linear SVM.ppt

Introduction to Information Retrieval
1
Linear classifiers: Which Hyperplane?
 Lots of possible solutions for a, b, c.
 Some methods find a separating hyperplane,
but not the optimal one [according to some
criterion of expected goodness]
 E.g., perceptron
 Support Vector Machine (SVM) finds an
optimal* solution.
 Maximizes the distance between the
hyperplane and the “difficult points” close to
decision boundary
 One intuition: if there are no points near the
decision surface, then there are no very
uncertain classification decisions
This line
represents the
decision
boundary:
ax + by − c = 0

2
Another intuition
 If you have to place a fat separator between classes,
you have less choices, and so the capacity of the
model has been decreased
Sec. 15.1

3
Support Vector Machine (SVM)
Support vectors
Maximizes
margin
 SVMs maximize the margin around
the separating hyperplane.
 A.k.a. large margin classifiers
 The decision function is fully
specified by a subset of training
samples, the support vectors.
 Solving SVMs is a quadratic
programming problem
 Seen by many as the most
successful current text
classification method*
*but other discriminative methods
often perform very similarly
Sec. 15.1
Narrower
margin

4
 w: decision hyperplane normal vector
 xi: data point i
 yi: class of data point i (+1 or -1) NB: Not 1/0
 Classifier is: f(xi) = sign(wTxi + b)
 Functional margin of xi is: yi (wTxi + b)
 But note that we can increase this margin simply by scaling w, b….
 Functional margin of dataset is twice the minimum
functional margin for any point
 The factor of 2 comes from measuring the whole width of the
margin
Maximum Margin: Formalization
Sec. 15.1

5
Geometric Margin
 Distance from example to the separator is
 Examples closest to the hyperplane are support vectors.
 Margin ρ of the separator is the width of separation between support vectors
of classes.
w
x
w b
y
r
T
+
=
r
ρ
x
x′
w
Derivation of finding r:
Dotted line x’−x is perpendicular to
decision boundary so parallel to w.
Unit vector is w/|w|, so line is rw/|w|.
x’ = x – yrw/|w|.
x’ satisfies wTx’+b = 0.
So wT(x –yrw/|w|) + b = 0
Recall that |w| = sqrt(wTw).
So wTx –yr|w| + b = 0
So, solving for r gives:
r = y(wTx + b)/|w|
Sec. 15.1

6
Linear SVM Mathematically
The linearly separable case
 Assume that all data is at least distance 1 from the hyperplane, then the
following two constraints follow for a training set {(xi ,yi)}
 For support vectors, the inequality becomes an equality
 Then, since each example’s distance from the hyperplane is
 The margin is:
wTxi + b ≥ 1 if yi = 1
wTxi + b ≤ −1 if yi = −1
w
2
=
r
w
x
w b
y
r
T
+
=
Sec. 15.1

7
Linear Support Vector Machine (SVM)
 Hyperplane
wT x + b = 0
 Extra scale constraint:
mini=1,…,n |wTxi + b| = 1
 This implies:
wT(xa–xb) = 2
ρ = ||xa–xb||2 = 2/||w||2
wT x + b = 0
wTxa + b = 1
wTxb + b = -1
ρ
Sec. 15.1

8
Linear SVMs Mathematically (cont.)
 Then we can formulate the quadratic optimization problem:
 A better formulation (min ||w|| = max 1/ ||w|| ):
Find w and b such that
is maximized; and for all {(xi , yi)}
wTxi + b ≥ 1 if yi=1; wTxi + b ≤ -1 if yi = -1
w
2
=
r
Φ(w) =½ wTw is minimized;
and for all {(xi ,yi)}: yi (wTxi + b) ≥ 1
Sec. 15.1

9
Solving the Optimization Problem
 This is now optimizing a quadratic function subject to linear constraints
 Quadratic optimization problems are a well-known class of mathematical
programming problem, and many (intricate) algorithms exist for solving them
(with many special ones built for SVMs)
 The solution involves constructing a dual problem where a Lagrange
multiplier αi is associated with every constraint in the primary problem:
Φ(w) =½ wTw is minimized;
and for all {(xi ,yi)}: yi (wTxi + b) ≥ 1
Find α1…αN such that
Q(α) =Σαi - ½ΣΣαiαjyiyjxi
Txj is maximized and
(1) Σαiyi = 0
(2) αi ≥ 0 for all αi
Sec. 15.1

