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# Large-scale Linear Classiﬁcation: Status and Challenges

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### Chih-Jen Lin

Department of Computer Science National Taiwan University

San Francisco Machine Learning Meetup, October 30, 2014

Chih-Jen Lin (National Taiwan Univ.) 1 / 43

(2)

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 2 / 43

(3)

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 3 / 43

(4)

Introduction

### Linear Classification

The model is a weight vector w (for binary classification)

The decision function is

sgn(wTx )

Although many new and advanced techniques are available (e.g., deep learning), linear classifiers remain to be useful because of their simplicity We will give an overview of this topic in this talk

Chih-Jen Lin (National Taiwan Univ.) 4 / 43

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### Linear and Kernel Classification

Linear Nonlinear

Linear: data in the original input space; nonlinear: data mapped to other spaces

Original: [height, weight]

Nonlinear: [height, weight, weight/height2] Kernel is one of the nonlinear methods

Chih-Jen Lin (National Taiwan Univ.) 5 / 43

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Introduction

### Linear and Nonlinear Classification

Methods such as SVM and logistic regression can be used in two ways

• Kernel methods: data mapped to another space x ⇒ φ(x )

φ(x )Tφ(y) easily calculated; no good control on φ(·)

• Linear classification + feature engineering:

Directly use x without mapping. But x may have been carefully generated. Full control on x

We will focus on the 2nd type of approaches in this talk

Chih-Jen Lin (National Taiwan Univ.) 6 / 43

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### Why Linear Classification?

• If φ(x ) is high dimensional, decision function sgn(wTφ(x ))

is expensive

• Kernel methods:

w ≡

l

X

i =1

αiφ(xi) for some α, K (xi, xj) ≡ φ(xi)Tφ(xj)

New decision function: sgn Pl

i =1αiK (xi, x )

• Special φ(x ) so calculating K (xi, xj) is easy. Example:

K (xi, xj) ≡ (xTi xj+ 1)2 = φ(xi)Tφ(xj), φ(x ) ∈ RO(n2)

Chih-Jen Lin (National Taiwan Univ.) 7 / 43

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Introduction

### Why Linear Classification? (Cont’d)

Prediction

wTx versus Xl

i =1αiK (xi, x ) If K (xi, xj) takes O(n), then

O(n) versus O(nl ) Kernel: cost related to size of training data Linear: cheaper and simpler

Chih-Jen Lin (National Taiwan Univ.) 8 / 43

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### Linear is Useful in Some Places

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

But training and testing are much faster Especially document classification

Number of features (bag-of-words model) very large Large and sparse data

Training millions of data in just a few seconds

Chih-Jen Lin (National Taiwan Univ.) 9 / 43

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Introduction

### Comparison Between Linear and Nonlinear (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

Chih-Jen Lin (National Taiwan Univ.) 10 / 43

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### Comparison Between Linear and Nonlinear (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

Chih-Jen Lin (National Taiwan Univ.) 10 / 43

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Introduction

### Comparison Between Linear and Nonlinear (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

Chih-Jen Lin (National Taiwan Univ.) 10 / 43

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### Binary Linear Classification

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

minw f (w ), f (w ) ≡ wTw 2 + C

l

X

i =1

ξ(w ; xi, yi)

wTw /2: regularization term (we have no time to talk about L1 regularization here)

ξ(w ; x , y ): loss function: we hope y wTx > 0 C : regularization parameter

Chih-Jen Lin (National Taiwan Univ.) 11 / 43

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Introduction

### Loss Functions

Some commonly used ones:

ξL1(w ; x , y ) ≡ max(0, 1 − y wTx ), (1) ξL2(w ; x , y ) ≡ max(0, 1 − y wTx )2, (2) ξLR(w ; x , y ) ≡ log(1 + e−y wTx). (3) SVM (Boser et al., 1992; Cortes and Vapnik, 1995):

(1)-(2)

Logistic regression (LR): (3)

Chih-Jen Lin (National Taiwan Univ.) 12 / 43

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### Loss Functions (Cont’d)

−y wTx ξ(w ; x , y )

ξL1

ξL2

ξLR

Their performance is usually similar

Optimization methods may be different because of differentiability

Chih-Jen Lin (National Taiwan Univ.) 13 / 43

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Optimization methods

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 14 / 43

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### Optimization Methods

Many unconstrained optimization methods can be applied

For kernel, optimization is over a variable α where

w =

l

X

i =1

αiφ(xi)

