COMS 4721: Machine Learning for Data Science Lecture 13, 3/2/2017

Transcription

1 COMS 4721: Machine Learning for Data Science Lecture 13, 3/2/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University

2 BOOSTING Robert E. Schapire and Yoav Freund, Boosting: Foundations and Algorithms, MIT Press, See this textbook for many more details. (I borrow some figures from that book.)

3 BAGGING CLASSIFIERS Algorithm: Bagging binary classifiers Given (x 1, y 1 ),..., (x n, y n ), x X, y { 1, 1} For b = 1,..., B Sample a bootstrap dataset Bb of size n. For each entry in B b, select (x i, y i) with probability 1. Some (xi, yi) will repeat and some won t appear in Bb. n Learn a classifier fb using data in B b. Define the classification rule to be ( B ) f bag (x 0 ) = sign f b (x 0 ). b=1 With bagging, we observe that a committee of classifiers votes on a label. Each classifier is learned on a bootstrap sample from the data set. Learning a collection of classifiers is referred to as an ensemble method.

4 BOOSTING How is it that a committee of blockheads can somehow arrive at highly reasoned decisions, despite the weak judgment of the individual members? Schapire & Freund, Boosting: Foundations and Algorithms Boosting is another powerful method for ensemble learning. It is similar to bagging in that a set of classifiers are combined to make a better one. It works for any classifier, but a weak one that is easy to learn is usually chosen. (weak = accuracy a little better than random guessing) Short history 1984 : Leslie Valiant and Michael Kearns ask if boosting is possible : Robert Schapire creates first boosting algorithm : Yoav Freund creates an optimal boosting algorithm : Freund and Schapire create AdaBoost (Adaptive Boosting), the major boosting algorithm.

5 BAGGING VS BOOSTING (OVERVIEW) Bootstrap sample f 3 (x) Weighted sample f 3 (x) Bootstrap sample f 2 (x) Weighted sample f 2 (x) Bootstrap sample f 1 (x) Weighted sample f 1 (x) Training sample Training sample Bagging Boosting

6 THE ADABOOST ALGORITHM (SAMPLING VERSION) Weighted sample weighted error ε 2 Weighted sample weighted error ε 1 Weighted sample Sample and classify B3 Sample and classify B2 Sample and classify B1 α 3, f 3 (x) α 2, f 2 (x) α 1, f 1 (x) Training sample Boosting ( T ) f boost (x 0 ) = sign α t f t (x 0 ) t=1

7 THE ADABOOST ALGORITHM (SAMPLING VERSION) Algorithm: Boosting a binary classifier Given (x 1, y 1 ),..., (x n, y n ), x X, y { 1, 1}, set w 1 (i) = 1 n for i = 1 : n For t = 1,..., T 1. Sample a bootstrap dataset B t of size n according to distribution w t. Notice we pick (x i, y i) with probability w t(i) and not 1 n. 2. Learn a classifier f t using data in B t. 3. Set ɛ t = ( n i=1 wt(i)1{yi ft(xi)} and αt = 1 ln 1 ɛ t 2 ɛ t ). 4. Scale ŵ t1(i) = w t(i)e αty i f t(x i ) and set w t1(i) = ŵt1(i) j ŵt1(j). Set the classification rule to be f boost (x 0 ) = sign ( T ) t=1 α t f t (x 0 ). Comment: Description usually simplified to learn classifier f t using distribution w t.

8 BOOSTING A DECISION STUMP (EXAMPLE 1) Original data Uniform distribution, w 1 Learn weak classifier Here: Use a decision stump x 1 > 1.7 ŷ = 1 ŷ = 3

9 BOOSTING A DECISION STUMP (EXAMPLE 1) Round 1 classifier Weighted error: ɛ 1 = 0.3 Weight update: α 1 = 0.42

10 BOOSTING A DECISION STUMP (EXAMPLE 1) Weighted data After round 1

11 BOOSTING A DECISION STUMP (EXAMPLE 1) Round 2 classifier Weighted error: ɛ 2 = 0.21 Weight update: α 2 = 0.65

12 BOOSTING A DECISION STUMP (EXAMPLE 1) Weighted data After round 2

13 BOOSTING A DECISION STUMP (EXAMPLE 1) Round 2 classifier Weighted error: ɛ 3 = 0.14 Weight update: α 3 = 0.92

14 BOOSTING A DECISION STUMP (EXAMPLE 1) Classifier after three rounds 0.42 x 0.65 x 0.92 x

15 BOOSTING A DECISION STUMP (EXAMPLE 2) Example problem Random guessing 50% error Decision stump 45.8% error Full decision tree 24.7% error Boosted stump 5.8% error

16 BOOSTING Point = one dataset. Location = error rate w/ and w/o boosting. The boosted version of the same classifier almost always produces better results.

17 BOOSTING (left) Boosting a bad classifier is often better than not boosting a good one. (right) Boosting a good classifier is often better, but can take more time.

18 BOOSTING AND FEATURE MAPS Q: What makes boosting work so well? A: This is a wellstudied question. We will present one analysis later, but we can also give intuition by tying it in with what we ve already learned. The classification for a new x 0 from boosting is ( T ) f boost (x 0 ) = sign t=1 α t f t (x 0 ). Define φ(x) = [ f 1 (x),..., f T (x)], where each f t (x) { 1, 1}. We can think of φ(x) as a high dimensional feature map of x. The vector α = [α 1,..., α T ] corresponds to a hyperplane. So the classifier can be written f boost (x 0 ) = sign(φ(x 0 ) α). Boosting learns the feature mapping and hyperplane simultaneously.

