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1、Machine Learning Foundations (_hx?)Lecture 2: Learning to Answer Yes/NoHsuan-Tien Lin (?0).twDepartment of Computer Science & Information EngineeringNationalUniversity( ?c'x? ? ?)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations0/22Learning to Answer Yes/No1 When Can Mac

2、hines Learn?2 Why Can Machines Learn?3 How Can Machines Learn?4 How Can Machines Learn Better?Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations1/22Lecture 2: Learning to Answer Yes/NoPerceptron Hypothesis Set Perceptron Learning Algorithm (PLA) Guarantee of PLANon-Separable DataLecture 1: The Le

3、arning ProblemA takes D and H to get gLearning to Answer Yes/NoPerceptron Hypothesis SetCredit Approval Problem RevisitedApplicant Informationunknown target functionf : X Y(ideal credit approval formula)learning algorithm A(historical records in bank)(learned formula to be used)hypothesis setH(set o

4、f candidate formula)what hypothesis set can we use?Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations2/22final hypothesisg ftraining examplesD : (x1, y1), · · · , (xN , yN )age23 yearsgenderfemaleannual salaryNTD 1,000,000year in residence1 yearyear in job0.5 yearcurrent debt200,00

5、0Learning to Answer Yes/NoPerceptron Hypothesis SetA Simple Hypothesis Set: the PerceptronFor x = (x1, x2, · · · , xd ) features of customer, compute aweighted score andapprove credit ifdeny credit ifXdi=1 di=1w x > thresholdi iXw x < thresholdi i Y: +1(good), 1(bad) , 0 ignored

6、linear formula h H are !dXh(x) = signwi xi thresholdi=1called perceptron hypothesis historicallyHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations3/22age23 yearsannual salaryNTD 1,000,000year in job0.5 yearcurrent debt200,000Learning to Answer Yes/NoPerceptron Hypothesis SetVector Form of Percept

7、ron Hypothesis !dXh(x)= signwi xithresholdi=1 !dX= signwi xi+ (threshold) · (+1)|wz0 | xz0 i=1 !dX= signwi xii=0= sign wT x each tall w represents a hypothesis h & is multiplied withtall x will usl versions to simplify notationwhat do perceptrons h look like?Hsuan-Tien Lin (NTU CSIE)Machine

8、 Learning Foundations4/22Learning to Answer Yes/NoPerceptron Hypothesis SetPerceptrons in R2h(x) = sign (w0 + w1x1 + w2x2)points on the plane (or points in Rd )customer features x: labels y :hypothesis h: (+1), × (-1)lines (or hyperplanes in Rd )positive on one side of a line, negative on the o

9、ther sidedifferent line classifies customers differentlyperceptrons linear (binary) classifiersHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations5/22Learning to Answer Yes/NoPerceptron Hypothesis SetFun TimeConsider using a perceptron to detect spam messages.Assume that eachis represented by the

10、frequency of keywordoccurrence, and output +1 indicates a spam. Which keywords below shall have large positive weights in a good perceptron for the task?1 coffee, tea, hamburger, steak2 free, drug, fantastic, deal3 machine, learning, statistics, textbook4 national, university, courseraHsuan-Tien Lin

11、 (NTU CSIE)Machine Learning Foundations6/22Learning to Answer Yes/NoPerceptron Hypothesis SetFun TimeConsider using a perceptron to detect spam messages.Assume that eachis represented by the frequency of keywordoccurrence, and output +1 indicates a spam. Which keywords belowshall have large positive

12、 weights in a good perceptron for the task?1 coffee, tea, hamburger, steak2 free, drug, fantastic, deal3 machine, learning, statistics, textbook4 national, university, courseraReference Answer: 2The occurrence of keywords with positive weights increase the spam score, and hence those keywords should

13、 often appear in spams.Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations6/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Select g from HH = all possible perceptrons, g =?want: g f (hard when f unknown)almost necessary: g f on D, ideallyg(xn) = f (xn) = yndifficult: H is of infinit

14、e sizeidea: start from some g0, and correct its mistakes on Dwill represent g0 by its weight vector w0Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations7/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Perceptron Learning Algorithmstart from some w0 (say, 0), and correct its mistake

15、s on Dw+y xxFor t = 0, 1, . . .1 find a mistake of wt called xn(t), yn(t) w Tsign w x=6ytn(t )n(t )w2 (try to) correct the mistake byw. . . until no more mistakes return last w (called wPLA) as gxxt +1 wt + yn(t )xn(t )Thats it!A fault confessed is half redressed. :-)Hsuan-Tien Lin (NTU CSIE)Machine

16、 Learning Foundations8/22y=+1w+yy=1Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Practical Implementation of PLAstart from some w0 (say, 0), and correct its mistakes on DCyclic PLAFor t = 0, 1, . . .1 find the next mistake of wt called xn(t), yn(t) Tsign w x=6ytn(t )n(t )2 correct the

17、mistake bywt +1 wt + yn(t )xn(t ). . . until a full cycle of not encountering mistakesnext can follow naïve cycle (1, · · · , N)or precomputed random cycleHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations9/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing i

18、s Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual

19、purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answe

20、r Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm w

21、ith < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE

22、)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algo

23、rithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi

24、 x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/

25、22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Seeing is Believingworked like a charm with < 20 lines!(note: made xi x0 = 1 for visual purpose)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations10/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Some Remaining Issues

26、 of PLAcorrect mistakes on D until no mistakesAlgorithmic: halt (with no mistake)?naïve cyclic: ?random cyclic: ? other variant: ?Learning: g f ?on D, if halt, yes (no mistake) outside D: ?if not halting: ?to be shown if (.), after enough corrections,any PLA variant haltsHsuan-Tien Lin (NTU CSI

