Large-scale Linear Classiﬁcation

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Large-scale Linear Classification

Chih-Jen Lin

Department of Computer Science National Taiwan University

International Winter School on Big Data, 2016

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Data Classification

Given training data in different classes (labels known)

Predict test data (labels unknown) Classic example: medical diagnosis

Find a patient’s blood pressure, weight, etc.

After several years, know if he/she recovers Build a machine learning model

New patient: find blood pressure, weight, etc Prediction

Training and testing

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Data Classification (Cont’d)

Among many classification methods, linear and kernel are two popular ones

They are very related

We will detailedly discuss linear classification and its connection to kernel

Talk slides:

http://www.csie.ntu.edu.tw/~cjlin/talks/

course-bilbao.pdf

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Outline

1 Linear classification

2 Kernel classification

3 Linear versus kernel classification

4 Solving optimization problems

5 Multi-core linear classification

6 Distributed linear classification

7 Discussion and conclusions

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Linear classification

Outline

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Linear classification

Outline

1 Linear classification Maximum margin

Regularization and losses Other derivations

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Linear classification Maximum margin

Outline

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Linear Classification

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

Consider a simple case with two classes:

Define an indicator vector y ∈ R^l y_i =

1 if xⁱ in class 1

−1 if xⁱ in class 2 A hyperplane to linearly separate all data

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4 4 4 4

4 4

4

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◦◦

4 4 4 4

4 4

4

w^Tx + b = h₊₁

−10

i

A separating hyperplane: w^Tx + b = 0 (w^Txⁱ) + b ≥ 1 if y_i = 1 (w^Txⁱ) + b ≤ −1 if y_i = −1

Decision function f (x) = sgn(w^Tx + b), x: test data

Many possible choices of w and b

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Maximal Margin

Maximizing the distance between w^Tx + b = 1 and

−1:

2/kwk = 2/√ w^Tw A quadratic programming problem

minw,b

1 2w^Tw

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

This is the basic formulation of support vector machines (Boser et al., 1992)

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Data May Not Be Linearly Separable

An example:

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4 4

4

4 4 4

We can never find a linear hyperplane to separate data

Remedy: allow training errors

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Data May Not Be Linearly Separable (Cont’d)

Standard SVM (Boser et al., 1992; Cortes and Vapnik, 1995)

w,b,minξ

1

2w^Tw +C

l

X

i =1

ξi

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

We explain later why this method is called support vector machine

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The Bias Term b

Recall the decision function is sgn(w^Tx + b) Sometimes the bias term b is omitted

sgn(w^Tx)

That is, the hyperplane always passes through the origin

This is fine if the number of features is not too small In our discussion, b is used for kernel, but omitted for linear (due to some historical reasons)

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Linear classification Regularization and losses

Outline

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Equivalent Optimization Problem

• Recall SVM optimization problem (without b) is minw,ξ

1

2w^Tw + C

l

X

i =1

ξ_i subject to y_iw^Txⁱ ≥ 1 − ξ_i,

ξ_i ≥ 0, i = 1, . . . , l .

• It is equivalent to minw

1

2w^Tw + C

l

X

i =1

max(0, 1 − y_iw^Txⁱ) (1)

• This reformulation is useful for subsequent discussion

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Equivalent Optimization Problem (Cont’d)

That is, at optimum,

ξ_i = max(0, 1 − y_iw^Txⁱ) Reason: from constraint

ξ_i ≥ 1 − y_iw^Txi and ξ_i ≥ 0 but we also want to minimize ξ_i

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Equivalent Optimization Problem (Cont’d)

We now derive the same optimization problem (1) from a different viewpoint

We now aim to minimize the training error minw (training errors)

To characterize the training error, we need a loss function ξ(w; x, y) for each instance (x, y) Ideally we should use 0–1 training loss:

ξ(w; x, y) =

(1 if yw^Tx < 0, 0 otherwise

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Equivalent Optimization Problem (Cont’d)

