Machine Learning A Geometric Approach

Transcription

1 Machine Learning A Geometric Approach Linear Classification: Perceptron Professor Liang Huang some slides from Alex Smola (CMU)

2 Perceptron Frank Rosenblatt

3 deep learning multilayer perceptron linear regression perceptron SVM CRF structured perceptron

4 Brief History of Perceptron 1959 osenblatt invention batch +max margin online 1962 Novikoff proof +kernels 1969* Minsky/Papert book killed it +soft-margin minibatch conservative updates DEAD inseparable case 1997 Cortes/Vapnik SVM online approx. max margin 1999 Freund/Schapire voted/avg: revived subgradient descent 2003 Crammer/Singer MIRA minibatch * Singer group Pegasos 2006 Singer group aggressive *mentioned in lectures but optional (others papers all covered in detail) AT&T Research 2002 Collins structured 2005* McDonald/Crammer/Pereira structured MIRA ex-at&t and students

5 Neurons Soma (CPU) Cell body - combines signals Dendrite (input bus) Combines the inputs from several other nerve cells Synapse (interface) Interface and parameter store between neurons Axon (output cable) May be up to 1m long and will transport the activation signal to neurons at different locations

6 Neurons x 1 x 2 x 3... x n w 1 w n synaptic weights output X f(x) = σ( w i x i = hw, xi) i

7 Frank Rosenblatt s Perceptron

8 Multilayer Perceptron (Neural Net)

9 Perceptron w/ bias Weighted linear x 1 x 2 x 3... x n combination Nonlinear decision function Linear offset (bias) synaptic weights w 1 output w n f(x) = (hw, xi + b) Linear separating hyperplanes Learning: w and b

10 Perceptron w/o bias Weighted linear x 0 = 1 x 1 x 2 x 3... x n combination w 0 w 1 w n Nonlinear decision function synaptic weights No Linear offset (bias): hyperplane through the origin output f(x) = (hw, xi + b) Linear separating hyperplanes Learning: w

11 Augmented Space 1 O can separate in 2D from the origin can t separate in 1D from the origin can separate in 3D from the origin can t separate in 2D from the origin

12 Perceptron Ham Spam

13 The Perceptron w/o bias initialize w = 0 and b =0 repeat if y i [hw, x i i + b] apple 0 then w w + y i x i and b b + y i end if until all classified correctly Nothing happens if classified correctly Weight vector is linear combination Classifier is linear combination of w = X i2i y i x i inner products f(x) = X i2i y i hx i,xi + b σ( )

14 The Perceptron w/ bias initialize w = 0 and b =0 repeat if y i [hw, x i i + b] apple 0 then w w + y i x i and b b + y i end if until all classified correctly Nothing happens if classified correctly Weight vector is linear combination Classifier is linear combination of w = X i2i y i x i inner products f(x) = X i2i y i hx i,xi + b σ( )

15 Demo x i w (bias=0)

16 Demo

17 Demo

18 Demo

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20 Convergence Theorem If there exists some oracle unit vector u : kuk = 1 y i (u x i ) for all i then the perceptron converges to a linear separator after a number of updates bounded by R 2 / 2 where R = maxkx i k i Dimensionality independent Order independent (but order matters in output) Dataset size independent Scales with difficulty of problem

21 Geometry of the Proof part 1: progress (alignment) on oracle projection assume w i is the weight vector before the ith update (on hx i, y i i) and assume initial w 0 = 0 w i+1 = w i + y i x i u w i+1 = u w i + y i (u x i ) u w i+1 u w i + u w i+1 i projection on u increases! y i (u x i ) x i w i+1 for all i (more agreement w/ oracle) u : kuk = 1 kw i+1 k = kukkw i+1 k u w i+1 i w i

22 Geometry of the Proof part 2: bound the norm of the weight vector w i+1 = w i + y i x i kw i+1 k 2 = kw i + y i x i k 2 = kw i k 2 + kx i k 2 + 2y i (w i x i ) apple kw i k 2 + R 2 mistake on x_i apple ir 2 (radius) x i w i+1 Combine with part 1 kw i+1 k = kukkw i+1 k u w i+1 i u : kuk = 1 i apple R 2 / 2 w i

23 Convergence Bound R 2 / 2 w is independent of: dimensionality number of examples starting weight vector order of examples constant learning rate and is dependent of: but test accuracy is dependent of: order of examples (shuffling helps) variable learning rate (1/total#error helps) can you still prove convergence? separation difficulty feature scale

24 Hardness margin vs. size hard easy

25 XOR XOR - not linearly separable Nonlinear separation is trivial Caveat from Perceptrons (Minsky & Papert, 1969) Finding the minimum error linear separator is NP hard (this killed Neural Networks in the 70s).

26 Brief History of Perceptron 1959 osenblatt invention batch +max margin online 1962 Novikoff proof +kernels 1969* Minsky/Papert book killed it +soft-margin minibatch conservative updates DEAD inseparable case 1997 Cortes/Vapnik SVM online approx. max margin 1999 Freund/Schapire voted/avg: revived subgradient descent 2003 Crammer/Singer MIRA minibatch * Singer group Pegasos 2006 Singer group aggressive *mentioned in lectures but optional (others papers all covered in detail) AT&T Research 2002 Collins structured 2005* McDonald/Crammer/Pereira structured MIRA ex-at&t and students

27 Extensions of Perceptron Problems with Perceptron doesn t converge with inseparable data update might often be too bold doesn t optimize margin is sensitive to the order of examples Ways to alleviate these problems voted perceptron and average perceptron MIRA (margin-infused relaxation algorithm)

