1. Machine Learning – Linear Classifiers and Boosting

2. Outline Different types of learning problems
Different types of learning algorithms
Supervised learning
Decision trees
Naïve Bayes
Perceptrons, Multi-layer Neural Networks
Boosting
Applications: learning to detect faces in images
Reading for today’s lecture:
Chapter 18.1 to 18.4 (inclusive) plus pages

3. Training Data for Supervised Learning

4. Classification Problem with Overlap

5. Inductive learning Let x represent the input vector of attributes
xj is the ith component of the vector x
value of the jth attribute, j = 1,…d
Let f(x) represent the value of the target variable for x
The implicit mapping from x to f(x) is unknown to us
We just have training data pairs, D = {x, f(x)} available
We want to learn a mapping from x to f, i.e.,
h(x; q) is “close” to f(x) for all training data points x
q are the parameters of our predictor h(..)
Examples:
h(x; q) = sign(w1x1 + w2x2+ w3)
hk(x) = (x1 OR x2) AND (x3 OR NOT(x4))

6. Decision Boundaries Decision Boundary Decision Region 1 Decision

7. Classification in Euclidean Space
A classifier is a partition of the space x into disjoint decision regions
Each region has a label attached
Regions with the same label need not be contiguous
For a new test point, find what decision region it is in, and predict the corresponding label
Decision boundaries = boundaries between decision regions
The “dual representation” of decision regions
We can characterize a classifier by the equations for its decision boundaries
Learning a classifier  searching for the decision boundaries that optimize our objective function

8. Example: Decision Trees
When applied to real-valued attributes, decision trees produce “axis-parallel” linear decision boundaries
Each internal node is a binary threshold of the form xj > t ?
converts each real-valued feature into a binary one
requires evaluation of N-1 possible threshold locations for N data points, for each real-valued attribute, for each internal node

9. Decision Tree Example
Debt
Income

10. Decision Tree Example
Debt
Income > t1 ?? t1
Income

11. Decision Tree Example
Debt
Income > t1 t2
Debt > t2 t1
Income ??

12. Decision Tree Example t2 t3 t1 Debt Income > t1 Debt > t2 Income

13. Decision Tree Example t2 t3 t1 Note: tree boundaries are
Debt
Income > t1 t2
Debt > t2 t3 t1
Income
Income > t3
Note: tree boundaries are
linear and axis-parallel

14. A Simple Classifier: Minimum Distance Classifier
Training
Separate training vectors by class
Compute the mean for each class, mk, k = 1,… m
Prediction
Compute the closest mean to a test vector x’ (using Euclidean distance)
Predict the corresponding class
In the 2-class case, the decision boundary is defined by the locus of the hyperplane that is halfway between the 2 means and is orthogonal to the line connecting them
This is a very simple-minded classifier – easy to think of cases where it will not work very well

15. Minimum Distance Classifier

16. Another Example: Nearest Neighbor Classifier
The nearest-neighbor classifier
Given a test point x’, compute the distance between x’ and each input data point
Find the closest neighbor in the training data
Assign x’ the class label of this neighbor
(sort of generalizes minimum distance classifier to exemplars)
If Euclidean distance is used as the distance measure (the most common choice), the nearest neighbor classifier results in piecewise linear decision boundaries
Many extensions
e.g., kNN, vote based on k-nearest neighbors

17. Local Decision Boundaries
Boundary? Points that are equidistant
between points of class 1 and 2
Note: locally the boundary is linear 1 2
Feature 2 1 2 ? 2 1
Feature 1

18. Finding the Decision Boundaries1 2
Feature 2 1 2 ? 2 1
Feature 1

19. Finding the Decision Boundaries1 2
Feature 2 1 2 ? 2 1
Feature 1

20. Finding the Decision Boundaries1 2
Feature 2 1 2 ? 2 1
Feature 1

21. Overall Boundary = Piecewise Linear
Decision Region
for Class 1
Decision Region
for Class 2 1 2
Feature 2 1 2 ? 2 1
Feature 1

22. Nearest-Neighbor Boundaries on this data set?

23. Linear Classifiers
Linear classifier  single linear decision boundary (for 2-class case)
We can always represent a linear decision boundary by a linear equation:
w1 x1 + w2 x2 + wd xd = S wj xj = wt x = 0
In d dimensions, this defines a (d-1) dimensional hyperplane
d=3, we get a plane; d=2, we get a line
For prediction we simply see if S wj xj > 0
The wi are the weights (parameters)
Learning consists of searching in the d-dimensional weight space for the set of weights (the linear boundary) that minimizes an error measure
Note that a minimum distance classifier is a special (restricted) case of a linear classifier