10
The Optimization Problem Solution
 The solution has the form:
 Each non-zero αi indicates that corresponding xi is a support vector.
 Then the classifying function will have the form:
 Notice that it relies on an inner product between the test point x and the
support vectors xi
 We will return to this later.
 Also keep in mind that solving the optimization problem involved
computing the inner products xi
Txj between all pairs of training points.
w =Σαiyixi b= yk- wTxk for any xk such that αk 0
f(x) = Σαiyixi
Tx + b
Sec. 15.1

11
Soft Margin Classification
 If the training data is not
linearly separable, slack
variables ξi can be added to
allow misclassification of
difficult or noisy examples.
 Allow some errors
 Let some points be moved
to where they belong, at a
cost
 Still, try to minimize training
set errors, and to place
hyperplane “far” from each
class (large margin)
ξj
ξi
Sec. 15.2.1

12
Soft Margin Classification
Mathematically
 The old formulation:
 The new formulation incorporating slack variables:
 Parameter C can be viewed as a way to control overfitting
 A regularization term
Φ(w) =½ wTw is minimized and for all {(xi ,yi)}
yi (wTxi + b) ≥ 1
Φ(w) =½ wTw + CΣξi is minimized and for all {(xi ,yi)}
yi (wTxi + b) ≥ 1- ξi and ξi ≥ 0 for all i
Sec. 15.2.1

13
Soft Margin Classification – Solution
 The dual problem for soft margin classification:
 Neither slack variables ξi nor their Lagrange multipliers appear in the dual
problem!
 Again, xi with non-zero αi will be support vectors.
 Solution to the dual problem is:
(1) Σαiyi = 0
(2) 0 ≤ αi ≤ C for all αi
w = Σαiyixi
b = yk(1- ξk) - wTxk where k = argmax αk’
k’ f(x) = Σαiyixi
Tx + b
w is not needed explicitly for
classification!
Sec. 15.2.1

14
Classification with SVMs
 Given a new point x, we can score its projection
onto the hyperplane normal:
 I.e., compute score: wTx + b = Σαiyixi
Tx + b
 Decide class based on whether < or > 0
 Can set confidence threshold t.
-1
0
1
Score > t: yes
Score < -t: no
Else: don’t know
Sec. 15.1

15
Linear SVMs: Summary
 The classifier is a separating hyperplane.
 The most “important” training points are the support vectors; they define
the hyperplane.
 Quadratic optimization algorithms can identify which training points xi are
support vectors with non-zero Lagrangian multipliers αi.
 Both in the dual formulation of the problem and in the solution, training
points appear only inside inner products:
(1) Σαiyi = 0
(2) 0 ≤ αi ≤ C for all αi
f(x) = Σαiyixi
Tx + b
Sec. 15.2.1

16
Non-linear SVMs
 Datasets that are linearly separable (with some noise) work out great:
 But what are we going to do if the dataset is just too hard?
 How about … mapping data to a higher-dimensional space:
0
x2
x
0 x
0 x
Sec. 15.2.3

17
Non-linear SVMs: Feature spaces
 General idea: the original feature space can always
be mapped to some higher-dimensional feature
space where the training set is separable:
Φ: x → φ(x)
Sec. 15.2.3

18
The “Kernel Trick”
 The linear classifier relies on an inner product between vectors K(xi,xj)=xi
Txj
 If every datapoint is mapped into high-dimensional space via some
transformation Φ: x → φ(x), the inner product becomes:
K(xi,xj)= φ(xi) Tφ(xj)
 A kernel function is some function that corresponds to an inner product in
some expanded feature space.
 Example:
2-dimensional vectors x=[x1 x2]; let K(xi,xj)=(1 + xi
Txj)2
,
Need to show that K(xi,xj)= φ(xi) Tφ(xj):
K(xi,xj)=(1 + xi
Txj)2
,= 1+ xi1
2xj1
2 + 2 xi1xj1 xi2xj2+ xi2
2xj2
2 + 2xi1xj1 + 2xi2xj2=
= [1 xi1
2 √2 xi1xi2 xi2
2 √2xi1 √2xi2]T [1 xj1
2 √2 xj1xj2 xj2
2 √2xj1 √2xj2]
= φ(xi)Tφ(xj) where φ(x) = [1 x1
2 √2 x1x2 x2
2 √2x1 √2x2]
Sec. 15.2.3

