An overview of Boosting. Yoav Freund UCSD

Transcription

1 An overview of Boosting Yoav Freund UCSD

2 Plan of talk Generative vs. non-generative modeling Boosting Alternating decision trees Boosting and over-fitting Applications 2

3 Toy Example Computer receives telephone call Measures Pitch of voice Decides gender of caller Male Human Voice Female 3

4 Generative modeling mean1 mean2 Probability var1 var2 Voice Pitch

5 Discriminative approach [Vapnik 85] No. of mistakes Voice Pitch

6 Ill-behaved data No. of mistakes Probability mean1 mean2 Voice Pitch

7 Traditional Statistics vs. Machine Learning Machine Learning Data Statistics Estimated world state Decision Theory Predictions Actions

8 Comparison of methodologies Model Generative Discriminative Goal Performance measure Mismatch problems Probability estimates Likelihood Outliers Classification rule Misclassification rate Misclassifications 8

9 Boosting

10 A weighted training set Feature vectors Binary labels {-1,+1} Positive weights ( x 1,y 1,w 1 ),( x 2,y 2,w 2 ),, ( x m,y m,w m ) 10

11 A weak learner Weighted training set ( ) ( x 1,y 1,w 1 ), x 2,y 2,w 2 ( ),, x m,y m,w m Weak Learner A weak rule h instances x 1,x 2,,x m h predictions ŷ 1, ŷ 2,, ŷ m ; ŷ i { 1,+1} The weak requirement: m y i ˆ y i w i i =1 m w i=1 i > γ > 0

12 The boosting process ( x 1,y 1,1), ( x 2, y 2,1),,( x n, y n,1) weak learner h ( x 1,y 1,w 1 ), ( x 2,y 2,w 2 ),,( x n, y n,w n ) ( x 1,y 1,w 1 ), ( x 2,y 2,w 2 ),,( x n,y n,w n ) weak learner h 2 h 3 T 1 T 1 T 1 ( x 1, y 1,w 1 ), ( x 2,y 2,w 2 ),,( x n,y n,w n ) h T... Prediction(x) = sign(f T (x))

13 AFormalDescriptionofBoosting [Slides from Rob Schapire] given training set (x 1, y 1 ),...,(x m, y m ) y i { 1, +1} correct label of instance x i X for t =1,...,T : construct distribution D t on {1,...,m} find weak classifier ( rule of thumb ) h t : X { 1, +1} with small error ϵ t on D t : ϵ t =Pr i Dt [h t (x i ) y i ] output final classifier H final

14 AdaBoost [Slides from Rob Schapire] [with Freund] constructing D t : D 1 (i) =1/m given D t and h t : D t+1 (i) = D t(i) Z t { e α t if y i = h t (x i ) e α t if y i h t (x i ) = D t(i) Z t exp( α t y i h t (x i )) where Z t =normalizationfactor ( ) 1 α t = 1 2 ln ϵt > 0 final classifier: ( ) H final (x) =sign α t h t (x) t ϵ t

15 Toy Example [Slides from Rob Schapire] D 1 weak classifiers = vertical or horizontal half-planes

16 Round 1 [Slides from Rob Schapire] h 1 D 2 ε 1 =0.30 α 1 =0.42

17 Round 2 [Slides from Rob Schapire] h 2 D 3 ε 2 =0.21 α 2 =0.65

18 Round 3 [Slides from Rob Schapire] h 3 ε 3 =0.14 α 3 =0.92

19 Final Classifier [Slides from Rob Schapire] H = sign final =

20 Analyzing the Training Error [with Freund] Theorem: write ϵ t as 1 2 γ t [ γ t = edge ] then [ 2 ] ϵ t (1 ϵ t ) training error(h final ) t = t exp 1 4γ 2 t ( 2 t [Slides from Rob Schapire] γ 2 t ) so: if t : γ t γ > 0 then training error(h final ) e 2γ2 T AdaBoost is adaptive: does not need to know γ or T apriori can exploit γ t γ

21 Boosting block diagram Strong Learner Accurate Rule Weak Learner Weak rule Example weights Booster

22 Boosting with specialists Specialists predict only when they are confident. In addition to {-1,+1} specialists use 0 to indicate no-prediction. As boosting allows both positive and negative weights: we restrict attention to specialists that output {0,1}. 22

23 Adaboost as variational optimization Weak Rule: h t : X {0,1} Label: y { 1,+1} Training set: {(x 1, y 1,1),(x 2, y 2,1),,(x n, y n + - α t = 1 2 ln ε + i:h t (x i )=1,y i =1 w t i t ε + w i i:h t (x i )=1,y i = 1 F t = F t 1 + α t h t

