Support Vector Machines

1
Support Vector Machines

• Adapted from Lectures by
• Raymond Mooney (UT Austin) and
• Andrew Moore (CMU)

2
Text classification

• Earlier Algorithms for text classification
• K Nearest Neighbor classification
• Simple, expensive at test time, low bias, high
  variance, non-linear
• Vector space classification using centroids and
  hyperplanes that split them
• Simple, linear classifier perhaps too simple
  high bias, low variance
• Today
• SVMs Some empirical evaluation and comparison
• Text-specific issues in classification

3
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
• Support Vector Machine (SVM) finds an optimal
  solution.
• Maximizes the distance between the hyperplane and
  the difficult points close to decision boundary
• Intuition Then there are fewer uncertain
  classification decisions

This line represents the decision boundary ax
by - c 0
15.0
4
Another intuition

• If we place a fat separator between classes, we
  have less choices, and so the capacity of the
  model has been decreased.

Define the margin of a linear classifier as the
width that the boundary could be increased by
before hitting a datapoint.
5
Decision hyperplane
Recall that a decision hyperplane can be defined by the intercept term b the normal vector w (weight vector), which is perpendicular to the hyperplane All points x on the hyperplane satisfy
6
Support Vector Machine (SVM)

• 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 most successful current text
  classification method

15.1
7
Robustness of SVMs

• If we make a small error in the location of the
  boundary, this approach minimizes
  misclassification.
• Recall that support vectors are datapoints that
  the margin pushes up against.
• The model is immune to removal of any non-support
  vector datapoints.
• Rocchio centroids depend on all points of a
  class.
• kNN boundaries depend on local neighborhoods.
• Empirically SVMs work well.

8
Maximum Margin Formalization

• w decision hyperplane normal
• 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 minimum
  functional margin for any point
• The factor of 2 comes from measuring the whole
  width of the margin

9
The planar decision surface in data-space for the
simple linear discriminant function
10
Functional Margin
We are confident in the classification of a point
if it is far away from the decision boundary.
Functional margin
The functional margin of the ith example xi w.r.t. the hyperplane The functional margin of a data set w.r.t. a decision surface is twice the functional margin of any of the points in the data set with minimal functional margin factor 2 comes from measuring across the whole width of the margin
But we can increase functional margin by scaling
w and b. So, we need to place some constraint on
the size of the w vector.
10
11
Geometric margin
Geometric margin of the classifier maximum
width of the band that can be drawn separating
the support vectors of the two classes.


The geometric margin is clearly invariant to
scaling of parameters if we replace w by 5 w
and b by 5b, then the geometric margin is the
same, because it is inherently normalized by the
length of w.
11
12
Geometric Margin

• Distance from example to the separator is
• Examples closest to the hyperplane are support
  vectors.
• Margin ? of the separator is the width of
  separation between support vectors of classes.

x
r
x'
13
Linear SVM Mathematically

• 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 examples distance from the
  hyperplane is
• The geometric margin is

wTxi b 1 if yi 1 wTxi b -1 if yi
-1
14
Linear Support Vector Machine (SVM)
wTxa b 1
?

• Hyperplane
• wT x b 0
• Extra scale constraint
• mini1,,n wTxi b 1
• This implies
• wT(xaxb) 2
• ? xaxb2 2/w2

wTxb b -1
wT x b 0
15
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 yi1 wTxi b -1 if yi -1
Find w and b such that F(w) ½ wTw is minimized
and for all (xi ,yi) yi (wTxi b) 1
16
Recapitulation

• We start with a training data set
• We feed the data through a quadratic optimization
  
• procedure to find the best separating
  hyperplane
• Given a new point to classify, the
  classification
• function computes the projection of the point
  onto
• the hyperplane normal.
• The sign of this function determines the class
• If the point is within the margin of the
  classifier, the
• classifier can return dont know.
• The value of may also be transformed
  into a
• probability of classification

16
17
Multiclass support vector machines

• SVMs inherently two-class classifiers.
• Most common technique in practice build C
  one-versus-rest classifiers (commonly referred to
  as one-versus-all or OVA classification), and
  choose the class which classifies the test data
  with greatest margin
• Another strategy build a set of one-versus-one
  classifiers, and choose the class that is
  selected by the most classifiers. While this
  involves building C(C - 1)/2 classifiers, the
  time for training classifiers may actually
  decrease, since the training data set for each
  classifier is much smaller.

