time ./libsvm-2.81/svm-train -m 0.01 a4a 11.463s

40M Cache

$ time ./libsvm-2.81/svm-train -m 40 a4a 7.817s

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Shrinking [Joachims, 1998]

Some bounded elements remain until the end Heuristically resized to a smaller problem

After certain iterations, most bounded elements identified and not changed [Lin, 2002]

So caching and shrinking are useful

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Caching: Issues

A simple way:

Store recently used columns What if in working set selection,

deliberately select some indices in cache Goal: minimize the total number of columns calculated

Difficult to connect algorithm and this goal

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SVM doesn’t Scale Up

Yes, if you use kernels

Training millions of data is time consuming

But other nonlinear methods face the same problem e.g., kernel logistic regression

Two possibilities

1 Linear SVMs: in some situations, can solve much larger problems

2 Approximation

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Training Linear SVMs

Linear kernel:

w,b,ξmin 1

2w^Tw+ C

l

X

i=1

ξi

subject to y_i(w^Tx_i + b) ≥ 1 − ξⁱ, ξi ≥ 0.

At optimum:

ξi = max 0, 1 − yⁱ(w^Txi + b)

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Remaining variables: w, b minw,b

1

2w^Tw+ C

l

X

i=1

max 0, 1 − yⁱ(w^Tx_i + b)

#variables = #features + 1 If #features small, easier to solve

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Traditional optimization methods can be applied Training time similar to methods such as logistic regression

What if #features and #instances both large?

Very challenging

Some language/document problems are of this type

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Decomposition Methods for Linear SVMs

Could we still solve the dual by decomposition methods?

Even if #features small

Slow convergence when C is large

$bsvm-train -b 500 -c 500 -t 0 australian_scale optimization finished, #iter = 260092

obj = -99310.588975, rho = 0.000000 K_ij = x^T_i x_j, rank ≤ #features

positive semi-definite only

Still a research topic in understanding this

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Decomposition Methods for Linear SVMs

But no need to use large C

C large enough, w the same [Keerthi and Lin, 2003]

decision function the same Remember

w =

l

X

i=1

αiy_ixi ∈ Rⁿ, b ∈ R¹

|# of 0 < αⁱ < C | ≤ n + 1

As C changes, optimal α share many elements at 0 and C

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Decomposition Methods for Linear SVMs (Cont’d)

Warm start very effective [Kao et al., 2004]

Starting from small C , faster convergence Using C = 1, 2, 4, 8, . . .

$bsvm-train -c 500 -t 0 australian_scale optimization finished, #iter = 10087 So decomposition methods can still handle large linear SVMs

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Approximations

#instances large and using nonlinear kernels Difficult to solve the dual

Subsampling

Simple and often effective

From this many more advanced techniques E.g., stratified subsampling

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Approximations (Cont’d)

Incremental way: (e.g., [Syed et al., 1999]) Data ⇒ 10 parts

train 1st part ⇒ SVs, train SVs + 2nd part, . . . Select good points first: KNN or heuristics e.g., [Bakır et al., 2005]

Hierarchical settings (e.g., [Yu et al., 2003]) Clustering training data to several groups SVM models built for each group

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Approximations (Cont’d)

Using only a subset to construct w

w = X

i∈B

αiy_iφ(xi).

Put this into the primal

αminB,b,ξ

1

2α^T_BQ_BBα_B + C

l

X

i=1

ξi

subject to Q_:,Bα_B + by ≥ e − ξ Without considering ξi, #variables = |B| + 1

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Approximations (Cont’d)

Selecting B:

random [Lee and Mangasarian, 2001], incremental [Keerthi et al., 2006], and many other ways

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Approximations (Cont’d)

All these approaches

some simple but some sophisticated In machine learning, very often

balance between simplification and performance

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Parameter/kernel selection and practical issues

Outline

Basic concepts

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Let’s Try a Practical Example

