Expectation Maximization Algorithm

Expectation Maximization Algorithm

Rong Jin

A Mixture Model Problem

Apparently, the dataset consists of two modes How can we automatically identify the two modes?

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Gaussian Mixture Model (GMM) Assume that the dataset is generated by two

mixed Gaussian distributions Gaussian model 1: Gaussian model 2:

If we know the memberships for each bin, estimating the two Gaussian models is easy.

How to estimate the two Gaussian models without knowing the memberships of bins?

1 1 1 1, ; p

2 2 2 2, ; p

EM Algorithm for GMM Let memberships to be hidden variables

EM algorithm for Gaussian mixture model Unknown memberships:

Unknown Gaussian models:

Learn these two sets of parameters iteratively

1 21 2 1 2{ , ,..., } , , , ,..., ,n n nmx x x x x xm m

1 21 2, , , ,..., ,n nx x xm m m

1 1 1 1

2 2 2 2

, ;

, ;

p

p

Start with A Random Guess

Random assign the memberships to each bin

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Start with A Random Guess

Random assign the memberships to each bin

Estimate the means and variance of each Gaussian model

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E-step Fixed the two Gaussian models Estimate the posterior for each data point

1 1 1 1

1 2 1 1 1 2 2 2

2 2 2 2

1 2 1 1 1 2 2 2

21

1 1 2211

( , ) ( | , )( , 1)( 1 | )

( ) ( , ) ( , ) ( | , ) ( | , )

( , ) ( | , )( , 2)( 2 | )

( ) ( , ) ( , ) ( | , ) ( | , )

1( | , ) exp

22

p x p x pp x mp m x

p x p x p x p x p p x p

p x p x pp x mp m x


xp x

221 1 22

22

1, ( | , ) exp

22

xp x

EM Algorithm for GMM Re-estimate the

memberships for each bin

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1 21

1 1 1 2 2 21

ˆ ˆ( 1 | ) log ( , ) ( 2 | ) log ( , )

ˆ ˆ( 1 | ) log log ( | , ) ( 2 | ) log log ( | , )

n

i i i i i ii

n

i i i i i ii

l p m x p x p m x p x

p m x p p x p m x p p x

22 21 1 1

1 1 1 1

1 1

221 1 1

2 2 2

1 1

ˆ ˆ ˆ( 1 | ) ( 1 | ) ( 1 | ), ,

ˆ ˆ( 1 | ) ( 1 | )

ˆ ˆ ˆ( 2 | ) ( 2 | ) ( 2 | ), ,

ˆ ˆ( 2 | ) ( 2 | )

n n ni i i i i i i ii i i

n ni i i ii i

n n ni i i i i i i ii i i

n ni i i ii i

p m x p m x x p m x xp

n p m x p m x

p m x p m x x p m x xp

n p m x p m x

22

M-Step Fixed the memberships Re-estimate the two model Gaussian

Weighted by posteriors

Weighted by posteriors

EM Algorithm for GMM Re-estimate the

memberships for each bin

Re-estimate the models

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At the 5-th Iteration Red Gaussian

component slowly shifts toward the left end of the x axis

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At the10-th Iteration

Red Gaussian component still slowly shifts toward the left end of the x axis

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component make more noticeable shift toward the left end of the x axis

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component is close to the desirable location

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At the 100-th Iteration The results are

almost identical to the ones for the 50-th iteration

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EM as A Bound Optimization EM algorithm in fact maximizes the log-likelihood function

of training data Likelihood for a data point x

Log-likelihood of training data

1 2 1 1 1 2 2 2

2 21 2

1 1 1 12 22 21 21 2

( ) ( , ) ( , ) ( | , ) ( | , )

1 1( | , ) exp , ( | , ) exp

2 22 2


x xp x p x

1 1 1 2 2 21 1log ( ) log ( | , ) ( | , )n nii il p x p x p p x p




1 2 1 1 1 2 2 2

2 21 2

1 1 1 12 22 21 21 2

( ) ( , ) ( , ) ( | , ) ( | , )