However, this function is discontinuous. The optimization problem becomes difficult

−yw^Tx ξ(w; x, y)

We need continuous approximations

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Common Loss Functions

Hinge loss (l1 loss)

ξ_L1(w; x, y) ≡ max(0, 1 − yw^Tx) (2) Squared hinge loss (l2 loss)

ξ_L2(w; x, y) ≡ max(0, 1 − yw^Tx)² (3) Logistic loss

ξ_LR(w; x, y) ≡ log(1 + e^−y^w^T^x) (4) SVM: (2)-(3). Logistic regression (LR): (4)

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Common Loss Functions (Cont’d)

−yw^Tx ξ(w; x, y)

ξ_L1 ξ_L2

ξ_LR

Logistic regression is very related to SVM Their performance is usually similar

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Common Loss Functions (Cont’d)

However, minimizing training losses may not give a good model for future prediction

Overfitting occurs

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Overfitting

See the illustration in the next slide For classification,

You can easily achieve 100% training accuracy This is useless

When training a data set, we should Avoid underfitting: small training error Avoid overfitting: small testing error

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l and s: training; and 4: testing

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Regularization

To minimize the training error we manipulate the w vector so that it fits the data

To avoid overfitting we need a way to make w’s values less extreme.

One idea is to make the objective function smoother

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General Form of Linear Classification

Training data {yi,xⁱ},xⁱ ∈ Rⁿ, i = 1, . . . , l , yi = ±1 l : # of data, n: # of features

minw f (w), f (w) ≡ w^Tw 2 + C

l

X

i =1

ξ(w; xⁱ, y_i) (5) w^Tw/2: regularization term

ξ(w; x, y): loss function C : regularization parameter

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General Form of Linear Classification (Cont’d)

If hinge loss

ξ_L1(w; x, y) ≡ max(0, 1 − yw^Tx) is used, then (5) goes back to the SVM problem described earlier (b omitted):

minw,ξ

1

2w^Tw + C

l

X

i =1

ξ_i subject to y_iw^Txⁱ ≥ 1 − ξ_i,

ξ_i ≥ 0, i = 1, . . . , l .

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Solving Optimization Problems

We have an unconstrained problem, so many

existing unconstrained optimization techniques can be used

However,

ξ_L1: not differentiable

ξ_L2: differentiable but not twice differentiable ξLR: twice differentiable

We may need different types of optimization methods

Details of solving optimization problems will be discussed later

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Linear classification Other derivations

Outline

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Logistic Regression

Logistic regression can be traced back to the 19th century

It’s mainly from statistics community, so many people wrongly think that this method is very different from SVM

Indeed from what we have shown they are very related.

Let’s see how to derive it from a statistical viewpoint

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Logistic Regression (Cont’d)

For a label-feature pair (y ,x), assume the probability model

p(y |x) = 1

1 + e^−y^w^T^x. Note that

p(1|x) + p(−1|x)

= 1

1 + e⁻^w^T^x + 1 1 + e^w^T^x

= e^w^T^x

1 + e^w^T^x + 1 1 + e^w^T^x

= 1

w is the parameter to be decided

Chih-Jen Lin (National Taiwan Univ.) 30 / 157

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Logistic Regression (Cont’d)

Idea of this model p(1|x) = 1

1 + e⁻^w^T^x

(→ 1 if w^Tx 0,

→ 0 if w^Tx 0 Assume training instances are

(yi,xⁱ), i = 1, . . . , l

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Logistic Regression (Cont’d)

Logistic regression finds w by maximizing the following likelihood

maxw l

Y

i =1

p (y_i|xⁱ) . (6) Negative log-likelihood

− log

l

Y

i =1

p (yi|xⁱ) = −

l

X

i =1

log p (yi|xⁱ)

=

l

X

i =1

log

1 + e^−yⁱ^w^T^xⁱ

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Logistic Regression (Cont’d)