19
Kernels
 Why use kernels?
 Make non-separable problem separable.
 Map data into better representational space
 Common kernels
 Linear
 Polynomial K(x,z) = (1+xTz)d
 Gives feature conjunctions
 Radial basis function (infinite dimensional space)
 Haven’t been very useful in text classification
Sec. 15.2.3

20
 Most (over)used data set
 21578 documents
 9603 training, 3299 test articles (ModApte/Lewis split)
 118 categories
 An article can be in more than one category
 Learn 118 binary category distinctions
 Average document: about 90 types, 200 tokens
 Average number of classes assigned
 1.24 for docs with at least one category
 Only about 10 out of 118 categories are large
Common categories
(#train, #test)
Evaluation: Classic Reuters-21578 Data Set
• Earn (2877, 1087)
• Acquisitions (1650, 179)
• Money-fx (538, 179)
• Grain (433, 149)
• Crude (389, 189)
• Trade (369,119)
• Interest (347, 131)
• Ship (197, 89)
• Wheat (212, 71)
• Corn (182, 56)
Sec. 15.2.4

21
Reuters Text Categorization data set
(Reuters-21578) document
<REUTERS TOPICS="YES" LEWISSPLIT="TRAIN" CGISPLIT="TRAINING-SET"
OLDID="12981" NEWID="798">
<DATE> 2-MAR-1987 16:51:43.42</DATE>
<TOPICS><D>livestock</D><D>hog</D></TOPICS>
<TITLE>AMERICAN PORK CONGRESS KICKS OFF TOMORROW</TITLE>
<DATELINE> CHICAGO, March 2 - </DATELINE><BODY>The American Pork Congress
kicks off tomorrow, March 3, in Indianapolis with 160 of the nations pork producers from 44
member states determining industry positions on a number of issues, according to the National Pork
Producers Council, NPPC.
Delegates to the three day Congress will be considering 26 resolutions concerning various issues,
including the future direction of farm policy and the tax law as it applies to the agriculture sector.
The delegates will also debate whether to endorse concepts of a national PRV (pseudorabies virus)
control and eradication program, the NPPC said.
A large trade show, in conjunction with the congress, will feature the latest in technology in all
areas of the industry, the NPPC added. Reuter
</BODY></TEXT></REUTERS>
Sec. 15.2.4

22
Per class evaluation measures
 Recall: Fraction of docs in class i
classified correctly:
 Precision: Fraction of docs assigned
class i that are actually about class i:
 Accuracy: (1 - error rate) Fraction of
docs classified correctly:
cii
i
å
cij
i
å
j
å
cii
c ji
j
å
cii
cij
j
å
Sec. 15.2.4

23
Micro- vs. Macro-Averaging
 If we have more than one class, how do we combine
multiple performance measures into one quantity?
 Macroaveraging: Compute performance for each
class, then average.
 Microaveraging: Collect decisions for all classes,
compute contingency table, evaluate.
Sec. 15.2.4

24
Micro- vs. Macro-Averaging: Example
Truth:
yes
Truth:
no
Classifi
er: yes
10 10
Classifi
er: no
10 970
Truth:
yes
Truth:
no
Classifi
er: yes
90 10
Classifi
er: no
10 890
Truth:
yes
Truth:
no
Classifier:
yes
100 20
Classifier:
no
20 1860
Class 1 Class 2 Micro Ave. Table
 Macroaveraged precision: (0.5 + 0.9)/2 = 0.7
 Microaveraged precision: 100/120 = .83
 Microaveraged score is dominated by score on
common classes
Sec. 15.2.4

25
Sec. 15.2.4

Precision-recall for category: Crude
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
LSVM
Decision Tree
Naïve Bayes
Rocchio
Precision
Recall
Dumais
(1998)

27
Precision-recall for category: Ship
Precision
Recall
Sec. 15.2.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
LSVM
Decision Tree
Naïve Bayes
Rocchio
Dumais
(1998)

28
Yang&Liu: SVM vs. Other Methods
Sec. 15.2.4

29
Good practice department:
Make a confusion matrix
 In a perfect classification, only the diagonal has non-zero entries
 Look at common confusions and how they might be addressed
53
Class assigned by classifier
Actual
Class
This (i, j) entry means 53 of the docs actually in
class i were put in class j by the classifier.
Sec. 15.2.4