A problem from astroparticle physics

1 1:2.6173e+01 2:5.88670e+01 3:-1.89469e-01 4:1.25122e+02 1 1:5.7073e+01 2:2.21404e+02 3:8.60795e-02 4:1.22911e+02 1 1:1.7259e+01 2:1.73436e+02 3:-1.29805e-01 4:1.25031e+02 1 1:2.1779e+01 2:1.24953e+02 3:1.53885e-01 4:1.52715e+02 1 1:9.1339e+01 2:2.93569e+02 3:1.42391e-01 4:1.60540e+02 1 1:5.5375e+01 2:1.79222e+02 3:1.65495e-01 4:1.11227e+02 1 1:2.9562e+01 2:1.91357e+02 3:9.90143e-02 4:1.03407e+02

Training and testing sets available: 3,089 and 4,000

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The Story Behind this Data Set

User:

I am using libsvm in a astroparticle physics application .. First, let me congratulate you to a really easy to use and nice package. Unfortunately, it gives me astonishingly bad results...

OK. Please send us your data

I am able to get 97% test accuracy. Is that good enough for you ?

User:

You earned a copy of my PhD thesis

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Training and Testing

Training

$./svm-train train.1

optimization finished, #iter = 6131 nSV = 3053, nBSV = 724

Total nSV = 3053 Testing

$./svm-predict test.1 train.1.model test.1.out Accuracy = 66.925% (2677/4000)

nSV and nBSV: number of SVs and bounded SVs (αi = C ).

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Why this Fails

After training, nearly 100% support vectors Training and testing accuracy different

$./svm-predict train.1 train.1.model o Accuracy = 99.7734% (3082/3089)

Most kernel elements:

K_ij = e^−kxⁱ^−x^j^k²^/4

(= 1 if i = j,

→ 0 if i 6= j.

Some features in rather large ranges

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Data Scaling

Without scaling

Attributes in greater numeric ranges may dominate Example:

height gender

x₁ 150 F

x₂ 180 M

x₃ 185 M

and

y₁ = 0, y2 = 1, y3 = 1.

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The separating hyperplane almost vertical

x₁

x₂x₃

Strongly depends on the first attribute; but second may be also important

Linearly scale the first to [0, 1] by:

1st attribute − 150 185 − 150 , Scaling generally helps, but not always

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Other ways for scaling

Needed for k Nearest Neighbor, Neural networks as well

unless the method is scale-invariant

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Data Scaling: Same Factors

A common mistake

$./svm-scale -l -1 -u 1 train.1 > train.1.scale

$./svm-scale -l -1 -u 1 test.1 > test.1.scale Same factor on training and testing

$./svm-scale -s range1 train.1 > train.1.scale

$./svm-scale -r range1 test.1 > test.1.scale

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After Data Scaling

Train scaled data and then prediction

$./svm-train train.1.scale

$./svm-predict test.1.scale train.1.scale.model test.1.predict

Accuracy = 96.15%

Training accuracy now is

$./svm-predict train.1.scale train.1.scale.model Accuracy = 96.439% (2979/3089)

Default parameter: C = 1, γ = 0.25

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Different Parameters

If we use C = 20, γ = 400

$./svm-train -c 20 -g 400 train.1.scale

$./svm-predict train.1.scale train.1.scale.model Accuracy = 100% (3089/3089)

100% training accuracy but

$./svm-predict test.1.scale train.1.scale.model Accuracy = 82.7% (3308/4000)

Very bad test accuracy Overfitting happens

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Overfitting

In theory

You can easily achieve 100% training accuracy This is useless

When training and predicting a data, we should Avoid underfitting: small training error

Avoid overfitting: small testing error

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● and ▲: training; and △: testing

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Parameter Selection

Is important

Now parameters are C, kernel parameters Example:

γ of e^−γkxⁱ^−x^j^k² a, b, d of (x^T_i x_j/a + b)^d How to select them?

So performance better?

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Parameter Selection (Cont’d)

Also how to select kernels?

e.g., RBF or polynomial

Moreover, how to select methods?

e.g., SVM or decision trees?