Logistic regression minw

l

X

i =1

log

. Regularized logistic regression

minw

1

2w^Tw + C

l

X

i =1

log

. (7) C : regularization parameter decided by users

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Discussion

We see that the same method can be derived from different ways

SVM

Maximal margin

Regularization and training losses LR

Regularization and training losses Maximum likelihood

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Kernel classification

Outline

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Kernel classification

Outline

2 Kernel classification Nonlinear mapping Kernel tricks

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Kernel classification Nonlinear mapping

Outline

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Data May Not Be Linearly Separable

This is an earlier example:

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◦

4 4

4

4 4 4

In addition to allowing training errors, what else can we do?

For this data set, shouldn’t we use a nonlinear classifier?

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Mapping Data to a Higher Dimensional Space

But modeling nonlinear curves is difficult. Instead, we map data to a higher dimensional space

φ(x) = [φ¹(x), φ²(x), . . .]^T. For example,

weight height²

is a useful new feature to check if a person overweights or not

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Kernel Support Vector Machines

Linear SVM:

w,b,ξmin 1

2w^Tw + C Xl i =1ξ_i subject to y_i(w^Txⁱ + b) ≥ 1 − ξ_i,

ξ_i ≥ 0, i = 1, . . . , l . Kernel SVM:

w,b,ξmin 1

2w^Tw + C X^l

i =1ξ_i

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

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Kernel Logistic Regression

minw,b

1

2w^Tw + C

l

X

i =1

log

1 + e^−yⁱ⁽^w^T^φ(^xⁱ^)+b)

.

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Difficulties After Mapping Data to a High-dimensional Space

# variables in w = dimensions of φ(x)

Infinite variables if φ(x) is infinite dimensional Cannot do an infinite-dimensional inner product for predicting a test instance

sgn(w^Tφ(x))

Use kernel trick to go back to a finite number of variables

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Kernel classification Kernel tricks

Outline

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Kernel Tricks

• It can be shown at optimum, w is a linear combination of training data

w = X^l

i =1y_iα_iφ(xⁱ)

Proofs not provided here. Later we will show that α is the solution of a dual problem

• Special φ(x) such that the decision function becomes sgn(w^Tφ(x)) = sgn

X^l

i =1y_iα_iφ(xi)^Tφ(x)

= sgn

X^l

i =1yiαiK (xⁱ,x)

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Kernel Tricks (Cont’d)

φ(xⁱ)^Tφ(x^j) needs a closed form Example: xi ∈ R³, φ(xi) ∈ R¹⁰ φ(xⁱ) = [1,√

2(x_i)₁,√

2(x_i)₂,√

2(x_i)₃, (x_i)²₁, (x_i)²₂, (x_i)²₃,√

2(x_i)₁(x_i)₂,√

2(x_i)₁(x_i)₃,√

2(x_i)₂(x_i)₃]^T Then φ(xⁱ)^Tφ(x^j) = (1 +x^Ti x^j)².

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

e^−γk^xⁱ⁻^x^j^k², (Radial Basis Function) (x^Ti xj/a + b)^d (Polynomial kernel)

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K (x, y) 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 + 2γx_ix_j

1! + (2γx_ix_j)²

2! + (2γx_ix_j)³

3! + · · ·

=e^−γxⁱ²^−γx^j² 1 · 1+

r2γ 1!x_i ·

r2γ 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(x_i)^Tφ(x_j), where

φ(x ) = e^−γx²

1,

r2γ 1!x ,

r(2γ)² 2! x²,

r(2γ)³

3! x³, · · ·

T

.