1 1( | , ) exp , ( | , ) exp

2 22 2


x xp x p x





1 2 1 1 1 2 2 2

2 21 2

1 1 1 12 22 21 21 2

( ) ( , ) ( , ) ( | , ) ( | , )

1 1( | , ) exp , ( | , ) exp

2 22 2


x xp x p x

1 2 1 1 1 2 2 21 1, log ( ) log ( | , ) ( | , )n nii il p x p x p p x p

Logarithm Bound Algorithm

1 2( , )l

• Start with initial guess

0 01 2,

0 01 2,



0 01 2 1 2 1 2( , ) ( , ) ( , )l l Q

• Come up with a lower bounded

0 01 2,

1 2 0,

1 2( , )l 0 0

1 2 1 2 1 2( , ) ( , ) ( , )l l Q

1 2

0 01 1 2 2

( , ) is a concave function

Touch point: ( , ) 0

Q

Q

Touch Point



0 01 2 1 2 1 2( , ) ( , ) ( , )l l Q


0 01 2,

1 2 0,

1 2( , )l 0 0

1 2 1 2 1 2( , ) ( , ) ( , )l l Q

1 2

0 01 1 2 2



Q

Q

• Search the optimal solution that maximizes 1 2( , )Q

1 11 2,



1 11 2 1 2 1 2( , ) ( , ) ( , )l l Q


0 01 2,

1 2 0,

1 2( , )l 0 0

1 2 1 2 1 2( , ) ( , ) ( , )l l Q

1 2

0 01 1 2 2



Q

Q

• Search the optimal solution that maximizes

• Repeat the procedure

1 2( , )Q 1 1

1 2, 2 21 2,

Logarithm Bound AlgorithmOptimal

Point

1 2( , )l

0 01 2, 1 1

1 2, 2 21 2, ,...



1 2 0,

0 01 2 1 2 1 2( , ) ( , ) ( , )l l Q

1 2

0 01 1 2 2



Q

Q

• Search the optimal solution that maximizes

• Repeat the procedure

• Converge to the local optimal

1 2( , )Q

EM as A Bound Optimization Parameter for previous iteration: Parameter for current iteration: Compute

' '1 2,

1 2,

1 2( , )Q

' '1 2 1 2 1 2

1 1 1 2 2 21 ' ' ' ' ' '

1 1 1 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

'2

( , ) ( , ) ( , )

( | , ) ( | , )log

( | , ) ( | , )

( | , ) ( | , )

( | , ) ( | , ) ( | , )log

( | ,

n i ii

i i

i i

i i i

i

Q l l

p x p p x p

p x p p x p

p x p p x p

p x p p x p p x p

p x

1 ' '2 2 2 2 2

' ' ' ' ' ' ' ' '1 1 1 2 2 2 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

' ' '2 2 2

' '1 1

) ( | , )

( | , ) ( | , ) ( | , )

( | , ) ( | , )log

( | , ) ( | , ) ( | , )

( | , )

( | , )

ni

i

i i i

i i

i i i

i

i

p p x p

p x p p x p p x p

p x p p x p

p x p p x p p x p

p x p

p x p

12 2 2

' ' ' ' ' ' '1 2 2 2 2 2 2

' '1 1 1 2 2 21 21 ' ' ' ' ' '

1 1 1 2 2 2

( | , )log

( | , ) ( | , )

( | , ) ( | , )( | ) log ( | ) log

( | , ) ( | , )

ni

i

i i

n i ii ii

i i

p x p

p x p p x p

p x p p x pp x p x

p x p p x p

' '1 2 1 2 1 2

1 1 1 2 2 21 ' ' ' ' ' '

1 1 1 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

'2

( , ) ( , ) ( , )

( | , ) ( | , )log

( | , ) ( | , )

( | , ) ( | , )

( | , ) ( | , ) ( | , )log

( | ,

n i ii

i i

i i

i i i

i

Q l l

p x p p x p

p x p p x p

p x p p x p

p x p p x p p x p

p x

1 ' '2 2 2 2 2

' ' ' ' ' ' ' ' '1 1 1 2 2 2 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