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Performance Evaluation

l training data, xi ∈ Rⁿ, yi ∈ {+1, −1}, i = 1, . . . , l, a learning machine:

x → f (x, α), f (x, α) = 1 or − 1.

Different α: different machines

The expected test error (generalized error) R(α) =

Z 1

2|y − f (x, α)|dP(x, y) y: class of x (i.e. 1 or -1)

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P(x, y ) unknown, empirical risk (training error):

R_emp(α) = 1 2l

l

X

i=1

|yⁱ − f (xⁱ, α)|

Training errors not important; only test errors count

1

2|yⁱ − f (xⁱ, α)| : loss, choose 0 ≤ η ≤ 1, with probability at least 1 − η:

R(α) ≤ R^emp(α) + another term A good classification method:

minimize both terms at the same time

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But Remp(α) → 0; another term → large SVM:

w,b,ξmin 1

2w^Tw+ C

l

X

i=1

ξi

subject to y_i(w^Tφ(xi) + b) ≥ 1 − ξⁱ, ξi ≥ 0, i = 1, . Pl

i=1ξi related to training error

w^Tw/2 relate to another term: called regularization term

C: balance between the two

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Performance Evaluation (Cont’d)

In practice

Available data ⇒ training and validation Train the training

Test the validation k-fold cross validation:

Data randomly separated to k groups

Each time k − 1 as training and one as testing

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Using CV on training + validation

Predict testing with the best parameters from CV

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CV and Test Accuracy

If we select parameters so that CV is the highest, Does CV represent future test accuracy ?

Slightly different

If we have enough parameters, we can achieve 100%

CV as well

e.g., more parameters than # of training data Available data with class labels

⇒ training, validation, testing

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Selecting Kernels

RBF, polynomial, or others?

or even combinations Two situations:

Too many kernels complicates the selection Design kernels suitable for target applications

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Selecting Kernels (Cont’d)

Contradicting but practically ok We have few general kernels

RBF, polynomial, etc. somewhat related Beginners’ don’t have many choices On the other hand

researchers design many special ones e.g., string kernels

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Selecting Kernels (Cont’d)

For beginners, use RBF first

Linear kernel: special case of RBF

Performance of linear the same as RBF under certain parameters [Keerthi and Lin, 2003]

Polynomial: numerical difficulties (< 1)^d → 0, (> 1)^d → ∞

More parameters than RBF

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A Simple Procedure

1 Conduct simple scaling on the data

2 Consider RBF kernel K (x, y) = e^−γkx−yk²

3 Use cross-validation to find the best parameter C and γ

4 Use the best C and γ to train the whole training set

5 Test

For beginners only, you can do a lot more

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Contour of Parameter Selection

d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2 d2

d2 98.8

98.6 98.4 98.2 98 97.8 97.6 97.4 97.2 97

1 2 3 4 5 6 7

lg(C)

-2 -1 0 1 2 3

lg(gamma)

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The good region of parameters is quite large SVM is sensitive to parameters, but not that sensitive

Sometimes default parameters work

but it’s good to select them if time is allowed

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Efficient Parameter Selection

CV on grid points may be time consuming OK if one or two parameters

But if more than two?

E.g., feature scaling:

K(x, y) = e⁻^Pⁿⁱ⁼¹^γⁱ^(xⁱ^−yⁱ⁾² Some features more important

Still a challenging research issue

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Remember given parameters C and γ, we solve SVM to obtain optimal w or α

Model a function of parameters

C,γmin1,...,γn

f(α(C , γ₁, . . . , γn), C , γ₁, . . . , γn) But usually non-convex

The function

from Bayesian frameworks (e.g., [Chu et al., 2003]) or

smoothing CV bound

CV(C , γ1, . . . , γn) ≤ f (α(C , γ¹, . . . , γn), C , γ1, . . . , γn)

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The minimization:

Gradient-type methods or

global optimization (e.g., genetic algorithms) The difficulty:

Certainly more efforts than one single γ But performance may be just similar?