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Linear versus kernel classification

Outline

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Linear versus kernel classification

Outline

3 Linear versus kernel classification Comparison on the cost

Numerical comparisons A real example

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Linear versus kernel classification Comparison on the cost

Outline

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Linear and Kernel Classification

Now we see that methods such as SVM and logistic regression can be used in two ways

Kernel methods: data mapped to a higher dimensional space

x ⇒ φ(x)

φ(xⁱ)^Tφ(x^j) easily calculated; little control on φ(·) Linear classification + feature engineering:

We have x without mapping. Alternatively, we can say that φ(x) is our x; full control on x or φ(x)

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Linear and Kernel Classification

The cost of using linear and kernel classification is different

Let’s check the prediction cost w^Tx versus Xl

i =1y_iα_iK (xi,x) If K (xⁱ,x^j) takes O(n), then

O(n) versus O(nl ) Linear is much cheaper

A similar difference occurs for training

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Linear and Kernel Classification (Cont’d)

In fact, linear is a special case of kernel

We can prove that accuracy of linear is the same as Gaussian (RBF) kernel under certain parameters (Keerthi and Lin, 2003)

Therefore, roughly we have

accuracy: kernel ≥ linear cost: kernel linear Speed is the reason to use linear

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Linear and Kernel Classification (Cont’d)

For some problems, accuracy by linear is as good as nonlinear

But training and testing are much faster

This particularly happens for document classification Number of features (bag-of-words model) very large Data very sparse (i.e., few non-zeros)

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Linear versus kernel classification Numerical comparisons

Outline

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Comparison Between Linear and Kernel (Training Time & Testing Accuracy)

Linear RBF Kernel

Data set Time Accuracy Time Accuracy

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

yahoo-japan 3.1 92.63 20,955.2 93.31 webspam 25.7 93.35 15,681.8 99.26 Size reasonably large: e.g., yahoo-japan: 140k instances and 830k features

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Comparison Between Linear and Kernel (Training Time & Testing Accuracy)

Linear RBF Kernel

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

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Comparison Between Linear and Kernel (Training Time & Testing Accuracy)

Linear RBF Kernel

MNIST38 0.1 96.82 38.1 99.70

ijcnn1 1.6 91.81 26.8 98.69

covtype 1.4 76.37 46,695.8 96.11

news20 1.1 96.95 383.2 96.90

real-sim 0.3 97.44 938.3 97.82

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Linear versus kernel classification A real example

Outline

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Linear Methods to Explicitly Train φ( x

ⁱ

)

We may directly train φ(xi), ∀i without using kernel This is possible only if φ(xⁱ) is not too high

dimensional

Next we show a real example of running a machine learning model is a small sensor hub

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Example: Classifier in a Small Device

In a sensor application (Yang, 2013), the classifier can use less than 16KB of RAM

Classifiers Test accuracy Model Size

Decision Tree 77.77 76.02KB

AdaBoost (10 trees) 78.84 1,500.54KB SVM (RBF kernel) 85.33 1,287.15KB Number of features: 5

We consider a degree-3 polynomial mapping dimensionality = 5 + 3

3

+ bias term = 57.

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Example: Classifier in a Small Device

One-against-one strategy for 5-class classification

5 2

× 57 × 4bytes = 2.28KB Assume single precision

Results

SVM method Test accuracy Model Size

RBF kernel 85.33 1,287.15KB

Polynomial kernel 84.79 2.28KB

Linear kernel 78.51 0.24KB

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Solving optimization problems

Outline

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Solving optimization problems

Outline

4 Solving optimization problems Kernel: decomposition methods Linear: coordinate descent method Linear: second-order methods Experiments

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Solving optimization problems Kernel: decomposition methods

Outline

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Dual Problem

Recall we said that the difficulty after mapping x to φ(x) is the huge number of variables

We mentioned

w =

l

X

i =1

α_iy_iφ(xⁱ) (8) and used kernels for prediction

Besides prediction, we must do training via kernels The most common way to train SVM via kernels is through its dual problem

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Dual Problem (Cont’d)