19 APPLICATION: FACE DETECTION

20 FACE DETECTION (VIOLA & JONES, 2001) Problem: Locate the faces in an image or video. Processing: Divide image into patches of different scales, e.g., 24 24, 48 48, etc. Extract features from each patch. Classify each patch as face or no face using a boosted decision stump. This can be done in realtime, for example by your digital camera (at 15 fps). One patch from a larger image. Mask it with many feature extractors. Each pattern gives one number, which is the sum of all pixels in black region minus sum of pixels in white region (total of 45,000 features).

21 FACE DETECTION (EXAMPLE RESULTS)

22 ANALYSIS OF BOOSTING

23 ANALYSIS OF BOOSTING Training error theorem We can use analysis to make a statement about the accuracy of boosting on the training data. Theorem: Under the AdaBoost framework, if ɛ t is the weighted error of classifier f t, then for the classifier f boost (x 0 ) = sign( T t=1 α tf t (x 0 )), training error = 1 n n i=1 ( 1{y i f boost (x i )} exp 2 T ( 1 2 ɛ t) ). 2 t=1 Even if each ɛ t is only a little better than random guessing, the sum over T classifiers can lead to a large negative value in the exponent when T is large. For example, if we set: ɛ t = 0.45, T = 1000 training error

24 PROOF OF THEOREM Setup We break the proof into three steps. It is an application of the fact that if a < b }{{} Step 2 and b < c }{{} Step 3 then a < c }{{} conclusion Step 1 calculates the value of b. Steps 2 and 3 prove the two inequalities. Also recall the following step from AdaBoost: Update ŵ t1 (i) = w t (i)e αtyi ft(xi). Normalize w t1 (i) = ŵt1(i) j ŵt1(j) Define Z t = j ŵt1(j).

25 PROOF OF THEOREM (a b c) Step 1 We first want to expand the equation of the weights to show that w T1 (i) = 1 n T e yi t=1 αt ft(xi) T t=1 Z := 1 t n yi ht (xi) e T t=1 Z t h T (x) := T α t f t (x i ) t=1 Derivation of Step 1: Notice the update rule: w t1 (i) = 1 αtyi ft(xi) w t (i)e Z t Do the same expansion for w t (i) and continue until reaching w 1 (i) = 1 n, f1(xi) e α1yi w T1 (i) = w 1 (i) e αt yi ft (xi) Z 1 Z T The product T t=1 Z t is b above. We use this form of w T1 (i) in Step 2.

26 PROOF OF THEOREM (a b c) Step 2 Next show the training error of f (T) boost (boosting after T steps) is T t=1 Z t. Currently we know w T1 (i) = 1 e y i h T (x i ) n T t=1 Zt w T1 (i) T t=1 Z t = 1 n e y i h T (x i ) & f (T) boost (x) = sign(h T(x)) Derivation of Step 2: Observe that 0 < e z1 and 1 < e z2 for any z 1 < 0 < z 2. Therefore 1 n 1{y i f (T) boost n (x i)} i=1 }{{} a = 1 n i=1 n i=1 yi ht (xi) e n T w T1 (i) Z t = t=1 a is the training error the quantity we care about. T t=1 Z t }{{} b

27 PROOF OF THEOREM (a b c) Step 3 The final step is to calculate an upper bound on Z t, and by extension T t=1 Z t. Derivation of Step 3: This step is slightly more involved. It also shows why α t := 1 2 ln ( 1 ɛt ɛ t ). Z t = = n i=1 αtyi ft(xi) w t (i)e i : y i=f t(x i) e αt w t (i) i : y i f t(x i) e αt w t (i) = e αt (1 ɛ t ) e αt ɛ t Remember we defined ɛ t = i : y i f t(x i) w t(i), the probability of error for w t.

28 PROOF OF THEOREM (a b c) Derivation of Step 3 (continued): Remember from Step 2 that training error = 1 n n 1{y i f boost (x i )} i=1 T Z t. t=1 and we just showed that Z t = e αt (1 ɛ t ) e αt ɛ t. We want the training error to be small, so we pick α t to minimize Z t. Minimizing, we get the value of α t used by AdaBoost: α t = 1 ( ) 1 2 ln ɛt. ɛ t Plugging this value back in gives Z t = 2 ɛ t (1 ɛ t ).

29 PROOF OF THEOREM (a b c) Derivation of Step 3 (continued): Next, rewrite Z t as Z t = 2 ɛ t (1 ɛ t ) = 1 4( 1 2 ɛ t) e x 1 x Then, use the inequality 1 x e x to conclude that Z t = ( 1 4( 1 2 ɛ t) 2) 1 2 ( e 4( 1 2 ɛt)2) 1 2 = e 2( 1 2 ɛt)2.

30 PROOF OF THEOREM Concluding the right inequality (a b c) Because both sides of Z t e 2( 1 2 ɛt)2 are positive, we can say that T t=1 Z t T t=1 e 2( 1 2 ɛt)2 = e 2 T t=1 ( 1 2 ɛt)2. This concludes the b c portion of the proof. Combining everything {}}{ 1 n training error = 1{y i f boost (x i )} n i=1 a b {}}{ T t=1 Z t c {}}{ e 2 T t=1 ( 1 2 ɛt)2. We set out to prove a < c and we did so by using b as a steppingstone.

31 TRAINING VS TESTING ERROR Q: Driving the training error to zero leads one to ask, does boosting overfit? A: Sometimes, but very often it doesn t! C4.5 (tree) testing error Error AdaBoost testing error AdaBoost training error Rounds of boosting