Logistic Regression. CAP 5610: Machine Learning Instructor: Guo-Jun QI
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1 Logstc Regresson CAP 561: achne Learnng Instructor: Guo-Jun QI
2 Bayes Classfer: A Generatve model odel the posteror dstrbuton P(Y X) Estmate class-condtonal dstrbuton P(X Y) for each Y Estmate pror dstrbuton P(Y) Predct the target value Y by axmum A Posteror (AP) Y = argmax Y P Y X = argmax Y P X Y P(Y) It s a generatve model, because Gven each class Y, we can draw (generate) a sample from P(Y X)
3 Bayes Classfer s the optmal classfer Any classfer can never yeld smaller predcton error than Bayes classfer
4 Drawback of a generatve model It s an ndrect method We need to estmate P(Y) and P(X Y) from the tranng set at frst For a classfcaton problem, we do not need any generatve nformaton contaned n P(X Y) Unnecessary assumptons are usually needed to estmate P(X Y) ave Bayes ndependence between features condtonal on the label Y How about drectly learn P(Y X)?
5 Dscrmnatve model Drectly model the posteror dstrbuton P(Y X) Assume some functonal forms for P(Y X), e.g., a lnear model Estmate the parameter of the proposed model drectly from tranng set, Opton 1: axmum lkelhood estmaton (LE) Opton 2: axmum A Posteror (AP)
6 Hnts from generatve model Suppose X s a vector of contnuous features: X=(X 1,, X ) Y s a bnary random varable {,1} ave Bayesan classfer: X s are ndependent gven Y P(Y) s a Bernoull dstrbuton Fnd the form for P(Y=1 X), by Bayes rule
7 Fnd P(Y=1 X) Bayes rule exp and ln wll cancel out. 1 P(Y ) P( X Y ) Assume P(X Y) are Gaussan 1 exp(ln ln ) wth the same varance but P( Y 1) 1 P( X Y 1) dfferent mean 1 1 P(Y ) ( X ;, ) (Y ) 1 exp(ln ln ) 1 exp(ln ) P( Y 1) ( X ;, ) PY ( 1) P X
8 Fnd P(Y X=1) Logstc functon P( Y 1 X ) 1 P(Y ) 1 exp( ln ) 2 PY ( 1) X exp( X )
9 Lnear classfer When Y= P( Y X ) 1 P( Y 1 X ) exp( X ) 1 1 exp( X ) 1 Log lkelhood rato: lnear n X and w log P( Y 1 X ) 1 log P( Y X ) exp( X ) 1 1 X
10 Logstc functon Smooth n b: dfferentable and contnuously dfferentable The logarthm of logstc functon s concave.
11 Extend to multple classes Y={,1,,C}, e.g., C=1 n IST dataset Assume P( Y c X ) exp( w X w ) 1 c c ormalzng P(Y X) such that the summaton over C classes s one. Softmax: the maxmum exponental functon (above) for class c wll have the largest posteror probablty P( Y c X ) exp( w X w ) c c Bnary logstc regresson s a specal case when C=2, and w 1 = C 1 exp( w X w ) c' 1 1 c' c'
12 Learnng logstc regresson model Gven a set of tranng example {(X (m),y (m) m=1,, }, LE corresponds to L ( m) ( m) (w) log P(Y X, w) ( m) ( m) ( m) ( m) ( m) ( m) Y log P(Y 1 X, w) (1 Y )log P(Y X, w) P(Y 1 X, w) Y log log (Y X, w) ( m) ( m) ( m) ( m) ( m) P ( m) ( m) P(Y X, w) ( m) ( m) ( m) Y wx w log 1 wx 1 1 exp( w )
13 LR Optmzaton problem The optmal soluton to w s * ( m) ( m) ( m) w w X w w X w 1 1 w arg max Y log 1 exp( ) The objectve functon s concave. Logstc functon s concave n ts logarthmc form. It has global optmal pont, avodng local optmum. Unfortunately, LR optmzaton problem does not have closed-form soluton.