The dual problem minα

1

2α^TQα −e^Tα

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

where Qij = yiyjφ(xⁱ)^Tφ(x^j) and e = [1, . . . , 1]^T From primal-dual relationship, at optimum (8) holds Dual problem has a finite number of variables

If no bias term b, then y^Tα = 0 disappears

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Example: Primal-dual Relationship

Consider the earlier example:

4

0

1

Now two data are x¹ = 1,x² = 0 with y = [+1, −1]^T The solution is (w , b) = (2, −1)

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Example: Primal-dual Relationship (Cont’d)

The dual objective function 1

2α₁ α₂1 0 0 0

α₁ α₂

−1 1α₁ α₂

= 1

2α²₁ − (α₁ + α₂)

In optimization, objective function means the function to be optimized

Constraints are

α₁ − α₂ = 0, 0 ≤ α₁, 0 ≤ α₂.

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Example: Primal-dual Relationship (Cont’d)

Substituting α₂ = α₁ into the objective function, 1

2α²₁ − 2α₁ has the smallest value at α₁ = 2.

Because [2, 2]^T satisfies constraints 0 ≤ α₁ and 0 ≤ α₂, it is optimal

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Example: Primal-dual Relationship (Cont’d)

Using the primal-dual relation w = y1α1x1 + y2α2x2

= 1 · 2 · 1 + (−1) · 2 · 0

= 2

This is the same as that by solving the primal problem.

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Decision function

At optimum

w = P^li =1αiyiφ(xⁱ) Decision function

w^Tφ(x) + b

= X^l

i =1α_iy_iφ(xⁱ)^Tφ(x) + b

= X^l

i =1α_iy_iK (xⁱ,x) + b Recall 0 ≤ α_i ≤ C in the dual problem

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Support Vectors

Only xⁱ of αi > 0 used ⇒ support vectors

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

-1.5 -1 -0.5 0 0.5 1

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Large Dense Quadratic Programming

minα

1

2α^TQα −e^Tα

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

Qij 6= 0, Q : an l by l fully dense matrix 50,000 training points: 50,000 variables:

(50, 000² × 8/2) bytes = 10GB RAM to store Q

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Large Dense Quadratic Programming (Cont’d)

Traditional optimization methods cannot be directly applied here because Q cannot even be stored

Currently, decomposition methods (a type of coordinate descent methods) are what used in practice

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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 Let the objective function be

f (α) = 1

2α^TQα −e^Tα

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Decomposition Methods (Cont’d)

Sub-problem on the variable dB

mindB

f ([α^α^BN] +_d

B

0 )

subject to −α_i ≤ d_i ≤ C − α_i, ∀i ∈ B d_i = 0, ∀i /∈ B,

y_B^Td^B = 0

The objective function of the sub-problem f ([^αα^B_N] +_d

B

0 )

=1

2d^TBQ_BBd^B + ∇_Bf (α)^Td^B + constant.

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Avoid Memory Problems

QBB is a sub-matrix of Q

Q_BB Q_BN Q_NB Q_NN

Note that

∇f (α) = Qα −e, ∇_Bf (α) = QB,:α −e^B

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Avoid Memory Problems (Cont’d)

Only B columns of Q are needed

In general |B| ≤ 10 is used. We need |B| ≥ 2 because of the linear constraint

y^T_Bd^B = 0

Calculated when used: trade time for space But is such an approach practical?

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How Decomposition Methods Perform?

Convergence not very fast. This is known because of using only first-order information

But, no need to have very accurate α decision function: X^l

i =1y_iα_iK (xⁱ,x) + b Prediction may still be correct with a rough α Further, in some situations,

# support vectors # training points Initial α¹ = 0, some instances never used

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How Decomposition Methods Perform?