27、E)Machine Learning Foundations11/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Fun TimeLets try to think about why Py work.Let n = n(t ), according to the rule of PLA below, which formula is true? Tsign w x=y ,wt +1 wt + ynxn6nntwTxn = yn1t +12 sign(wT xn) = ynt +13 ynwT x y w xTnnnt

28、 +1t4 ynwT xn < ynwT xnt +1tHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations12/22Learning to Answer Yes/NoPerceptron Learning Algorithm (PLA)Fun TimeLets try to think about why Py work.Let n = n(t ), according to the rule of PLA below, which formula is true? Tsign w x=y ,wt+1 wt + ynxn6nntwT

29、xn = yn1t +12 sign(wT xn) = ynt +13 ynwT x y w xTnnnt +1t4 ynwT xn < ynwT xnt +1tReference Answer: 3Simply multiply the second part of the rule by ynxn. The result shows that the rule somewhat tries to correct the mistake.Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations12/22Learning to Answe

30、r Yes/NoGuarantee of PLALinear Separability if PLA halts (i.e. no more mistakes),(necessary condition) D allows some w to make no mistake call such D linear separable(linear separable)(not linear separable)(not linear separable)assume linear separable D,does PLA always halt?Hsuan-Tien Lin (NTU CSIE)

31、Machine Learning Foundations13/22Learning to Answer Yes/NoGuarantee of PLAPLA Fact: wt Gets More Aligned with wflinear separable D exists perfect wf such that yn = sign(wT xn)f wf perfect hence every xn correctly away from line:TTyw xmin y w x> 0n(t )n(t )nnffnwT wt by updating with any xn(t), yn

32、(t) f TTw w= ww + yxt +1tn(t )ffn(t )TT wf wt+ min ynwf xnn+ 0.> wT wtfwt appears more aligned with wf after update(really?)Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations14/22Learning to Answer Yes/NoGuarantee of PLAPLA Fact: wt Does Not Grow Too Fastwt changed only when mistake TT sign w

33、x6= y yw x 0n(t )n(t )n(t )n(t )tt mistake limits kwt k2 growth, even when updating with longest xn22kwt+1kkwt + yn(t )xn(t )k=2T2kw k + 2yw x+ kyxktn(t )tn(t )n(t ) n(t )22kwt k+ 0 + kyn(t )xn(t )k22kwt k + max ky x kn nnstart from w0 = 0, after T mistake corrections,wT wTfT · constantkwf k kw

34、T kHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations15/22Learning to Answer Yes/NoGuarantee of PLAFun TimeLets upper-bound T , the number of mistakes that PLA corrects. wTDefine R2 = max k2f xnk = min yxnnkwf knnWe want to show that T . Express the upper bound by the twoterms above.1 R/2 R2/23 R

35、/24 2/R2Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations16/22Learning to Answer Yes/NoGuarantee of PLAFun TimeLets upper-bound T , the number of mistakes that PLA corrects. wTDefine R2 = max k2f xnk = min yxnnkwf knnWe want to show that T . Express the upper bound by the twoterms above.1 R/2 R2

36、/23 R/24 2/R2Reference Answer:2wTwtTheum value ofis 1. Since Tfkwf k kwt kmistake corrections increase the innerproduct by T · constant, theum2number of corrected mistakes is 1/constant .Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations16/22Learning to Answer Yes/NoNon-Separable DataMore ab

37、out PLAGuaranteeas long as linear separable and correct by mistakeinner product of wf and wt grows fast; length of wt grows slowly PLA lines are more and more aligned with wf haltsProssimple to implement, fast, works in any dimension dCons assumes linear separable D to haltproperty unknown in advanc

38、e (no need for PLA if we know wf ) not fully sure how long halting takes ( depends on wf )though practically fastwhat if D not linear separable?Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations17/22Learning to Answer Yes/NoNon-Separable DataLearning with Noisy Dataunknown target functionf : X Y+

39、 noise(ideal credit approval formula)learning algorithm A(historical records in bank)(learned formula to be used)hypothesis setH(set of candidate formula)how to at least get g f on noisy D?Hsuan-Tien Lin (NTU CSIE)Machine Learning Foundations18/22final hypothesisg ftraining examplesD : (x1, y1),

40、83; · · , (xN , yN )Learning to Answer Yes/NoNon-Separable DataLine with Noise Toleranceassume little noise: yn = f (xn) usuallyif so, g f on D yn = g(xn) usuallyhow aboutNrzXTw argminy =6sign(w x )gnnwn=1NP-hard to solve, unfortunatelycan we modify PLA to get an approximately good g?Hsuan

41、-Tien Lin (NTU CSIE)Machine Learning Foundations19/22Learning to Answer Yes/NoNon-Separable DataPockgorithmmodify PLA algorithm (black lines) by keeping best weights in pocketinitialize pocket weights wFor t = 0, 1, · · ·1 find a (random) mistake of wt called (xn(t), yn(t)2 (try to) c

42、orrect the mistake bywt +1 wt + yn(t )xn(t )3 if wt+1 makes fewer mistakes than w , replace w by wt+1.until enough iterationsreturn w (called wPOCKET) as ga simple modification of PLA to find (somewhat) best weightsHsuan-Tien Lin (NTU CSIE)Machine Learning Foundations20/22Learning to Answer Yes/NoNo

43、n-Separable DataFun TimeShould we use pocket or PLA?Since we do not know whether D is linear separable in advance, wemay decide to just go with pocket instead of PLA. If D is actually linearseparable, whats the difference between the two?1 pocket on D is slower than PLA2 pocket on D is faster than PLA3 pocke