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CHNG 2HC C GIM STSUPERVISED LEARNING

Ni dung

2.1 Khi nim 2.2 Hi quy 2.2.1 Gradient descent 2.2.2 Phng php o hm 2.2.3 Phng php xc sut 2.2.4 Hi quy trng s cc b 2.2.5 M hnh tuyn tnh tng qut 2.2.6 Phn loi nhiu lp

KHI NIM Hc c gim st :

u vo l b d liu cng vi cc gi tr u ra ng tng ng. Hc mi quan h gia u vo v gi tr u ra T to ra cc gi tr u ra ng cho cc d liu u vo mi

KHI NIM

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KHI NIM Mt s k hiu

m : s lng d liu hun luyn x : gi tr cc bin u vo/ gi tr cc c tnh ca 1 mu y : gi tr u ra/ch tng ng

Cp (x, y) m t 1 mu hun luyn Trong tp d liu hun luyn , l mu hun luyn

th i

Hi quy bt u hc ta phi chn cch biu din gi thuyt

VD. Trong bi ton d on gi phng ta chn

: din tch phng : l cc tham s biu din mi quan h gia v

Trong trng hp mu c nhiu thuc tnh (, , . . , ) . .

(vi 1)

(vi , , . . , , v 1, , . . , )

Ta phi chn tham s sao cho gn nht vi

Hi quy Hm chi ph

1

2

Biu din mc gn ca trn ton b tp hun luyn

Ta chn tham s sao cho lm ti thiu hm

Hi quy

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Thut ton Gradient Descent Thut ton Gradient Descent Thut ton gradient descent

Bt u vi 1 gi tr khi to ngu nhin ca

Lp li bc cp nht

Vi 0, . . , Cho ti khi hi t.

l hng s dng, l tc hc, v thng chn l gi tr nh

Thut ton Gradient Descent Vich1muhunluyn

1

2

Lut cp nht (vi 1 mu)

Thut ton Gradient Descent

Tng qut cho mu hun luyn

Lut cp nht trn c lp i lp li cho n khi hi t

Thut ton trn c gi l : batch gradient descent

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Thut ton Gradient Descent Thut ton stochastic gradient descent

(incremental gradient descent)

Lp for 1 to do

Thut ton Gradient Descent So snh batch gradient descent v stochastic gradient

descent: C hai u l thut ton tm kim cc b batch gradient descent phi duyt ton b cc mu

hun luyn trong mi bc lp (chi ph cao nu kch thc tp hun luyn ln).

stochastic gradient descent duyt ln lt tng mu, nn tm c ti im cc tr nhanh hn. Tuy nhin c th n khng th hi t c n im cc tr m ch dao ng xung quanh.

Thut ton Gradient Descent Thut ton Gradient Descent K hiu matrix

Lutcpnhtcathuttongradientdescent

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Phng php o hm Tm s dng phng php o hm Tp hun luyn u vo

Vi 1 ( 1, . . , )

Phng php o hm Gi tr ch

, . . ,

Cc tham s , . . ,

Ta c

Phng php o hm Vi l vector bt k th

Biu din li

1

2

1

2

Phng php o hm

1

2

1

2

. . ccbcbini

1

2 2

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Phng php o hm S dng phng trnh chun tc (Normal Equation)

ti thiu ta gn o hm bc nht ca n bngkhng

0

Tathu c

Hi quy Hi quy tuyn tnh

Hi quy Hi quy a thc

Hi quy a thc bc hai :

Hi quy a thc bc k : . .

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Hi quy Tm thng qua cch biu din bng xc sut

l h s li Gi s l cc gi tr c lp v c cng phn b (IID),

v n tun theo phn phi chun Gaussian ~0,

1

2exp

2

Suy ra

; 1

2exp

2

Hi quy

Normal distribution

Hi quy Cho cc mu hun luyn , v , hi phn phi ca l g ?

Xc sut ca d liu c cho bi |; Mi quan h gia , v c biu din bng hm ca l

; , |;

V ta gi s l c lp (do l c lp khi bit ), ta c th vit li hm kh nng (likelihood) nh sau:

;

1

2exp

2

Hi quy Chng ta mun chn sao cho kh nng ny l ln nht

(maximum likelihood): tm sao cho

1

2exp

2

t max.

Mo: tm sao cho log t max th n cng lm t max.

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Hi quy

log log1

2exp

2

log1

2exp

2

log1

2

1

1

2

Tng ng vi tm ti thiu1

2

Hi quy

Nhn xt: Vi gi s v xc sut trn d liu th hi quy bnh

phng sai s nh nht tng ng vi tm c lng kh nng ln nht ca

Vic la chn khng ph thuc vo trong gi s m hnh xc sut ca d liu.

Hi quy Hi quy tuyn tnh trng s cc b (Locally weighted

linear regression)

Hi quy tuyn tnh trng s cc b

Hi quy tuyn tnh trng s cc b tm ti thiu

l s khng m c gi l trng s

exp

2

l tham s rng bng thng (bandwidth)

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Hi quy Thut ton hc tham s - parametric algorithm (vd hi quy

khng trng s): ta cn mt s lng hu hn, c nh cctham s . Sau khi ta tm c b gi tr tham s ph hpth ta khng cn lu gi li b d liu hun luyn don na.

Thut ton hc khng tham s - non-parametric algorithm(vd. Hi quy tuyn tnh trng s cc b): d on trnghp mi ta lun cn lu gi cc d liu trong tp hunluyn. Ta khng c mt m hnh tng qut d onchung.