24 AdaBoost as variational optimization. x = input, a scalar, y = output, -1 or +1 (x1,y1),(x2,y2),,(xn,yn) = training set Ft-1 = Strong rule after t-1 boosting iterations. ht = Weak rule produced at iteration t ht Ft-1 x 24

25 AdaBoost as variational optimization. Ft-1 = Strong rule after t-1 boosting iterations. ht = Weak rule produced at iteration t F t 1 (x) + 0h t (x) ht Ft-1 x 25

26 AdaBoost as variational optimization. Ft-1 = Strong rule after t-1 boosting iterations. ht = Weak rule produced at iteration t F t 1 (x) + 0.4h t (x) ht Ft-1 x 26

27 AdaBoost as variational optimization. Ft-1 = Strong rule after t-1 boosting iterations. ht = Weak rule produced at iteration t F t (x) = F t 1 (x) + 0.8h t (x) ht Ft-1 x 27

28 Margins Fix a set of weak rules:! h(x) = (h 1 (x),h 2 (x),,h T (x)) - - Represent the i'th example x i with outputs of the weak rules:! h i Labels: y { 1,+1} Training set: (! h 1, y 1 ),(! h 2, y 2 ),,(! h n, y n ) Goal : Find a weight vector! α = (α 1,,α T ) that minimizes number of training mistakes! α (! h i, y i ) ! α reflect Mistake (y i! hi, y i ) - - Correct Project Cumulative # examples Mistakes margin! y i " hi " α Correct Margin

29 Boosting as gradient descent Our goal is to minimize the number of mistakes (0-1 loss) Unfortunately, the step function has deriv. zero at all points other than at zero where the derivative is undefined. Ada boost uses an upper bound on the 0-1 loss. Adaboost : e margin Loss Adaboost minimizes the exponential loss which is an upper bound. Finds the vector \alpha through coordinate-wise gradient descent. Weak rules are added one at a time. 0-1 loss Mistakes Correct Margin

30 Adaboost as gradient descent Weak rules are added one at a time. Fixing the alphas for 1..t-1, find alpha for the new rule (ht) Weak rule defines how each example would move = increase or decrease the margin. 0-1 loss new margin of (x i, y i ) is y i F t 1 (x i ) + α y i h t (x i ) new total loss = n exp y i F t 1 (x i ) + αh t (x i ) i=1 derivative of total loss wrt α : n α α =0 i=1 ( ( )) ( ( )) exp y i F t 1 (x i ) + αh t (x i ) weight of example (x i,y i ) n!## "## $ = y i h t (x i ) exp( -y i F t-1 (x i )) i=1 Mistakes Correct Margin

31 Logitboost as gradient descent Also called Gentle-Boost and Logit Boost, Hastie, Freedman & Tibshirani margin=m Logitboost loss Instead of the loss e m define loss to be log 1+ e m ( ) ( ) e m ( ) m When margin 0 : log 1+ e m When margin 0 :log 1+ e m Logitboost weight Adaboost (loss and weight) new margin of (x i, y i ) is y i F t 1 (x i ) + α y i h t (x i ) new total loss = n log 1+ exp y i ( F t 1 (x i ) + αh t (x i i=1 derivative of total loss wrt α : n α α =0 i=1 log 1+ exp y i F t 1 (x i ) + αh t (x i ) ( ( ( )) weight of example (x i,y i )!### "### $ n exp( -y = y i h t (x i ) i F t-1 (x i )) 1+ exp -y i F t-1 (x i ) i=1 ( ) Mistakes Correct Margin

32 Noise resistance Adaboost: perform well when a achievable error rate is close to zero (almost consistent case). Errors = examples with negative margins, get very large weights, can overfit. Logitboost: Inferior to adaboost when achievable error rate is close to zero. Often better than Adaboost when achievable error rate is high. Weight on any example never larger than 1. 32

33 Gradient-Boost / AnyBoost A general recipe for learning by incremental optimization Applies to any (differentiable) loss function. labeled example is (x, y) (can be anything). t prediction is of the form: F t (x) = α i h i (x) i=1 n Loss: L( F t (x), y) Total Loss so far is: L F t 1 (x j ), y j j=1 Weight of example (x, y) : z L ( F t 1(x) + z, y) ( ) At each iteration: calc example weights call weak learner with weighted example Add generated weak rule to create new rule: F t = F t 1 + α t h t 33

34 Beach Summer Winter Styles of weak learners Simple (stumps)!# Generalist "# $ α -1 Winter Season Summer +1 Specialists!##" ## $ Season Season = α α Complex (fully grown trees) α -1 Prison Location Winter Season +1 Summer Ski Slope Everything in between (neural networks, Nearest neighbors, Naive Bayes,..)