' ' '2 2 2

' '1 1

) ( | , )

( | , ) ( | , ) ( | , )

( | , ) ( | , )log

( | , ) ( | , ) ( | , )

( | , )

( | , )

ni

i

i i i

i i

i i i

i

i

p p x p

p x p p x p p x p

p x p p x p

p x p p x p p x p

p x p

p x p

12 2 2

' ' ' ' ' ' '1 2 2 2 2 2 2

' '1 1 1 2 2 21 21 ' ' ' ' ' '

1 1 1 2 2 2

( | , )log

( | , ) ( | , )

( | , ) ( | , )( | ) log ( | ) log

( | , ) ( | , )

ni

i

i i

n i ii ii

i i

p x p

p x p p x p

p x p p x pp x p x

p x p p x p

' '1 2 1 2 1 2

1 1 1 2 2 21 ' ' ' ' ' '

1 1 1 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

'2

( , ) ( , ) ( , )

( | , ) ( | , )log

( | , ) ( | , )

( | , ) ( | , )

( | , ) ( | , ) ( | , )log

( | ,

n i ii

i i

i i

i i i

i

Q l l

p x p p x p

p x p p x p

p x p p x p

p x p p x p p x p

p x

1 ' '2 2 2 2 2

' ' ' ' ' ' ' ' '1 1 1 2 2 2 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

' ' '2 2 2

' '1 1

) ( | , )

( | , ) ( | , ) ( | , )

( | , ) ( | , )log

( | , ) ( | , ) ( | , )

( | , )

( | , )

ni

i

i i i

i i

i i i

i

i

p p x p

p x p p x p p x p

p x p p x p

p x p p x p p x p

p x p

p x p

12 2 2

' ' ' ' ' ' '1 2 2 2 2 2 2

' '1 1 1 2 2 21 21 ' ' ' ' ' '

1 1 1 2 2 2

( | , )log

( | , ) ( | , )

( | , ) ( | , )( | ) log ( | ) log

( | , ) ( | , )

ni

i

i i

n i ii ii

i i

p x p

p x p p x p

p x p p x pp x p x

p x p p x p

Concave property of logarithm function

log( (1 ) ) log (1 ) log

0 1, , 0

p p p p

p

' '1 2 1 2 1 2

1 1 1 2 2 21 ' ' ' ' ' '

1 1 1 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

'2

( , ) ( , ) ( , )

( | , ) ( | , )log

( | , ) ( | , )

( | , ) ( | , )

( | , ) ( | , ) ( | , )log

( | ,

n i ii

i i

i i

i i i

i

Q l l

p x p p x p

p x p p x p

p x p p x p

p x p p x p p x p

p x

1 ' '2 2 2 2 2

' ' ' ' ' ' ' ' '1 1 1 2 2 2 2 2 2

' ' '1 1 1 1 1 1

' ' ' ' ' ' ' ' '1 1 1 2 2 2 1 1 1

' ' '2 2 2

' '1 1

) ( | , )

( | , ) ( | , ) ( | , )

( | , ) ( | , )log

( | , ) ( | , ) ( | , )

( | , )

( | , )

ni

i

i i i

i i

i i i

i

i

p p x p

p x p p x p p x p

p x p p x p

p x p p x p p x p

p x p

p x p

12 2 2

' ' ' ' ' ' '1 2 2 2 2 2 2

' ' ' '1 1 1 2 2 21 1 2 1 1 21 ' ' ' ' ' '

1 1 1 2 2 2

( | , )log

( | , ) ( | , )

( | , ) ( | , )( 1 | ; , ) log ( 2 | ; , ) log

( | , ) ( | , )

ni

i

i i

n i ii ii

i i

p x p

p x p p x p

p x p p x pp m x p m x

p x p p x p

Definition of posterior' ' '

' '1 1 11 1 2' ' ' ' ' '

1 1 1 2 2 2

( | , )( 1 | ; , )

( | , ) ( | , )i

ii i

p x pp m x

p x p p x p

Log-Likelihood of EM Alg.