27. The Perceptron Classifier (pages 740-743 in text)
The perceptron classifier is just another name for a linear classifier for 2-class data, i.e.,
output(x) = sign( S wj xj )
Loosely motivated by a simple model of how neurons fire
For mathematical convenience, class labels are +1 for one class and -1 for the other
Two major types of algorithms for training perceptrons
Objective function = classification accuracy (“error counting”)
Objective function = squared error (use gradient descent)
Gradient descent is generally faster and more efficient – but there is a problem!

28. The Sigmoid Function s(f) f Sigmoid function is defined as
s[ f ] = [ 2 / ( 1 + exp[- f ] ) ] - 1
Derivative of sigmoid
¶s/df [ f ] = .5 * ( s[f]+1 ) * ( 1-s[f] )

29. Two different types of perceptron output
x-axis below is f(x) = f = weighted sum of inputs
y-axis is the perceptron output
Thresholded output,
takes values +1 or -1
o(f) f
s(f)
Sigmoid output, takes
real values between -1 and +1
The sigmoid is in effect an approximation
to the threshold function above, but
has a gradient that we can use for learning f

30. Squared Error for Perceptron with Sigmoidal Output
Squared error = E[w] = Si [ s(f[x(i)]) - t(i) ]2
where x(i) is the ith input vector in the training data, i=1,..N
t(i) is the ith target value (-1 or 1)
f[x(i)] = S wj xj is the weighted sum of inputs
s(f[x(i)]) is the sigmoid of the weighted sum
Note that everything is fixed (once we have the training data) except for the weights w
So we want to minimize E[w] as a function of w

31. Gradient Descent Learning of Weights
Gradient Descent Rule:
w new = w old h D ( E[w] )
where
D (E[w]) is the gradient of the error function E wrt weights, and
h is the learning rate (small, positive)
Notes:
1. This moves us downhill in direction D ( E[w] ) (steepest downhill)
2. How far we go is determined by the value of h

32. Gradient Descent Update Equation
From basic calculus, for perceptron with sigmoid, and squared error objective function, gradient for a single input x(i) is
D ( E[w] ) = - ( t(i) – s[f(i)] ) ¶s[f(i)] xj(i)
Gradient descent weight update rule:
wj = wj + h ( t(i) – s[f(i)] ) ¶s[f(i)] xj(i)
can rewrite as:
wj = wj + h * error * c * xj(i)

33. Pseudo-code for Perceptron Training
Initialize each wj (e.g.,randomly)
While (termination condition not satisfied)
for i = 1: N % loop over data points (an iteration)
for j= 1 : d % loop over weights
deltawj = h ( t(i) – s[f(i)] ) ¶s[f(i)] xj(i)
wj = wj + deltawj
end
calculate termination condition
Inputs: N features, N targets (class labels), learning rate h
Outputs: a set of learned weights

34. Comments on Perceptron Learning
Iteration = one pass through all of the data
Algorithm presented = incremental gradient descent
Weights are updated after visiting each input example
Alternatives
Batch: update weights after each iteration (typically slower)
Rate of convergence
convergence is guaranteed as long as learning rate is small enough
But if we make it too small, learning will be *very* slow
But if learning rate is too large, we move further, but can overshoot the solution and oscillate, and not converge at all

35. Learning to Detect Faces A Large-Scale Application of Machine Learning (This material is not in the text: for further information see the paper by P. Viola and M. Jones, International Journal of Computer Vision, 2004

36. Viola-Jones Face Detection Algorithm
Overview :
Viola Jones technique overview
Features
Integral Images
Feature Extraction
Weak Classifiers
Boosting and classifier evaluation
Cascade of boosted classifiers
Example Results

37. Viola Jones Technique Overview
Three major contributions/phases of the algorithm :
Feature extraction
Learning using boosting and decision stumps
Multi-scale detection algorithm
Feature extraction and feature evaluation.
Rectangular features are used, with a new image representation their calculation is very fast.
Classifier learning using a method called boosting
A combination of simple classifiers is very effective

38. Features Four basic types. They are easy to calculate.
The white areas are subtracted from the black ones.
A special representation of the sample called the integral image makes feature extraction faster.