28 Voted/Avged Perceptron motivation: updates on later examples taking over! voted perceptron (Freund and Schapire, 1999) record the weight vector after each example in D (not just after each update) and vote on a new example using D models shown to have better generalization power averaged perceptron (from the same paper) an approximation of voted perceptron just use the average of all weight vectors can be implemented efficiently

29 Voted Perceptron

30 Voted/Avged Perceptron (low dim - less separable) test error

31 Voted/Avged Perceptron (high dim - more separable) test error

32 Averaged Perceptron voted perceptron is not scalable and does not output a single model avg perceptron is an approximation of voted perceptron initialize w = 0 and w 0 b =0 repeat c c +1 if y i [hw, x i i + b] apple 0 then w w + y i x i and b b + y i wend 0 ifw 0 + w until all classified correctly output w 0 /c after each example, not each update 32

33 Efficient Implementation of Averaging naive implementation (running sum) doesn t scale very clever trick from Daume (2006, PhD thesis) initialize w = 0 and w a b =0 repeat c c +1 if y i [hw, x i i + b] apple 0 then w w + y i x i and b b + y i end w a if w a + cy i x i until all classified correctly output w w a /c w t w (0) = w (1) = w (2) = w (3) = w (1) w (1) w (2) w (1) w (2) w (3) t w w (4) = w (1) w (2) w (3) w (4) 33

34 MIRA perceptron often makes too bold updates but hard to tune learning rate the smallest update to correct the mistake? w i+1 = w i + y i w i x i kx i k 2 x i easy to show: y i (w i+1 x i ) = y i (w i + y i margin-infused relaxation algorithm (MIRA) w i x i kx i k 2 x i ) x i = 1 kx i k w i x i 1 kx i k perceptron overcorrects this mistake w i x i perceptron MIRA 1 w i x i kx i k

35 Perceptron x i perceptron w perceptron undercorrects this mistake (bias=0)

36 MIRA min kw 0 wk 2 w 0 s.t. w 0 x 1 minimal change to ensure margin MIRA x i MIRA makes sure after update, dotproduct w x_i = 1 MIRA 1-step SVM perceptron margin of 1/ x_i w perceptron undercorrects this mistake (bias=0)

37 Aggressive MIRA aggressive version of MIRA also update if correct but margin isn t big enough functional margin: y i (w x i ) geometric margin: y i (w x i ) kwk update if functional margin is <=p (0<=p<1) update rule is same as MIRA called p-aggressive MIRA (MIRA: p=0) larger p leads to a larger geometric margin but slower convergence

38 Aggressive MIRA p=0.2 perceptron p=0.9

39 Demo perceptron vs. 0.2-aggressive vs. 0.9-aggressive p=0.2 perceptron p=0.9

40 Demo perceptron vs. 0.2-aggressive vs. 0.9-aggressive why does this dataset so slow to converge? perceptron: 22, p=0.2: 87, p=0.9: 2,518 epochs answer: margin shrinks in augmented space! 1 small margin in 2D O big margin in 1D

41 Demo perceptron vs. 0.2-aggressive vs. 0.9-aggressive why does this dataset so fast to converge? perceptron: 3, p=0.2: 1, p=0.9: 5 epochs answer: margin shrinks in augmented space! 1 ok margin in 2D O big margin in 1D

42 What if data is not separable in practice, data is almost always inseparable wait, what does that mean?? perceptron cycling theorem (1970) weights will remain bounded and not diverge use dev set for when to stop (prevents overfitting) higher-order features by combining atomic ones kernels => separable in higher dimensions

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44 Solving XOR (x 1,x 2 ) (x 1,x 2,x 1 x 2 ) XOR not linearly separable Mapping into 3 dimensions makes it easily solvable

45 Useful Engineering Tips: averaging, shuffling, variable learning rate, fixing feature scale averaging helps significantly; MIRA helps a tiny little bit perceptron < MIRA < avg. perceptron avg. MIRA shuffling the data helps hugely if classes were ordered shuffling before each epoch helps a little bit variable learning rate often helps a little 1/(total#updates) or 1/(total#examples) helps any requirement in order to converge? how to prove convergence now? centering of each feature dim helps why? => R smaller, margin bigger unit variance also helps (why?) 0-mean, 1-var => each feature a unit Gaussian 1 O 1 O

46 Useful Engineering Tips: categorical=>binary, feature bucketing (binning/quantization) Age, Workclass, Education, Marital_Status, Occupation, Race, Sex, Hours, Country, Target 40, Private, Doctorate, Married-civ-spouse, Prof-specialty, White, Male, 60, United-States, >50K HW1 Adult income dataset: <=50K, or >50K? 2 numerical features age and hours-per-week option 1: treat them as numerical features but is older and more hours always better? option 2: treat them as binary features e.g., age=22, hours=38,... option 3: bin them (e.g., age=0-25, hours= ) 7 categorical features: convert to binary features country, race, occupation, etc. e.g., country:united_states, education:doctorate,... optional: you can probably add a numerical feature edu_level perceptron: ~20% error, avg. perceptron: ~16% error

47 Brief History of Perceptron 1959 osenblatt invention batch +max margin online 1962 Novikoff proof +kernels 1969* Minsky/Papert book killed it +soft-margin minibatch conservative updates DEAD inseparable case 1997 Cortes/Vapnik SVM online approx. max margin 1999 Freund/Schapire voted/avg: revived subgradient descent 2003 Crammer/Singer MIRA minibatch * Singer group Pegasos 2006 Singer group aggressive *mentioned in lectures but optional (others papers all covered in detail) AT&T Research 2002 Collins structured 2005* McDonald/Crammer/Pereira structured MIRA ex-at&t and students