We cannot minimize over w because it may be infinite dimensional

However, for linear, minimizing over w or α is ok

Chih-Jen Lin (National Taiwan Univ.) 15 / 43

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Optimization methods

### Optimization Methods (Cont’d)

Among unconstrained optimization methods,

Low-order methods: quickly get a model, but slow final convergence

High-order methods: more robust and useful for ill-conditioned situations

We will quickly discuss some examples and show both types of optimization methods are useful for linear classification

Chih-Jen Lin (National Taiwan Univ.) 16 / 43

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### Optimization: 2nd Order Methods

Newton direction (if twice differentiable) mins ∇f (wk)Ts + 1

2sT2f (wk)s This is the same as solving Newton linear system

2f (wk)s = −∇f (wk)

Hessian matrix ∇2f (wk) too large to be stored

2f (wk) : n × n, n : number of features But Hessian has a special form

2f (w ) = I + CXTDX ,

Chih-Jen Lin (National Taiwan Univ.) 17 / 43

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Optimization methods

### Optimization: 2nd Order Methods (Cont’d)

X : data matrix. D diagonal.

Using Conjugate Gradient (CG) to solve the linear system. Only Hessian-vector products are needed

2f (w )s = s + C · XT(D(X s)) Therefore, we have a Hessian-free approach

Chih-Jen Lin (National Taiwan Univ.) 18 / 43

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### Optimization: 1st Order Methods

We consider L1-loss and the dual SVM problem minα f (α)

subject to 0 ≤ αi ≤ C , ∀i , where

f (α) ≡ 1

TQα − eTα and

Qij = yiyjxTi xj, e = [1, . . . , 1]T We will apply coordinate descent (CD) methods The situation for L2 or LR loss is very similar

Chih-Jen Lin (National Taiwan Univ.) 19 / 43

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Optimization methods

### 1st Order Methods (Cont’d)

Coordinate descent: a simple and classic technique Change one variable at a time

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

d f (α + d ei) = 1

2Qiid2 + ∇if (α)d + constant Without constraints

optimal d = −∇if (α) Qii

Now 0 ≤ αi + d ≤ C αi ← min

 max



αi − ∇if (α) Qii , 0

 , C



Chih-Jen Lin (National Taiwan Univ.) 20 / 43

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### Comparisons

L2-loss SVM is used

DCDL2: Dual coordinate descent DCDL2-S: DCDL2 with shrinking PCD: Primal coordinate descent TRON: Trust region Newton method This result is from Hsieh et al. (2008)

Chih-Jen Lin (National Taiwan Univ.) 21 / 43

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Optimization methods

### Objective values (Time in Seconds)

news20 rcv1

yahoo-japan yahoo-korea

Chih-Jen Lin (National Taiwan Univ.) 22 / 43

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### Low- versus High-order Methods

• We saw that low-order methods are efficient to give a model. However, high-order methods may be useful for difficult situations

• An example: # instance: 32,561, # features: 123

Objective value Accuracy

# features is small ⇒ solving primal is more suitable

Chih-Jen Lin (National Taiwan Univ.) 23 / 43

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Sample applications

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

Dependency parsing using feature combination Transportation-mode detection in a sensor hub

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 24 / 43

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### Outline

1 Introduction

2 Optimization methods

3 Sample applications

Dependency parsing using feature combination Transportation-mode detection in a sensor hub

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 25 / 43

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Sample applications Dependency parsing using feature combination

### Dependency Parsing: an NLP Application

Kernel Linear

RBF Poly-2 Linear Poly-2 Training time 3h34m53s 3h21m51s 3m36s 3m43s

Parsing speed 0.7x 1x 1652x 103x

UAS 89.92 91.67 89.11 91.71

LAS 88.55 90.60 88.07 90.71

We get faster training/testing, while maintain good accuracy

But how to achieve this?

Chih-Jen Lin (National Taiwan Univ.) 26 / 43

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i

### )

Example: low-degree polynomial mapping:

φ(x ) = [1, x1, . . . , xn, x12, . . . , xn2, x1x2, . . . , xn−1xn]T For this mapping, # features = O(n2)

Recall O(n) for linear versus O(nl ) for kernel Now O(n2) versus O(nl )

Sparse data

n ⇒ ¯n, average # non-zeros for sparse data

¯

n  n ⇒ O( ¯n2) may be much smaller than O(l ¯n)

Chih-Jen Lin (National Taiwan Univ.) 27 / 43

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Sample applications Dependency parsing using feature combination

### Handing High Dimensionality of φ(x )

A multi-class problem with sparse data

n Dim. of φ(x ) l n w ’s # nonzeros¯ 46,155 1,065,165,090 204,582 13.3 1,438,456

¯

n: average # nonzeros per instance Degree-2 polynomial is used

Dimensionality of w is very high, but w is sparse Some training feature columns of xixj are entirely zero