0 10 20 30 40 50 60 70 80 90 100-410

-405

-400

-395

-390

-385

-380

-375

Iteration

Logl

ikel

hood

Saddle points

Maximize GMM Model

What is the global optimal solution to GMM?

Maximizing the objective function of GMM is ill-posed problem


2 21 2

1 1 1 12 22 21 21 2

1 1( | , ) exp , ( | , ) exp

2 22 2

x xp x p x

11 1 1 1 2 1 2, 0, , 1, 0.5

nii x

x p pn

Maximize GMM Model

What is the global optimal solution to GMM?

Maximizing the objective function of GMM is ill-posed problem


2 21 2

1 1 1 12 22 21 21 2

1 1( | , ) exp , ( | , ) exp

2 22 2

x xp x p x

11 1 1 1 2 1 2, 0, , 1, 0.5

nii x

x p pn

Identify Hidden Variables For certain learning problems, identifying hidden variables is

not a easy task Consider a simple translation model

For a pair of English and Chinese sentences:

A simple translation model is

The log-likelihood of training corpus

1 2 1 2: ( , ,..., ) : ( , ,..., )s le e e e c c c c

11 1Pr( | ) Pr( | ) Pr( | )s s tj j kkj je c e c e c

1 1, ,..., ,n ne c e c

, ,1 1 1 1log Pr( | ) log Pr( | )i in n e ci i i j i ki i j kl e c e c

Identify Hidden Variables Consider a simple case

Alignment variable a(i)

Rewrite

1 2

1 2

: ( )

: ( )

e e e

c c c

2 211

1 1 2 1 1 2 2 2

1 1 2 2 1 2 2 1

Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

j kkje c e c

e c e c e c e c

e c e c e c e c

:

map a position in English sentence to a position in Chinese sentence

a

1 (1) 2 (2)Pr( | ) Pr( | ) Pr( | )a aae c e c e c



Rewrite

1 2

1 2

: ( )

: ( )

e e e

c c c

2 211

1 1 2 1 1 2 2 2

1 1 2 2 1 2 2 1

Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

j kkje c e c

e c e c e c e c

e c e c e c e c

:


a




Rewrite

1 2

1 2

: ( )

: ( )

e e e

c c c

2 211

1 1 2 1 1 2 2 2

1 1 2 2 1 2 2 1

Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

j kkje c e c

e c e c e c e c

e c e c e c e c

:


a




Rewrite

1 2

1 2

: ( )

: ( )

e e e

c c c

2 211

1 1 2 1 1 2 2 2

1 1 2 2 1 2 2 1

Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

Pr( | ) Pr( | ) Pr( | ) Pr( | )

j kkje c e c

e c e c e c e c

e c e c e c e c

:


a


EM Algorithm for A Translation Model Introduce an alignment variable for each translation pair

EM algorithm for the translation model E-step: compute the posterior for each alignment variable M-step: estimate the translation probability Pr(e|c)

1 1 2 21 2, , , , , ,..., , , ,n n na a ae c e c e c

Pr( | , )j j ja e c

| | | |

, , ( ) , , ( )1 1| | | |

', , '( ) , ,' 1

1 1

Pr( | ) Pr( | )Pr( , , )

Pr( | , )Pr( ', , )

Pr( | ) Pr( | )

j j

j ji

e e

j k j a k j k j a kj j k k

j j e ej j ca

j k j a k j s j ta tk k

e c e ca e c

a e ca e c

e c e c

EM Algorithm for A Translation Model Introduce an alignment variable for each translation pair

EM algorithm for the translation model E-step: compute the posterior for each alignment variable M-step: estimate the translation probability Pr(e|c)

1 1 2 21 2, , , , , ,..., , , ,n n na a ae c e c e c

Pr( | , )j j ja e c

| | | |

, , ( ) , , ( )1 1| | | |

', , '( ) , ,' 1

1 1

Pr( | ) Pr( | )Pr( , , )

