CHAPTER 2: Supervised Learning

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1 HATER 2: Supervsed Learnng

2 Learnng a lass from Eamples lass of a famly car redcon: Is car a famly car? Knowledge eracon: Wha do people epec from a famly car? Oupu: osve (+) and negave ( ) eamples Inpu represenaon: : prce, 2 : engne power 2

3 Tranng se X X N {,r } r f 0 f s posve s negave 2 3

4 lass p prce p AND e engne power e 2 2 4

5 Hypohess class H h( ) f 0 f h says s posve h says s negave Error of h on H E( h X ) N h r 5

6 S, G, and he Verson Space mos specfc hypohess, S mos general hypohess, G h H, beween S and G s conssen and mae up he verson space (Mchell, 997) 6

7 Margn hoose h wh larges margn 7

8 V Dmenson N pons can be labeled n 2 N ways as +/ H shaers N f here ess h H conssen for any of hese: V(H ) = N An as-algned recangle shaers 4 pons only! 8

9 robably Appromaely orrec (A) Learnng How many ranng eamples N should we have, such ha wh probably a leas δ, h has error a mos ε? (Blumer e al., 989) Each srp s a mos ε/4 r ha we mss a srp ε/4 r ha N nsances mss a srp ( ε/4) N r ha N nsances mss 4 srps 4( ε/4) N 4( ε/4) N δ and ( ) ep( ) 4ep( εn/4) δ and N (4/ε)log(4/δ) 9

10 Nose and Model ompley Use he smpler one because Smpler o use (lower compuaonal compley) Easer o ran (lower space compley) Easer o eplan (more nerpreable) Generalzes beer (lower varance - Occam s razor) 0

11 Mulple lasses, =,...,K N,r } { X, f f j r j 0, f f j h j 0 Tran hypoheses h (), =,...,K:

12 Regresson X r r f N, r g w w0 g 2 w2 w w0 E N N gx r g E N N w w r w w, 0 X

13 Model Selecon & Generalzaon Learnng s an ll-posed problem; daa s no suffcen o fnd a unque soluon The need for nducve bas, assumpons abou H Generalzaon: How well a model performs on new daa Overfng: H more comple han or f Underfng: H less comple han or f 3

14 Trple Trade-Off There s a rade-off beween hree facors (Deerch, 2003):. ompley of H, c (H), 2. Tranng se sze, N, 3. Generalzaon error, E, on new daa As N,E As c (H),frs Eand hen E 4

15 ross-valdaon To esmae generalzaon error, we need daa unseen durng ranng. We spl he daa as Tranng se (50%) Valdaon se (25%) Tes (publcaon) se (25%) Resamplng when here s few daa 5

16 Dmensons of a Supervsed Learner. Model: g 2. Loss funcon: 3. Opmzaon procedure: E X L r, g * arg mne X 6

17 HATER 3: Bayesan Decson Theory

18 robably and Inference Resul of ossng a con s {Heads,Tals} Random var X {,0} Bernoull: {X=} = p X o ( p o )( X) Sample: X = { } N = Esmaon: p o = # {Heads}/#{Tosses} = / N redcon of ne oss: Heads f p o > ½, Tals oherwse 8

19 lassfcaon red scorng: Inpus are ncome and savngs. Oupu s low-rs vs hgh-rs Inpu: = [, 2 ] T,Oupu: Î {0,} redcon: choose or choose f 0 oherwse f ( ( 0 oherwse,, 2 2 ) ) 0. 5 ( 0, 2 ) 9

20 Bayes Rule p p p p p p 20 poseror lelhood pror evdence

21 Bayes Rule: K>2 lasses K p p p p ma f choose and K 0 2

22 Losses and Rss Acons: α Loss of α when he sae s : λ Epeced rs (Duda and Har, 973) R K choose f R mn R 22

23 Losses and Rss: 0/ Loss f f 0 K R 23 For mnmum rs, choose he mos probable class

24 Losses and Rss: Rejec 0 0 oherwse f f, K K K R R oherwse rejec and f choose 24

25 Dscrmnan Funcons choose g f p g ma R g g,,, K K decson regons R,...,R K R g ma g 25

26 K=2 lasses Dchoomzer (K=2) vs olychoomzer (K>2) g() = g () g 2 () f g choose 0 2 oherwse Log odds: log 2 26

27 Uly Theory rob of sae gven edence : (S ) Uly of α when sae s : U Epeced uly: EU U S hoose α f EU ma EU j j 27

28 Assocaon Rules Assocaon rule: X Y eople who buy/clc/vs/enjoy X are also lely o buy/clc/vs/enjoy Y. A rule mples assocaon, no necessarly causaon. 28

29 Assocaon measures Suppor (X Y): X, Y # cusomers who bough # cusomers X and Y onfdence (X Y): Y X Lf (X Y): X, Y ( X) ( Y) ( Y X) ( Y) X, Y ( X) # cusomers who bough X and # cusomers who bough X Y 29

30 Apror algorhm (Agrawal e al., 996) For (X,Y,Z), a 3-em se, o be frequen (have enough suppor), (X,Y), (X,Z), and (Y,Z) should be frequen. If (X,Y) s no frequen, none of s superses can be frequen. Once we fnd he frequen -em ses, we conver hem o rules: X, Y Z,... and X Y, Z,... 30

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