14 Gradent Ascent ethod Gradent descent method s an teratve algorthm hll clmbng method to fnd the peak pont of a mountan At each pont, compute ts gradent L L L L,,..., w w w 1 Gradent s a vector that ponts to the steepest drecton clmbng up the mountan. At each pont, w s updated so t moves a sze of step λ n the gradent drecton w w L( w)
15 Gradent Ascent ethod LR Loglkelhood: ( m) ( m) ( m) 1 1 L( W ) Y w X w log 1 exp( w X w ) Dervatve to each weght w L w ( m) ( m) Y P( Y 1 X ) L w ( m) ( m) ( m) ( m) Y X X P( Y 1 X ) Update rule ( m) ( m) ( Y ( 1 )) w w P Y X ( m) ( m) ( m) ( m) ( Y ( 1 )) w w X X P Y X
16 Tranng algorthm Input: a set of tranng example {(X (m),y (m) m=1,, } Intalze w=[w, w 1,, w ] Repeat ( m) ( m) ( Y ( 1 )) w w P Y X ( m) ( m) ( m) ( m) ( Y ( 1 )) w w X X P Y X (optonal) check f the loglkelhood ncreases after each update. If not, shrnk the step sze λ Untl convergence
17 Tranng algorthm Input: a set of tranng example {(X (m),y (m) m=1,, } Intalze w=[w, w 1,, w ] Repeat ( m) ( m) ( Y ( 1 )) w w P Y X ( m) ( m) ( m) ( m) ( Y ( 1 )) w w X X P Y X (optonal) check f the loglkelhood ncreases after each update. If not, shrnk the step sze λ Untl convergence
18 Test algorthm Computng the log lkelhood rato P( Y 1 X ) 1 f(x) log log P( Y X ) exp( X ) 1 1 X If >, X s postve (Y=1) If <, X s negatve (Y=) X 2 R 1 f(x) > Lnear decson boundar R 2 f(x) < X 1
19 Stochastc Gradent Ascent ethod akng the learnng algorthm scalable to bg data Typcal gradent ascent method Each update needs to go over all examples to compute the dervatves L w ( m) ( m) Y P( Y 1 X ) L w ( m) ( m) ( m) ( m) Y X X P( Y 1 X ) It wll save us a huge amount of computatons f only a small porton of examples are used
20 Stochastc Gradent Ascent ethod A random subset Ω of examples are drawn from tranng set, whch are used to compute dervatves L w ( m) ( m) Y P( Y 1 X ) L w ( m) ( m) ( m) ( m) Y X X P( Y 1 X ) L w m ( m) ( m) Y P( Y 1 X ) L w ( m) ( m) ( m) ( m) Y X X P( Y 1 X ) Especally, only one example s drawn from tranng set to compute the dervatve m
21 Stochastc Gradent Ascent ethod Pros: Each round of update only needs very small number of examples Tme complexty: ore rapd update Space complexty: Only upload the examples drawn n each round to the memory Cons: Cannot guarantee convergence Practce: very effcent
22 Opton 2: axmum A Posteror (AP) Add a pror dstrbuton on w P(w) = (, σ 2 I) Posteror dstrbuton on w can be gven by ( m) ( m) log P( w D) log P( w) log P(Y X, w) log P( w) L( w) The optmal soluton 1 w argmax L(w) log P( w) argmax L(w) w 2 * 2 w w 2 2
23 L 2 regularzer L 2 Regularzed loglkelhood objectve functon 1 w argmax L(w) log P( w) argmax L(w) w 2 * 2 w w 2 2 Role of a regularzer Reduce the overfttng, by mposng a pror knowledge on w keepng the weghts closer to zero
24 Phlosophy behnd L 2 regularzer Why should we prefer a zero weght w n LR? axmum entropy prncple: do OT make a bas towards one of two outcomes unless you have enough nformaton (least bas). A con flppng: how do you predct the outcome f you do not know any attrbutes about the con? When w =, P(Y=1 X)=P(Y= X)=.5 (equal probablty) P( Y 1 X ) exp( X )
25 Comparson between ave Bayes and LR umber of model parameters ave Bayes: for each attrbute, two parameters (mean and varance), for two classes; two class pror dstrbuton 4+2 LR: weght coeffcents ore parameters usually mean more tranng examples to estmate them. Assumptons: ave Bayes: condtonal ndependence gven the class label LR: no such ndependence assumpton (why?) Wth weaker assumpton, LR may outperform ave Bayes f the assumpton does not hold.
26 Summary Derve LR from ave Bayes wth Gaussan class-condtonal dstrbuton LE: Estmate the coeffcent weghts n LR Gradent ascent method Stochastc gradent ascent method AP: add a pror on w A L 2 regularzer ext lecture: lnear regresson for contnuous output varable (PRL Chapter 3)
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