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Kernel Combination

How about using

t₁K₁ + t2K₂ + · · · + t^rKr, where

t₁ + · · · + t^r = 1 as the kernel

Related to parameter selection

t₁e^−γ¹^kx−yk+ · · · + t^re^−γ^r^kx−yk If γ1 good ⇒ t¹ close to 1, others close to 0

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[Lanckriet et al., 2004] form a convex f(α(t₁, . . . , tr), t₁, . . . , tr) when C is fixed

Semi-definite programming problem But computational cost is also high Need more empirical studies

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Design Kernels

Still a research issue

e.g., in bioinformatics and vision, many new kernels But, should be careful if the function is a valid one

K(x, y) = φ(x)^Tφ(y)

For example, any two strings s1, s2 we can define edit distance

e^−γedit(s¹^,s²⁾

It’s not a valid kernel [Cortes et al., 2003]

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Mercer condition

What kind of Kij can be represented as φ(xi)^Tφ(xj)?

K(x, y) = φ(x)^Tφ(y) if and only if ∀g s.t.

R g (x)²d x finite

⇒ R K (x, y)g(x)g(y)dxdy ≥ 0 A condition developed early last century However, still not easy to check

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Multi-class classification

Outline

Basic concepts

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Multi-class Classification

k classes

One-against-the rest: Train k binary SVMs:

1st class vs. (2 − k)th class 2nd class vs. (1, 3 − k)th class

...

k decision functions

(w¹)^Tφ(x) + b1

...

(w^k)^Tφ(x) + bk

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Prediction:

arg max

j (w^j)^Tφ(x) + bj

Reason: If the 1st class, then we should have (w¹)^Tφ(x) + b1 ≥ +1

(w²)^Tφ(x) + b2 ≤ −1 ...

(w^k)^Tφ(x) + bk ≤ −1

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Multi-class Classification (Cont’d)

One-against-one: train k(k − 1)/2 binary SVMs (1, 2), (1, 3), . . . , (1, k), (2, 3), (2, 4), . . . , (k − 1, k) If 4 classes ⇒ 6 binary SVMs

y_i = 1 yi = −1 Decision functions class 1 class 2 f¹²(x) = (w¹²)^Tx+ b¹² class 1 class 3 f¹³(x) = (w¹³)^Tx+ b¹³ class 1 class 4 f¹⁴(x) = (w¹⁴)^Tx+ b¹⁴ class 2 class 3 f²³(x) = (w²³)^Tx+ b²³ class 2 class 4 f²⁴(x) = (w²⁴)^Tx+ b²⁴ class 3 class 4 f³⁴(x) = (w³⁴)^Tx+ b³⁴

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For a testing data, predicting all binary SVMs Classes winner

1 2 1

1 3 1

1 4 1

2 3 2

2 4 4

3 4 3

Select the one with the largest vote class 1 2 3 4

# votes 3 1 1 1 May use decision values as well

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More Complicated Forms

For example,

[Vapnik, 1998, Weston and Watkins, 1999]:

w,b,ξmin 1 2

k

X

m=1

w_m^Tw_m+ C

l

X

i=1

X

m6=yi

ξ_i^m w^T_y

iφ(xi) + by_i ≥ w^Tmφ(xi) + bm + 2 − ξi^m, ξ_i^m ≥ 0, i = 1, . . . , l, m ∈ {1, . . . , k}\yⁱ. y_i: class of xi

kl constraints

Dual: kl variables; very large

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There are many other methods

A comparison in [Hsu and Lin, 2002]

Accuracy similar for many problems But 1-against-1 fastest for training

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Why 1vs1 Faster in Training

1 vs. 1

k(k − 1)/2 problems, each 2l/k data on average 1 vs. all

k problems, each l data

If solving the optimization problem:

polynomial of the size with degree d Their complexities

k(k − 1)

2 O 2l k

d

vs. kO(l^d)

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Outline

Basic concepts

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Future Directions

I mentioned quite a few. Here are others.