Pr( | , )Pr( ', , )

Pr( | ) Pr( | )

j j

j ji

e e

j k j a k j k j a kj j k k

j j e ej j ca

j k j a k j s j ta tk k

e c e ca e c

a e ca e c

e c e c

We are luck here. In general, this step can be extremely difficult and usually requires approximate approaches

Compute Pr(e|c) First compute

,

| |

, ,11^

| |

, ,11

Pr( | ; , ) ( ) ( ) Pr( | , ) ( ( ) )

Pr( , , ) ( ( ) )( ) ( )

Pr( , )

Pr( | ) Pr( | )

( ) ( )

Pr( | )

(

ji

j k

ji

i i i i i ia

i iai i

i i

ec

j s j ttk e e

i i ec

j s j ttk

e c e c e e c c a e c a e c

a e c a e ce e c c

e c

e c e c

e e c c

e c

,1

Pr( | )) ( )

Pr( | )ii i c

j tt

e ce e c c

e c

Pr( | ; , )i ie c e c

Compute Pr(e|c) First compute

,

| |

, ,11^

| |

, ,11

Pr( | ; , ) ( ) ( ) Pr( | , ) ( ( ) )

Pr( , , ) ( ( ) )( ) ( )

Pr( , )

Pr( | ) Pr( | )

( ) ( )

Pr( | )

(

ji

j k

ji

i i i i i ia

i iai i

i i

ec

j s j ttk e e

i i ec

j s j ttk

e c e c e e c c a e c a e c

a e c a e ce e c c

e c

e c e c

e e c c

e c

,1

Pr( | )) ( )

Pr( | )ii i c

j tt

e ce e c c

e c

Pr( | ; , )i ie c e c

1Pr( | ) Pr( | ; , )ni iie c e c e c

, ,1 1 1 1

, ,1 1 1 1

Pr( | ) for the current iteration

' Pr'( | ) for the previous iteration

( ) log Pr( | ; ) log Pr( | )

( ') log Pr( | ; ') log Pr'( | )

i i

i i

n n e ci i i j i ki i j k

n n e ci i i j i ki i j k

e c

e c

l e c e c

l e c e c

θ

θ

θ θ

θ θ

, ,11 1

, ,1

Pr( | )( , ') ( ) ( ') log

Pr'( | )

i

i

i

ci j i kn e k

i j ci j i ll

e cQ l l

e c

θ θ θ θ

Bound Optimization for A Translation Model

Bound Optimization for A Translation Model

, ,11 1

, ,1

, , , ,1 1

1 , ,, ,1

, ,

,

Pr( | )( , ') ( ) ( ') log

Pr'( | )

Pr'( | ) Pr( | )log

Pr'( | )Pr'( | )

Pr'( | )

Pr'( |

i

i

i

ii

i

ci j i kn e k

i j ci j i ll

ci j i k i j i kn e

i j ck i j i ki j i ll

i j i k

i j

e cQ l l

e c

e c e c

e ce c

e c

e

θ θ θ θ

, ,1 1 1

, ,,1

Pr( | )log

Pr'( | ))i i

i

i j i kn e ci j k c

i j i ki ll

e c

e cc

1,1

Pr'( | )Pr( | ) ( ) ( )

Pr'( | )i

ni ii c

j tt

e ce c e e c c

e c

Iterative Scaling Maximum entropy model

1

exp( )exp( )( | ; ) , ( ) log

exp( ) exp( )iN i yy

train iy i yy y

x wx wp y x l D

x w x w

Iterative scaling All features Sum of features are constant

, 0i jx

,1d

i jj x g

Iterative Scaling Compute the empirical mean for each feature of every class,

i.e., for every j and every class y

Start w1 ,w2 …, wc = 0 Repeat

Compute p(y|x) for each training data point (xi, yi) using w from the previous iteration

Compute the mean of each feature of every class using the estimated probabilities, i.e., for every j and every y

Compute for every j and every y

Update w as

, ,1( , )

Ny j i j ii

e x y y N

, ,1( | )