39. Integral images Summed area tables
A representation that means any rectangle’s values can be calculated in four accesses of the integral image.

40. Fast Computation of Pixel Sums

41. Features are extracted from sub windows of a sample image.
Feature Extraction
Features are extracted from sub windows of a sample image.
The base size for a sub window is 24 by 24 pixels.
Each of the four feature types are scaled and shifted across all possible combinations
In a 24 pixel by 24 pixel sub window there are ~160,000 possible features to be calculated.

42. Learning with many features
We have 160,000 features – how can we learn a classifier with only a few hundred training examples without overfitting?
Idea:
Learn a single very simple classifier (a “weak classifier”)
Classify the data
Look at where it makes errors
Reweight the data so that the inputs where we made errors get higher weight in the learning process
Now learn a 2nd simple classifier on the weighted data
Combine the 1st and 2nd classifier and weight the data according to where they make errors
Learn a 3rd classifier on the weighted data
… and so on until we learn T simple classifiers
Final classifier is the combination of all T classifiers
This procedure is called “Boosting” – works very well in practice.

43. “Decision Stumps”
Decision stumps = decision tree with only a single root node
Certainly a very weak learner!
Say the attributes are real-valued
Decision stump algorithm looks at all possible thresholds for each attribute
Selects the one with the max information gain
Resulting classifier is a simple threshold on a single feature
Outputs a +1 if the attribute is above a certain threshold
Outputs a -1 if the attribute is below the threshold
Note: can restrict the search for to the n-1 “midpoint” locations between a sorted list of attribute values for each feature. So complexity is n log n per attribute.
Note this is exactly equivalent to learning a perceptron with a single intercept term (so we could also learn these stumps via gradient descent and mean squared error)

44. Boosting Example

45. First classifier

46. First 2 classifiers

47. First 3 classifiers

48. Final Classifier learned by Boosting

49. Final Classifier learned by Boosting

50. Boosting with Decision Stumps
Viola-Jones algorithm
With K attributes (e.g., K = 160,000) we have 160,000 different decision stumps to choose from
At each stage of boosting
given reweighted data from previous stage
Train all K (160,000) single-feature perceptrons
Select the single best classifier at this stage
Combine it with the other previously selected classifiers
Reweight the data
Learn all K classifiers again, select the best, combine, reweight
Repeat until you have T classifiers selected
Very computationally intensive
Learning K decision stumps T times
E.g., K = 160,000 and T = 1000

51. How is classifier combining done?
At each stage we select the best classifier on the current iteration and combine it with the set of classifiers learned so far
How are the classifiers combined?
Take the weight*feature for each classifier, sum these up, and compare to a threshold (very simple)
Boosting algorithm automatically provides the appropriate weight for each classifier and the threshold
This version of boosting is known as the AdaBoost algorithm
Some nice mathematical theory shows that it is in fact a very powerful machine learning technique

52. Reduction in Error as Boosting adds Classifiers

53. Useful Features Learned by Boosting

54. A Cascade of Classifiers

55. Detection in Real Images
Basic classifier operates on 24 x 24 subwindows
Scaling:
Scale the detector (rather than the images)
Features can easily be evaluated at any scale
Scale by factors of 1.25
Location:
Move detector around the image (e.g., 1 pixel increments)
Final Detections
A real face may result in multiple nearby detections
Postprocess detected subwindows to combine overlapping detections into a single detection

56. Training
Examples of 24x24 images with faces

57. Small set of 111 Training Images

58. Sample results using the Viola-Jones Detector
Notice detection at multiple scales

59. More Detection Examples

60. Practical implementation
Details discussed in Viola-Jones paper
Training time = weeks (with 5k faces and 9.5k non-faces)
Final detector has 38 layers in the cascade, 6060 features
700 Mhz processor:
Can process a 384 x 288 image in seconds (in 2003 when paper was written)

61. Summary Learning Different approaches to learning
Given a training data set, a class of models, and an error function, this is essentially a search or optimization problem
Different approaches to learning
Divide-and-conquer: decision trees
Global decision boundary learning: perceptrons
Constructing classifiers incrementally: boosting
Learning to recognize faces
Viola-Jones algorithm: state-of-the-art face detector, entirely learned from data, using boosting+decision-stumps