(Cont’d)

An example of training 50,000 instances using the software LIBSVM (|B| = 2)

$svm-train -c 16 -g 4 -m 400 22features Total nSV = 3370

Time 79.524s

This was done on a typical desktop

Calculating the whole Q takes more time

#SVs = 3,370 50,000

A good case where some remain at zero all the time

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Solving optimization problems Linear: coordinate descent method

Outline

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Coordinate Descent Methods for Linear Classification

We consider L1-loss SVM as an example here The same method can be extended to L2 and logistic loss

More details in Hsieh et al. (2008); Yu et al. (2011)

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SVM Dual (Linear without Kernel)

From primal dual relationship minα f (α)

subject to 0 ≤ α_i ≤ C , ∀i , where

f (α) ≡ 1

2α^TQα −e^Tα and

Q_ij = y_iy_jx^Ti x^j, e = [1, . . . , 1]^T

No linear constraint y^Tα = 0 because of no bias term b

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Dual Coordinate Descent

Very simple: minimizing one variable at a time While α not optimal

For i = 1, . . . , l minαi

f (. . . , α_i, . . .) A classic optimization technique

Traced back to Hildreth (1957) if constraints are not considered

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The Procedure

Given current α. Let ei = [0, . . . , 0, 1, 0, . . . , 0]^T. min

d f (α + deⁱ) = 1

2Q_iid² + ∇_if (α)d + constant This sub-problem is a special case of the earlier sub-problem of the decomposition method for kernel classifiers

That is, the working set B = {i } Without constraints

optimal d = −∇_if (α) Q_ii

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The Procedure (Cont’d)

Now 0 ≤ α_i + d ≤ C α_i ← min

max

α_i − ∇_if (α) Qii

, 0

, C

Note that

∇_if (α) = (Qα)_i − 1 = X^l

j =1Q_ijα_j − 1

= X^l

j =1y_iy_jx^Ti xjα_j − 1

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The Procedure (Cont’d)

Directly calculating gradients costs O(ln) l :# data, n: # features

This is the case for kernel classifiers For linear SVM, define

u ≡ X^l

j =1y_jα_jx^j, Easy gradient calculation: costs O(n)

∇_if (α) = yiu^Txⁱ − 1

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The Procedure (Cont’d)

All we need is to maintain u u = X^l

j =1y_jα_jx^j, If

¯

α_i : old ; α_i : new then

u ← u + (αi − ¯αi)yixⁱ. Also costs O(n)

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Algorithm: Dual Coordinate Descent

Given initial α and find u = X

i

y_iα_ixi.

While α is not optimal (Outer iteration) For i = 1, . . . , l (Inner iteration)

(a) ¯α_i ← α_i

(b) G = yiu^Txⁱ − 1 (c) If α_i can be changed

α_i ← min(max(α_i − G /Q_ii, 0), C ) u ← u + (α_i − ¯α_i)y_ixⁱ

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Difference from the Kernel Case

• We have seen that coordinate-descent type of

methods are used for both linear and kernel classifiers

• Recall the i -th element of gradient costs O(n) by

∇_if (α) =

l

X

j =1

y_iy_jx^Ti x^jα_j − 1 = (y_ixⁱ)^T

l

X

j =1

y_jx^jα_j

− 1

= (y_ixⁱ)^Tu− 1 but we cannot do this for kernel because

K (xⁱ,x^j) = φ(xⁱ)^Tφ(x^j) cannot be separated

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Difference from the Kernel Case (Cont’d)

If using kernel, the cost of calculating ∇_if (α) must be O(ln)

However, if O(ln) cost is spent, the whole ∇f (α) can be maintained (details not shown here)

In contrast, the setting of using u knows ∇if (α) rather than the whole ∇f (α)

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Difference from the Kernel Case (Cont’d)

In existing coordinate descent methods for kernel classifiers, people also use ∇f (α) information to select variables (i.e., select the set B) for update In optimization there are two types of coordinate descent methods:

sequential or random selection of variables greedy selection of variables

To do greedy selection, usually the whole gradient must be available

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Difference from the Kernel Case (Cont’d)

Existing coordinate descent methods for linear ⇒ related to sequential or random selection