Ny j i j ii

m x p y x N

, , ,j y j y j yw w w

, , ,1

log logj y j y j yw e mg

Iterative Scaling

1 2

' ' '1 2

1

, ,..., : parameters for the current iteration

' , ,..., : parameters for the last iteration

exp( )( | ; )

exp( )

exp( )( ) log ( | ; ) log

exp( )i

c

c

y

yy

N i y

i ii yy

w w w

w w w

x wp y x

x w

x wl p y x

x w

1

'

'1 1

exp( )( ') log ( | ; ') log

exp( )i

N

N N i y

i ii yy

x wl p y x

x w

'

'1

exp( ) exp( )( ) ( ') log

exp( ) exp( )i iN i y i y

ii y i yy y

x w x wl l

x w x w

Iterative Scaling

'

'1

' '1

exp( ) exp( )( ) ( ') log

exp( ) exp( )

log exp( ) log exp( )

i i

i i

N i y i y

ii y i yy y

Ni y y i y i yi y y

x w x wl l

x w x w

x w w x w x w

Can we use the concave property of logarithm function?

No, we can’t because we need a lower bound

Iterative Scaling

log 1 log exp( ) exp( ) 1i y i yy yx x x w x w

' '1

' '1

( ) ( ')


log exp( ) exp( ) 1

i i

i i

Ni y y i y i yi y y

Ni y y i y i yi y y

l l

x w w x w x w

x w w x w x w

• Weights still couple with each other

• Still need further decomposition

yw

Iterative Scaling

,, , , ,1 1 1

,1

, ,, , ,1 1 1

,1

exp( ) exp exp

exp exp

d d di ji y i j y j y j i kdj j k

i kk

d d di j i jy j i k y jdj k j

i kk

xx w x w w x

x

x xw x gw

gx

' '1

,' ', , , ,1

( ) ( ')

log exp( ) exp( ) 1

log exp( ) exp( ) 1

i i

i i

Ni y y i y i yi y y

N i ji j y j y j i y y ji j y y j

l l

x w w x w x w

xx w w x w gw

g

exp exp for , 0, 1i i i i i ii i iq p p q i p p

Iterative Scaling

,' ', , , ,1

,' ', , , ,1 1

( , ') log exp( ) exp( ) 1

log exp( ) 1 , exp( )

i i

N i ji j y j y j i y y ji j y y j

N N i ji y i j y j y j i y ji y i y j

xQ x w w x w gw

g

xx w x w w y y gw

g

, , ,1,

,1,

,1

( , '), exp( ) 0

,log

Ni j i i j y ji y j

y j

Ni j ii y j

y j Ni ji y j

Qx y y x gw

w

x y yw

x

Wait a minute, this can not be right! What happens?



0 01 2 1 2 1 2( , ) ( , ) ( , )l l Q


0 01 2,

1 2 0,

1 2( , )l 0 0

1 2 1 2 1 2( , ) ( , ) ( , )l l Q

1 2

0 01 1 2 2



Q

Q

• Search the optimal solution that maximizes 1 2( , )Q

1 11 2,

Iterative Scaling

,' ', , , ,1 1

( , ')

log exp( ) 1 , exp( )N N i j

i y i j y j y j i y ji y i y j

Q

xx w x w w y y gw

g

,' ' ' ', , , ,1 1

,' ',1 1

( ', ')

log exp( ) 1 , exp( )

log exp( ) 1 exp( )

0

N N i ji y i j y j y j i y ji y i y j

N N i ji y y ji y i y j

Q

xx w x w w y y gw

g

xx w gw

g

Where does it go wrong?