17
18
Walkthrough example building an SVM over the
data set shown in the figure

• Working geometrically
• The maximum margin weight vector will be parallel
  to the shortest line connecting points of the two
  classes, that is, the line between (1, 1) and
• (2, 3), giving a weight vector of (1,2).
• The optimal decision surface is orthogonal to
  that line and intersects it at the halfway point.
  Therefore, it passes through (1.5, 2).
• So, the SVM decision boundary is
• y x1 2x2 - 5.5

18
19
Walkthrough example building an SVM over the
data set shown in the figure

• Working algebraically
• With the constraint sign
• , we seek to
• minimize
• We know that the solution is
• for some a. So
• a 2a b -1, 2a 6a b 1
• Hence, a 2/5 and b -11/5.
• So the optimal hyperplane is given by
  and b -11/5.
• The margin ? is

19
20
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

x2
x
0
15.2.3
21
Nonlinear SVMs The Clever Bit!

• Project the linearly inseparable data to high
  dimensional space where it is linearly separable
  and then we can use linear SVM

22
Not linearly separable data.
Linearly separable data.
Angular degree (phase)
polar coordinates
0
5
Distance from center (radius)
Need to transform the coordinates polar
coordinates, kernel transformation into higher
dimensional space (support vector machines).
23
Non-linear SVMs Feature spaces
F x ? f(x)
24
(contd)

• Kernel functions and the kernel trick are used to
  transform data into a different linearly
  separable feature space

25
Mathematical Details SKIP
26
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 problems, and
  many (rather intricate) algorithms exist for
  solving them
• The solution involves constructing a dual problem
  where a Lagrange multiplier ai is associated with
  every constraint in the primary problem

Find w and b such that F(w) ½ wTw is minimized
and for all (xi ,yi) yi (wTxi b) 1
Find a1aN such that Q(a) Sai -
½SSaiajyiyjxiTxj is maximized and (1) Saiyi
0 (2) ai 0 for all ai
27
The Optimization Problem Solution

• The solution has the form
• Each non-zero ai 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.
• Also keep in mind that solving the optimization
  problem involved computing the inner products
  xiTxj between all pairs of training points.

w Saiyixi b yk- wTxk for any xk
such that ak? 0
f(x) SaiyixiTx b
28
Soft Margin Classification

• If the training set is not linearly separable,
  slack variables ?i 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)

?i
?j
15.2.1
29
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

Find w and b such that F(w) ½ wTw is minimized
and for all (xi ,yi) yi (wTxi b) 1
Find w and b such that F(w) ½ wTw CS?i is
minimized and for all (xi ,yi) yi (wTxi b)
1- ?i and ?i 0 for all i
30
Soft Margin Classification Solution

• The dual problem for soft margin classification
• Neither slack variables ?i nor their Lagrange
  multipliers appear in the dual problem!
• Again, xi with non-zero ai will be support
  vectors.
• Solution to the dual problem is

Find a1aN such that Q(a) Sai -
½SSaiajyiyjxiTxj is maximized and (1) Saiyi
0 (2) 0 ai C for all ai
But w not needed explicitly for classification!
w Saiyixi b yk(1- ?k) - wTxk
where k argmax ak
f(x) SaiyixiTx b
k
31
Classification with SVMs

• Given a new point x (x1,x2), score its
  projection onto the hyperplane normal
• In 2 dims score w1x1w2x2b.
• I.e., compute score wx b SaiyixiTx b
• Set confidence threshold t.

Score gt t yes Score lt -t no Else dont know
7
5
3
32
Linear SVMs Summary

• The classifier is a separating hyperplane.
• Most important training points are support
  vectors they define the hyperplane.
• Quadratic optimization algorithms can identify
  which training points xi are support vectors with
  non-zero Lagrangian multipliers ai.
• Both in the dual formulation of the problem and
  in the solution training points appear only
  inside inner products

f(x) SaiyixiTx b
Find a1aN such that Q(a) Sai -
½SSaiajyiyjxiTxj is maximized and (1) Saiyi
0 (2) 0 ai C for all ai
33
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

F x ? f(x)
34
The Kernel Trick

• The linear classifier relies on an inner product
  between vectors K(xi,xj)xiTxj
• If every datapoint is mapped into
  high-dimensional space via some transformation F
  x ? f(x), the inner product becomes
• K(xi,xj) f(xi) Tf(xj)
• A kernel function is some function that
  corresponds to an inner product in some expanded
  feature space.
• Example
• 2-dimensional vectors xx1 x2 let
  K(xi,xj)(1 xiTxj)2,
• Need to show that K(xi,xj) f(xi) Tf(xj)
• K(xi,xj)(1 xiTxj)2, 1 xi12xj12 2 xi1xj1
  xi2xj2 xi22xj22 2xi1xj1 2xi2xj2
• 1 xi12 v2 xi1xi2 xi22 v2xi1
  v2xi2T 1 xj12 v2 xj1xj2 xj22 v2xj1 v2xj2
  