Existing coordinate descent methods for kernel ⇒ related to greedy selection

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Bias Term b and Linear Constraint in Dual

In our discussion, b is used for kernel but not linear Mainly history reason

For kernel SVM, we can also omit b to get rid of the linear constraint y^Tα = 0

Then for kernel decomposition method, |B| = 1 can also be possible

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Solving optimization problems Linear: second-order methods

Outline

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Optimization for Linear and Kernel Cases

Recall that

w =

l

X

i =1

y_iα_iφ(xi)

Kernel: can only solve an optimization problem of α Linear: can solve either w or α

We will show an example to minimize over w

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Newton Method

Let’s minimize a twice-differentiable function minw f (w)

For example, logistic regression has minw

1

2w^Tw + C

l

X

i =1

log

. Newton direction at iterate w^k

mins ∇f (w^k)^Ts + 1

2s^T∇²f (w^k)s

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Truncated Newton Method

The above sub-problem is equivalent to solving Newton linear system

∇²f (w^k)s = −∇f (w^k) Approximately solving the linear system ⇒ truncated Newton

However, Hessian matrix ∇²f (w^k) is too large to be stored

∇²f (w^k) : n × n, n : number of features For document data, n can be millions or more

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Using Special Properties of Data Classification

But Hessian has a special form

∇²f (w) = I + CX^TDX , D diagonal. For logistic regression,

D_ii = e^−yⁱ^w^T^xⁱ 1 + e^−yⁱ^w^T^xⁱ X : data, # instances × # features

X = [x¹, . . . ,x^l]^T

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Using Special Properties of Data Classification (Cont’d)

Using Conjugate Gradient (CG) to solve the linear system.

CG is an iterative procedure. Each CG step mainly needs one Hessian-vector product

∇²f (w)d = d + C · X^T(D(Xd)) Therefore, we have a Hessian-free approach

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Using Special Properties of Data Classification (Cont’d)

Now the procedure has two layers of iterations Outer: Newton iterations

Inner: CG iterations per Newton iteration Past machine learning works used Hessian-free approaches include, for example, (Keerthi and DeCoste, 2005; Lin et al., 2008)

Second-order information used: faster convergence than first-order methods

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Solving optimization problems Experiments

Outline

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Comparisons

L2-loss SVM is used

DCDL2: Dual coordinate descent (Hsieh et al., 2008)

DCDL2-S: DCDL2 with shrinking (Hsieh et al., 2008)

PCD: Primal coordinate descent (Chang et al., 2008)

TRON: Trust region Newton method (Lin et al., 2008)

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Objective values (Time in Seconds)

news20 rcv1

yahoo-japan yahoo-korea

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Analysis

Dual coordinate descents are very effective if # data and # features are both large

Useful for document classification Half million data in a few seconds However, it is less effective if

# features small: should solve primal; or large penalty parameter C ; problems are more ill-conditioned

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An Example When # Features Small

# instance: 32,561, # features: 123

Objective value Accuracy

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Multi-core linear classification

Outline

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Outline

Parallel matrix-vector multiplications Experiments

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Multi-core Linear Classification

Parallelization in shared-memory system: use the power of multi-core CPU if data can fit in memory Example: we can parallelize the 2nd-order method (i.e., the Newton method) discussed earlier.

We discuss the study in Lee et al. (2015)

Recall the bottleneck is the Hessian-vector product

∇²f (w)d = d + C · X^T(D(Xd)) See the analysis in the next slide

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Matrix-vector Multiplications: More Than 90% of the Training Time

Data set #instances #features ratio

kddb 19,264,097 29,890,095 82.11%

url combined 2,396,130 3,231,961 94.83%

webspam 350,000 16,609,143 97.95%

rcv1 binary 677,399 47,236 97.88%

covtype binary 581,012 54 89.20%

epsilon normalized 400,000 2,000 99.88%

rcv1 518,571 47,236 97.04%

covtype 581,012 54 89.06%

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Matrix-vector Multiplications: More Than 90% of the Training Time (Cont’d)