Better ways to handle unbalanced data

i.e., some classes few data, some classes a lot Multi-label classification

An instance associated with ≥ 2 labels e.g., a document in both politics, sports Structural data sets

An instance may not be a vector e.g., a tree from a sentence

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Conclusions

Dealing with data is interesting especially if you get good accuracy

Some basic understandings are essential when applying classification methods

SVM is a rather mature topic

but still quite a few interesting research issues

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References I

Bakır, G. H., Bottou, L., and Weston, J. (2005).

Breaking svm complexity with cross-training.

In Saul, L. K., Weiss, Y., and Bottou, L., editors, Advances in Neural Information Processing Systems 17, pages 81–88. MIT Press, Cambridge, MA.

Boser, B., Guyon, I., and Vapnik, V. (1992).

A training algorithm for optimal margin classifiers.

In Proceedings of the Fifth Annual Workshop on Computational Learning Theory, pages 144–152. ACM Press.

Chu, W., Keerthi, S., and Ong, C. (2003).

Bayesian trigonometric support vector classifier.

Neural Computation, 15(9):2227–2254.

Cortes, C., Haffner, P., and Mohri, M. (2003).

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In Proceedings of the 16th Annual Conference on Learning Theory, pages 41–56.

Cortes, C. and Vapnik, V. (1995).

Support-vector network.

Machine Learning, 20:273–297.

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References II

Cristianini, N. and Shawe-Taylor, J. (2000).

An Introduction to Support Vector Machines.

Cambridge University Press, Cambridge, UK.

Hsu, C.-W. and Lin, C.-J. (2002).

A comparison of methods for multi-class support vector machines.

IEEE Transactions on Neural Networks, 13(2):415–425.

Joachims, T. (1998).

Making large-scale SVM learning practical.

In Sch¨olkopf, B., Burges, C. J. C., and Smola, A. J., editors, Advances in Kernel Methods - Support Vector Learning, Cambridge, MA. MIT Press.

Kao, W.-C., Chung, K.-M., Sun, C.-L., and Lin, C.-J. (2004).

Decomposition methods for linear support vector machines.

Keerthi, S. S., Chapelle, O., and DeCoste, D. (2006).

Building support vector machines with reduced classifier complexity.

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References III

Keerthi, S. S. and Lin, C.-J. (2003).

Asymptotic behaviors of support vector machines with Gaussian kernel.

Lanckriet, G., Cristianini, N., Bartlett, P., El Ghaoui, L., and Jordan, M. (2004).

Learning the Kernel Matrix with Semidefinite Programming.

Journal of Machine Learning Research, 5:27–72.

Lee, Y.-J. and Mangasarian, O. L. (2001).

RSVM: Reduced support vector machines.

In Proceedings of the First SIAM International Conference on Data Mining.

Lin, C.-J. (2001).

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Lin, C.-J. (2002).

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IEEE Transactions on Neural Networks, 13(5):1045–1052.

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References IV

Osuna, E., Freund, R., and Girosi, F. (1997).

Training support vector machines: An application to face detection.

In Proceedings of CVPR’97, pages 130–136, New York, NY. IEEE.

Platt, J. C. (1998).

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In Sch¨olkopf, B., Burges, C. J. C., and Smola, A. J., editors, Advances in Kernel Methods - Support Vector Learning, Cambridge, MA. MIT Press.

Sch¨olkopf, B. and Smola, A. J. (2002).

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MIT Press.

Syed, N. A., Liu, H., and Sung, K. K. (1999).

Incremental learning with support vector machines.

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Vapnik, V. (1998).

Statistical Learning Theory.

Wiley, New York, NY.

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References V

Weston, J. and Watkins, C. (1999).

Multi-class support vector machines.

In Verleysen, M., editor, Proceedings of ESANN99, Brussels. D. Facto Press.

Yu, H., Yang, J., and Han, J. (2003).

Classifying large data sets using svms with hierarchical clusters.

In KDD ’03: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 306–315, New York, NY, USA. ACM Press.

Support Vector Machines