30
The Real World
P. Jackson and I. Moulinier. 2002. Natural Language Processing for Online Applications
 “There is no question concerning the commercial value of
being able to classify documents automatically by content.
There are myriad potential applications of such a capability
for corporate intranets, government departments, and
Internet publishers”
 “Understanding the data is one of the keys to successful
categorization, yet this is an area in which most categorization
tool vendors are extremely weak. Many of the ‘one size fits
all’ tools on the market have not been tested on a wide range
of content types.”
Sec. 15.3

31
The Real World
 Gee, I’m building a text classifier for real, now!
 What should I do?
 How much training data do you have?
 None
 Very little
 Quite a lot
 A huge amount and its growing
Sec. 15.3.1

32
Manually written rules
 No training data, adequate editorial staff?
 Never forget the hand-written rules solution!
 If (wheat or grain) and not (whole or bread) then
 Categorize as grain
 In practice, rules get a lot bigger than this
 Can also be phrased using tf or tf.idf weights
 With careful crafting (human tuning on development
data) performance is high:
 Construe: 94% recall, 84% precision over 675 categories
(Hayes and Weinstein 1990)
 Amount of work required is huge
 Estimate 2 days per class … plus maintenance
Sec. 15.3.1

33
Very little data?
 If you’re just doing supervised classification, you
should stick to something high bias
 There are theoretical results that Naïve Bayes should do
well in such circumstances (Ng and Jordan 2002 NIPS)
 The interesting theoretical answer is to explore semi-
supervised training methods:
 Bootstrapping, EM over unlabeled documents, …
 The practical answer is to get more labeled data as
soon as you can
 How can you insert yourself into a process where humans
will be willing to label data for you??
Sec. 15.3.1

34
A reasonable amount of data?
 Perfect!
 We can use all our clever classifiers
 Roll out the SVM!
 But if you are using an SVM/NB etc., you should
probably be prepared with the “hybrid” solution
where there is a Boolean overlay
 Or else to use user-interpretable Boolean-like models like
decision trees
 Users like to hack, and management likes to be able to
implement quick fixes immediately
Sec. 15.3.1

35
A huge amount of data?
 This is great in theory for doing accurate
classification…
 But it could easily mean that expensive methods like
SVMs (train time) or kNN (test time) are quite
impractical
 Naïve Bayes can come back into its own again!
 Or other advanced methods with linear training/test
complexity like regularized logistic regression (though
much more expensive to train)
Sec. 15.3.1

36
Accuracy as a function of data size
 With enough data the choice
of classifier may not matter
much, and the best choice
may be unclear
 Data: Brill and Banko on
context-sensitive spelling
correction
 But the fact that you have to
keep doubling your data to
improve performance is a
little unpleasant
Sec. 15.3.1

37
How many categories?
 A few (well separated ones)?
 Easy!
 A zillion closely related ones?
 Think: Yahoo! Directory, Library of Congress classification,
legal applications
 Quickly gets difficult!
 Classifier combination is always a useful technique
 Voting, bagging, or boosting multiple classifiers
 Much literature on hierarchical classification
 Mileage fairly unclear, but helps a bit (Tie-Yan Liu et al. 2005)
 May need a hybrid automatic/manual solution
Sec. 15.3.2

38
How can one tweak performance?
 Aim to exploit any domain-specific useful features
that give special meanings or that zone the data
 E.g., an author byline or mail headers
 Aim to collapse things that would be treated as
different but shouldn’t be.
 E.g., part numbers, chemical formulas
 Does putting in “hacks” help?
 You bet!
 Feature design and non-linear weighting is very important in the
performance of real-world systems
Sec. 15.3.2

39
Upweighting
 You can get a lot of value by differentially weighting
contributions from different document zones:
 That is, you count as two instances of a word when
you see it in, say, the abstract
 Upweighting title words helps (Cohen & Singer 1996)
 Doubling the weighting on the title words is a good rule of thumb
 Upweighting the first sentence of each paragraph helps
(Murata, 1999)
 Upweighting sentences that contain title words helps (Ko
et al, 2002)
Sec. 15.3.2