This result is by Newton methods using one core We should parallelize matrix-vector multiplications For ∇²f (w)d we must calculate

u = Xd (9)

u ← Du (10)

¯u = X^Tu, where u = DXd (11) Because D is diagonal, (10) is easy

We will discuss the parallelization of (9) and (11)

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Multi-core linear classification Parallel matrix-vector multiplications

Outline

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Parallel X d Operation

Assume that X is in a row-oriented sparse format

X =





 x^T1

...

x^Tl





 and u = Xd =





 x^T1 d

...

x^Tl d





 we have the following simple loop

1: for i = 1, . . . , l do

2: u_i = x^Ti d

3: end for

Because the l inner products are independent, we can easily parallelize the loop by, for example, OpenMP

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Parallel X

^T

u Operation

For the other matrix-vector multiplication

¯

u = X^Tu, where u = DXd, we have

¯

u = u₁x¹ + · · · + u_lx^l.

Because matrix X is row-oriented, accessing columns in X^T is much easier than rows We can use the following loop

1: for i = 1, . . . , l do

2: ¯u ← ¯u + u_ixi 3: end for

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Parallel X

^T

u Operation (Cont’d)

There is no need to store a separate X^T

However, it is possible that threads on u_i₁xi1 and u_i₂xⁱ2 want to update the same component ¯u_s at the same time:

1: for i = 1, . . . , l do in parallel

2: for (xⁱ)s 6= 0 do

3: u¯_s ← ¯u_s + u_i(xi)_s

4: end for

5: end for

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Atomic Operations for Parallel X

^T

u

An atomic operation can avoid other threads to write ¯u_s at the same time.

1: for i = 1, . . . , l do in parallel

2: for (xi)_s 6= 0 do

3: atomic: ¯u_s ← ¯u_s + u_i(xⁱ)_s

4: end for

5: end for

However, waiting time can be a serious problem

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Reduce Operations for Parallel X

^T

u

Another method is using temporary arrays

maintained by each thread, and summing up them in the end

That is, store uˆ^p = X

i

{u_ixi | i run by thread p}

and then

¯

u = X

p

ˆ u^p

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Atomic Operation: Almost No Speedup

Reduce operations are superior to atomic operations

1 2 4# threads6 8 10 12 0

2 4 6 8 10

Speedup

OMP-array OMP-atomic

1 2 4# threads6 8 10 12 0

1 2 3 4 5

Speedup

OMP-array OMP-atomic

rcv1 binary covtype binary Subsequently we use the reduce operations

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Existing Algorithms for Sparse Matrix-vector Product

This is always an important research issue in numerical analysis

Instead of our direct implementation to parallelize loops, in the next slides we will consider two existing methods

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Recursive Sparse Blocks (Martone, 2014)

RSB (Recursive Sparse Blocks) is an effective format for fast parallel sparse matrix-vector multiplications It recursively partitions a matrix to be like the figure

Locality of memory references improved, but the construction time is not negligible

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Recursive Sparse Blocks (Cont’d)

Parallel, efficient sparse matrix-vector operations Improve locality of memory references

But the initial construction time is about 20

multiplications, which is not negligible in some cases We will show the result in the experiment part

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Intel MKL

Intel Math Kernel Library (MKL) is a commercial library including optimized routines for linear algebra (Intel)

It supports fast matrix-vector multiplications for different sparse formats.

We consider the row-oriented format to store X .

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Multi-core linear classification Experiments

Outline

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Experiments

Baseline: Single core version in LIBLINEAR 1.96 OpenMP to parallelize loops

MKL: Intel MKL version 11.2 RSB: librsb version 1.2.0

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Speedup of X d: All are Excellent

rcv1 binary webspam kddb

url combined covtype binary rcv1