• f(xi) Tf(xj) where f(x) 1
  x12 v2 x1x2 x22 v2x1 v2x2

35
Kernels

• Why use kernels?
• Make non-separable problem separable.
• Map data into better representational space
• Common kernels
• Linear
• Polynomial K(x,z) (1xTz)d
• Radial basis function (infinite dimensional
  space)

36
Evaluation Classic Reuters Data Set

• Most (over)used data set
• 21578 documents
• 9603 training, 3299 test articles (ModApte 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

• 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)

Common categories (train, test)
37
Reuters Text Categorization data set
(Reuters-21578) document
ltREUTERS TOPICS"YES" LEWISSPLIT"TRAIN"
CGISPLIT"TRAINING-SET" OLDID"12981"
NEWID"798"gt ltDATEgt 2-MAR-1987 165143.42lt/DATEgt
ltTOPICSgtltDgtlivestocklt/DgtltDgthoglt/Dgtlt/TOPICSgt ltTITLE
gtAMERICAN PORK CONGRESS KICKS OFF
TOMORROWlt/TITLEgt ltDATELINEgt CHICAGO, March 2 -
lt/DATELINEgtltBODYgtThe 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 3lt/BODYgtlt/TEXTgtlt/REUTERSgt
38
New Reuters RCV1 810,000 docs

• Top topics in Reuters RCV1

39
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
• Correct rate (1- error rate) Fraction of docs
  classified correctly

40
Dumais et al. 1998 Reuters - Accuracy
Recall labeled in category among those stories
that are really in category
Precision really in category among those
stories labeled in category
Break Even (Recall Precision) / 2
41
Results for Kernels (Joachims 1998)
42
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.

43
SKIP
44
Micro- vs. Macro-Averaging Example
Class 1
Class 2
Micro.Av. Table
Truth yes Truth no
Classifier yes 10 10
Classifier no 10 970
Truth yes Truth no
Classifier yes 90 10
Classifier no 10 890
Truth yes Truth no
Classifier yes 100 20
Classifier no 20 1860

• Macroaveraged precision (0.5 0.9)/2 0.7
• Microaveraged precision 100/120 .83
• Why this difference?

45
(No Transcript)
46
Reuters ROC - Category Grain
Recall
LSVM Decision Tree Naïve Bayes Find Similar
Precision
Recall labeled in category among those stories
that are really in category
Precision really in category among those
stories labeled in category
47
ROC for Category - Crude
Recall
LSVM Decision Tree Naïve Bayes Find Similar
Precision
48
ROC for Category - Ship
Recall
LSVM Decision Tree Naïve Bayes Find Similar
Precision
49
YangLiu SVM vs. Other Methods
50
Good practice departmentConfusion matrix
This (i, j) entry means 53 of the docs actually
in class i were put in class j by the classifier.
Class assigned by classifier
Actual Class
53

• In a perfect classification, only the diagonal
  has non-zero entries

51
The Real World

• P. Jackson and I. Moulinier 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.

52
The Real World

• Gee, Im 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

53
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

54
Very little data?

• If youre 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??

55
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

56
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)

57
A huge amount of data?

• 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

58
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
• May need a hybrid automatic/manual solution

59
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 shouldnt be.
• E.g., part numbers, chemical formulas

60
Does putting in hacks help?

• You bet!
• You can get a lot of value by differentially
  weighting contributions from different document
  zones
• 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)

61
Two techniques for zones

• Have a completely separate set of
  features/parameters for different zones like the
  title
• 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

62
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.

63
Does stemming/lowercasing/ help?

• As always its 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.

64
Measuring ClassificationFigures 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

65
A common problem Concept Drift

• Categories change over time
• Example president of the united states
• 1999 clinton is great feature
• 2002 clinton is bad feature
• One measure of a text classification system is
  how well it protects against concept drift.
• Can favor simpler models like Naïve Bayes
• Feature selection can be bad in protecting
  against concept drift

66
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 SVMlight
• SVMlight is accurate and fast and free (for
  research)
• Now lots of software libsvm, TinySVM, .
• Comparative evaluation of methods
• Real world exploit domain specific structure!