40
Two techniques for zones
1. Have a completely separate set of
features/parameters for different zones like the title
2. Use the same features (pooling/tying their
parameters) across zones, but upweight the
contribution of different zones
 Commonly the second method is more successful: it
costs you nothing in terms of sparsifying the data,
but can give a very useful performance boost
 Which is best is a contingent fact about the data
Sec. 15.3.2

41
Text Summarization techniques in text
classification
 Text Summarization: Process of extracting key pieces
from text, normally by features on sentences
reflecting position and content
 Much of this work can be used to suggest weightings
for terms in text categorization
 See: Kolcz, Prabakarmurthi, and Kalita, CIKM 2001: Summarization
as feature selection for text categorization
 Categorizing purely with title,
 Categorizing with first paragraph only
 Categorizing with paragraph with most keywords
 Categorizing with first and last paragraphs, etc.
Sec. 15.3.2

42
Does stemming/lowercasing/… help?
 As always, it’s hard to tell, and empirical evaluation is
normally the gold standard
 But note that the role of tools like stemming is rather
different for TextCat vs. IR:
 For IR, you often want to collapse forms of the verb
oxygenate and oxygenation, since all of those documents
will be relevant to a query for oxygenation
 For TextCat, with sufficient training data, stemming does
no good. It only helps in compensating for data sparseness
(which can be severe in TextCat applications). Overly
aggressive stemming can easily degrade performance.
Sec. 15.3.2

43
Measuring Classification
Figures of Merit
 Not just accuracy; in the real world, there are
economic measures:
 Your choices are:
 Do no classification
 That has a cost (hard to compute)
 Do it all manually
 Has an easy-to-compute cost if doing it like that now
 Do it all with an automatic classifier
 Mistakes have a cost
 Do it with a combination of automatic classification and manual
review of uncertain/difficult/”new” cases
 Commonly the last method is most cost efficient and is
adopted

44
A common problem: Concept Drift
 Categories change over time
 Example: “president of the united states”
 1999: clinton is great feature
 2010: clinton is bad feature
 One measure of a text classification system is how
well it protects against concept drift.
 Favors simpler models like Naïve Bayes
 Feature selection: can be bad in protecting against
concept drift

45
Summary
 Support vector machines (SVM)
 Choose hyperplane based on support vectors
 Support vector = “critical” point close to decision boundary
 (Degree-1) SVMs are linear classifiers.
 Kernels: powerful and elegant way to define similarity metric
 Perhaps best performing text classifier
 But there are other methods that perform about as well as SVM, such
as regularized logistic regression (Zhang & Oles 2001)
 Partly popular due to availability of good software
 SVMlight is accurate and fast – and free (for research)
 Now lots of good software: libsvm, TinySVM, ….
 Comparative evaluation of methods
 Real world: exploit domain specific structure!

Resources for today’s lecture
 Christopher J. C. Burges. 1998. A Tutorial on Support Vector Machines for Pattern Recognition
 S. T. Dumais. 1998. Using SVMs for text categorization, IEEE Intelligent Systems, 13(4)
 S. T. Dumais, J. Platt, D. Heckerman and M. Sahami. 1998. Inductive learning algorithms and
representations for text categorization. CIKM ’98, pp. 148-155.
 Yiming Yang, Xin Liu. 1999. A re-examination of text categorization methods. 22nd Annual
International SIGIR
 Tong Zhang, Frank J. Oles. 2001. Text Categorization Based on Regularized Linear
Classification Methods. Information Retrieval 4(1): 5-31
 Trevor Hastie, Robert Tibshirani and Jerome Friedman. Elements of Statistical Learning: Data
Mining, Inference and Prediction. Springer-Verlag, New York.
 T. Joachims, Learning to Classify Text using Support Vector Machines. Kluwer, 2002.
 Fan Li, Yiming Yang. 2003. A Loss Function Analysis for Classification Methods in Text
Categorization. ICML 2003: 472-479.
 Tie-Yan Liu, Yiming Yang, Hao Wan, et al. 2005. Support Vector Machines Classification with
Very Large Scale Taxonomy, SIGKDD Explorations, 7(1): 36-43.
 ‘Classic’ Reuters-21578 data set: http://www.daviddlewis.com /resources
/testcollections/reuters21578/
Ch. 15

linear SVM.ppt

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linear SVM.ppt