35 Alternating Decision Trees Joint work with Llew Mason

36 Decision Trees Y X>3 +1 no -1 yes Y>5 5-1 no -1 yes X

37 Decision tree as a sum -0.2 Y no X>3 yes sign no Y>5 yes X

38 An alternating decision tree Y sign no Y<1 X>3 0.0 yes +0.7 no -0.1 yes no Y>5 yes X

39 Example: Medical Diagnostics Cleve dataset from UC Irvine database. Heart disease diagnostics (+1=healthy,-1=sick) 13 features from tests (real valued and discrete). 303 instances.

40 Adtree for Cleveland heart-disease diagnostics problem

41 Cross-validated accuracy Learning algorithm Number of splits Average test error Test error variance ADtree % 0.6% C % 0.5% C5.0 + boosting Boost Stumps % 0.5% % 0.8%

42 Applications of Boosting

43 Applications of Boosting Academic research Applied research Commercial deployment

44 Academic research % test error rates Database Other Boosting Error reduction Cleveland 27.2 (DT) % Promoters 22.0 (DT) % Letter 13.8 (DT) % Reuters 4 5.8, 6.0, ~60% Reuters , 12.1, ~40%

45 Schapire, Singer, Gorin 98 Applied research AT&T, How may I help you? Classify voice requests Voice -> text -> category Fourteen categories Area code, AT&T service, billing credit, calling card, collect, competitor - competitor, - dial assistance, - directory, - how to dial, - person to person, - rate, - third party, - time charge

46 Example transcribed sentences Yes I d like to place a collect call long distance please Ø collect Operator I need to make a call but I need to bill it to my office Ø third party Yes I d like to place a call on my master card please Ø calling card I just called a number in Sioux city and I musta rang the wrong number because I got the wrong party and I would like to have that taken off my bill Ø billing credit

47 Weak rules generated by boostexter Categories Calling card Collect call Third party Weak Rule Word occurs Word does not occur

48 Results 7844 training examples hand transcribed 1000 test examples hand / machine transcribed Accuracy with 20% rejected Machine transcribed: 75% Hand transcribed: 90%

49 Freund, Mason, Rogers, Pregibon, Cortes 2000 Commercial deployment Distinguish business/residence customers Using statistics from call-detail records Alternating decision trees Combines very simple rules Can over-fit, cross validation used to stop

50 Massive datasets (for 1997) 260M calls / day 230M telephone numbers Label unknown for ~30% Hancock: software for computing statistical signatures. 100K randomly selected training examples, ~10K is enough Training takes about 2 hours. Generated classifier has to be both accurate and efficient

51 Alternating tree for buizocity

52 Alternating Tree (Detail)

53 Precision/recall graphs Accuracy Score

54 Face Detection Viola and Jones / 2001

55 Face Detection / Viola and Jones Live demo in 2001 conference. Higher accuracy than manually designed state of art Real -time using 2001 laptop Many Uses Standard feature in cameras User Interfaces Security Systems Video Compression Image Database Analysis

56 Face Detection as a Filtering process Larger Scale Smallest Scale 50,000 Locations/Scales Most Negative

57 Classifier is Learned from Labeled Data Example: 28x28 image patch Label: Face / Non face 5000 faces, 10 8 non faces Faces are normalized Scale, translation Rotation remains

58 Image Features Rectangle filters Similar to Haar wavelets Papageorgiou, et al. h t (x i ) = 1 if f t(x i ) > θ t 0 otherwise Very fast to compute using integral image. Unique Binary Features Combined using adaboost

59 Cascaded boosting Features combined using Adaboost Find best weak rule by exhaustive search. Ran for 2 days on a 250 node cluster For detection, features combined in a cascade: IMAGE SUB-WINDOW 50% 1 Feature 5 Features F NON-FACE 20% 2% 20 Features Runs in real time, 15 FPS, on laptop F NON-FACE F NON-FACE FACE 59

60 Paul Viola and Mike Jones 60

61 Summary Boosting is a computational method for learning accurate classifiers Resistance to over-fit explained by margins Boosting has been applied successfully to a variety of classification problems