PHN LOI V HI QUYLOGIC

Phn loi Xt trng hp phn loi nh phn

0,1

VD. Phn loi th rc, phn loi bi bo a thch,

1

0

Hi quy logic B qua vic l cc gi tr ri rc, s dng m hnh hi

quy trc d on vi u vo . Hiu qu rt ti Cc gi tr 1 hoc 0 ca khng c xt

Ci tin: thay i dng ca

1

1

gi l hm logistic hoc hm sigmoid

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Hi quy logic Hi quy logic

1

1

o hm ca hm sigmoid

1

1

1

1

1

1 1

1

1

1

Hi quy logic

Tm ? Xy dng hm kh nng (likelihood) Tm sao cho hm kh nng t gi tr ln nht trn

tp hun luyn(Xy dng thut ton lp gradient descent tm )

Xy dng hm kh nng (likelihood) Gi s

1 ;

0 ; 1

Vit li ;

1

Gi s m mu hun luyn c sinh ra c lp

; ;

1

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Tm Tm sao cho lm ti a logarithm ca hm kh nng log

log 1 log1

Lm th no ti a gi tr ca ? S dng thut ton lp gradient ascent

Lut cp nht:

Tm

1

1

1

1

1

1

1

1 1

1 1

Gradient ascent

Lut cp nht vi gradient ascent

Thut ton khc tm max Thut ton lp Newton: s dng tm im 0 ca hm.

:

Tm sao cho 0

Lut cp nht trong bc lp:

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Phng php Newton-Raphson Phng php Fischer scoring p dng vo tm gi tr sao cho cc i.

Tm im khng ca hm

Lut cp nht :

Hay :

Trong

M hnh tuyn tnh tng qut

Trong cc phn trc ta c |; ~, trong hi quy |; ~Bernoulli trong hi quy logic

Tt c cc trng hp ny u thuc v mt h m hnh l m hnh tuyn tnh tng qut (Generalized Linear Models)

M hnh tuyn tnh tng qut H phn phi dng hm m

;

Trong : : tham s t nhin (natural parameter) : sufficient statistic (thng trong cc trng hp

phn phi c xem xt th ) : log partition function : hng s chun ha ( tng cc ; theo

bng 1)

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H phn phi dng hm m Phn phi Bernoulli:

; 1

exp log 1 log1

exp log

1 log1

Ta c:

log

,suyra

log 1 log1

1

H phn phi dng hm m Phn phi Gaussian:

Nhn xt: trong hi quy tuyn tnh vic la chn khng ph thuc vo do ta c th chn ty (VD. Chn 1)

; 1

2exp

1

2

1

2exp

1

2 exp

1

2

H phn phi dng hm m Phn phi Gaussian

Ta c:

exp

Xy dng mt phn phi dng hm m

Gi s: |; : cho , phn phi ca

tun theo mt h phn phi dng hm m vi tham s

Cho , ta cn d on gi tr (thng th ). C ngha l ta d on gi tr u ra bng gi thuyt ca chng ta tha mn |

Tham s t nhin v u vo c quan h tuyn tnh:

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Xy dng mt phn phi dng hm m Hi quy logic:

0,1 nn ta chn h Bernoulli m hnh phn phi c iu kin ca theo .

Trong phn phi Bernoulli theo h hm m ta c

Trong phn phi Bernoulli |;

Gi thuyt

|; 1

1

1

1

Phn loi nhiu lp Phn loi nhiu lp vi softmax regression:

VD. Phn loi th thnh cc loi {c nhn, cng vic, th rc}

Mi u ra nhn 1 trong gi tr c th 1,2, . . , u ra vn l gi tr ri rc nhng c th nhn nhiu hn

2 gi tr

Ta s m hnh ha n nh l phn phi a thc (nhiu bin)

Phn loi nhiu lp Phn loi nhiu lp Ta s dng cc tham s , . . , tham s ha cho u ra c th ( l xc sut u ra l lp )

Tuy nhin ta c 1, do ch cn 1 tham s , . . , v 1

biu din phn phi a thc di dng phn phi h hm m ta nh ngha

1

100. .0

2

010. .0

1

000. .1

000. .0

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Phn loi nhiu lp l thnh phn th ca

Hm 1. 1 1. V d 1 3 3 1 1 0. V d 1 3 5 0

Mi quan h ca v s l 1

Ta c

Phn loi nhiu lp ;

. .

. .

. .

exp log log . . log

1

log

exp log

log

. .

log

log

exp

Phn loi nhiu lp Trong

log

log. .

log

log

1

Phn loi nhiu lp l vector trong log

vi 1, . . , 1

V log

0

Ta c:

1

Suy ra

v

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Phn loi nhiu lp Theo gi s 3 ta c (vi 1, . . , 1) trong

l cc tham s ca m hnh nh ngha 0 v th 0 Gi s v xc sut c iu kin ca vi l

;

Softmax regression l trng hp tng qut ca logistic regression

Phn loi nhiu lp Gi thuyt s cho u ra

|;

1 11 2

. .1 1

;

. .

. .

Phn loi nhiu lpVi tp mu gm m mu hun luyn, ta c hm logarithm ca hm kh nng log | ;

log

Ta c th tm cc tham s lm t gi tr ln nht bng cch s dng gradient ascent hoc phng php Newton