Hashing techniques are used to handle sparse w

Chih-Jen Lin (National Taiwan Univ.) 28 / 43

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### Discussion

See more details in Chang et al. (2010) If φ(x ) is too high dimensional, people have proposed projection or hashing techniques to use fewer features as approximations

Examples: Kar and Karnick (2012); Pham and Pagh (2013)

This has been used in computational advertising (Chapelle et al., 2014)

Chih-Jen Lin (National Taiwan Univ.) 29 / 43

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Sample applications Transportation-mode detection in a sensor hub

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

Dependency parsing using feature combination Transportation-mode detection in a sensor hub

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 30 / 43

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### Example: Classifier in a Small Device

In a sensor application (Yu et al., 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.

Chih-Jen Lin (National Taiwan Univ.) 31 / 43

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Sample applications Transportation-mode detection in a sensor hub

### 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

Chih-Jen Lin (National Taiwan Univ.) 32 / 43

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### Outline

1 Introduction

2 Optimization methods

3 Sample applications

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 33 / 43

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Big-data linear classification

### Big-data Linear Classification

Nowadays data can be easily larger than memory capacity

Disk-level linear classification: Yu et al. (2012) and subsequent developments

Distributed linear classification: recently an active research topic

Example: we can parallelize the 2nd-order method discussed earlier. Recall the Hessian-vector product

2f (w )s = s + C · XT(D(X s))

Chih-Jen Lin (National Taiwan Univ.) 34 / 43

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### Parallel Hessian-vector Product

Hessian-vector products are the computational bottleneck

XTDX s

Data matrix X is now distributedly stored

X1 X2

. . . Xp node 1

node 2

node p

XTDX s = X1TD1X1s + · · · + XpTDpXps

Chih-Jen Lin (National Taiwan Univ.) 35 / 43

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Big-data linear classification

### Instance-wise and Feature-wise Data Splits

Xiw,1 Xiw,2 Xiw,3

Xfw,1Xfw,2Xfw,3

Instance-wise Feature-wise

We won’t have time to get into details. But their communication cost is different

Data moved per Hessian-vector product Instance-wise: O(n), Feature-wise: O(l )

Chih-Jen Lin (National Taiwan Univ.) 36 / 43

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### Discussion: Dostributed Training or Not?

One can always subsample data to one machine for deep analysis

Deciding to do distributed classification or not is an issue

In some areas distributed training has been successfully applied

One example is CTR (click-through rate) prediction in computational advertising

Chih-Jen Lin (National Taiwan Univ.) 37 / 43

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Big-data linear classification

### Discussion: Platform Issues

For the above-mentioned Newton methods, we have MPI and Spark implementations

We are preparing the integration to Spark MLlib Other existing distributed linear classifiers include Vowpal Wabbit from Yahoo!/Microsoft and Sibyl from Google

Platforms such as Spark are still being rapidly changed. This is a bit annoying

A carefully implementation may sometimes thousands times faster than a casual one

Chih-Jen Lin (National Taiwan Univ.) 38 / 43

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### Discussion: Design of Distributed Algorithms

On one computer, often we do batch rather than online learning

Online and streaming learning may be more useful for big-data applications

The example (Newton method) we showed is a synchronous parallel algorithms

Maybe asynchronous ones are better for big data?

Chih-Jen Lin (National Taiwan Univ.) 39 / 43

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Conclusions

### Outline

1 Introduction

2 Optimization methods

3 Sample applications

4 Big-data linear classification

5 Conclusions

Chih-Jen Lin (National Taiwan Univ.) 40 / 43

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### Resources on Linear Classification

• Since 2007, we have been actively developing the software LIBLINEAR for linear classification www.csie.ntu.edu.tw/~cjlin/liblinear

• A distributed extension (MPI and Spark) is now available

• An earlier survey on linear classification is Yuan et al.

(2012)

Recent Advances of Large-scale Linear Classification.

Proceedings of IEEE, 2012

It contains many references on this subject

Chih-Jen Lin (National Taiwan Univ.) 41 / 43

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Conclusions

### Conclusions

Linear classification is an old topic; but recently there are new and interesting applications

Kernel methods are still useful for many

applications, but linear classification + feature engineering are suitable for some others

Linear classification will continue to be used in situations ranging from small-model to big-data applications

Chih-Jen Lin (National Taiwan Univ.) 42 / 43

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### Acknowledgments

Many students have contributed to our research on large-scale linear classification

We also thank the partial support from National Science Council of Taiwan

Chih-Jen Lin (National Taiwan Univ.) 43 / 43

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