Iterative Scaling

log 1 log exp( ) exp( ) 1i y i yy yx x x w x w

' '1

' '1

( ) ( ')


log exp( ) exp( ) 1

i i

i i

Ni y y i y i yi y y

Ni y y i y i yi y y

l l

x w w x w x w

x w w x w x w

Not zero when = ’

' '

exp( ) exp( )log 1 log 1

exp( ) exp( )

i yy i y

yi y i yy y

x w x wx x

x w x w

Iterative Scaling

''1

'

'1

1

( ) ( ')

exp( )1

exp( )

exp( ) exp( )1

exp( )

( | ; ') exp( ) 1

i i

i

i

N i yi y yi y

i yy

N i y i yi yi y

i yy

Ni y i i yi y

l l

x wx w w

x w

x w xx

x w

x p y x x

' '

exp( ) exp( )log 1 log 1

exp( ) exp( )

i yy i y

yi y i yy y

x w x wx x

x w x w

Definition of conditional exponential model

'y y yw w

Iterative Scaling

,, , , ,1 1 1

,1

, ,, , ,1 1 1

,1

exp( ) exp exp

exp exp


i kk

d d di j i jy j i k y jdj k j

i kk

xx x x

x

x xx g

gx

1

,, , ,1

,, , ,1

( ) ( ') ( | ; ') exp( ) 1

( | ; ') exp( ) 1

( , ) ( | ; ') exp( ) 1

i

i

Ni y i i yi y

N i ji j y j i y ji j y j

N i ji j y j i i y ji j y

l l x p y x x

xx p y x g

g

xx y y p y x g

g

Iterative Scaling

,, , ,1

( , ') ( , ) ( | ; ') exp( ) 1N i j

i j y j i i y ji j y

xQ x y y p y x g

g

, , , ,1,

, ,' 1, , ,

,1

( , ')( , ) ( | ; ') exp( ) 0

( , )1log

( | ; ')

Ni j y j i i i j y ji

y j

Ni j y j ii

y j y j y j Ni i ji

Qx y y p y x x g

x y yw w

g p y x x

Iterative Scaling How about ? ,1 constantd

i j ij x g

,, , , ,1 1 1

,1

, ,, , ,1 1 1

,1

exp( ) exp exp

exp exp


i kk

d d di j i jy j i k i y jdj k j

ii kk

xx x x

x

x xx g

gx

,, , ,1

( , ') ( , ) ( | ; ') exp( ) 1N i j

i j y j i i i y ji j yi

xQ x y y p y x g

g

, , , ,1,

( , ')( , ) ( | ; ') exp( ) 0

Ni j y j i i i j i y ji

y j

Qx y y p y x x g

Is this solution unique?

Iterative Scaling How about negative features?

, ,1

,, , , ,1 1

exp( ) exp

1exp exp

di y i j y jj

d d i jy j i k y j i kj j

x x

xd x d x

d d

, , , ,1

1( , ') ( , ) ( | ; ') exp( ) 1

Ni j y j i i i j y ji j y

Q x y y p y x x dd

, , , ,1,

( , ')( , ) ( | ; ') exp( ) 0

Ni j y j i i y j i ji

y j

Qx y y p y x d x

Faster Iterative Scaling The lower bound may not be tight given all the

coupling between weights is removed

A tighter bound can be derived by not fully decoupling the correlation between weights

,, , ,1

,1

( , ') ( , ) ( | ; ') exp( ) 1

( )

N i ji j y j i i i y ji j y

i

Ny ji j y

xQ x y y p y x g

g

q

Univariate functions!

,,, ,

,

( , ') , log ( | ) y j igi ji i j y j

j i y i yi

xQ y y x p y x e

g

Faster Iterative Scaling

Log-likelihood

Bad News You may feel great after the struggle of the derivation. However, is iterative scaling a true great idea? Given there have been so many studies in optimization, we

should try out existing methods.

Comparing Improved Iterative Scaling to Newton’s Method

Dataset Instances Features

Rule 29,602 246

Lex 42,509 135,182

Summary 24,044 198,467

Shallow 8,625,782 264,142

Dataset Iterations Time (s)

Rule 823 42.48

81 1.13

Lex 241 102.18

176 20.02

Summary 626 208.22

69 8.52

Shallow 3216 71053.12

421 2420.30

Limited-memory Quasi-Newton method

Improved iterative scaling

Try out the standard numerical methods before you get excited

about your algorithm

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Expectation Maximization Algorithm

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