Unconstrained Optimization

Rong Jin

Logistic Regression2

2 1 1

1( ) ( ) log

1 exp ( )N m

reg train train ii il D l D w s w

y b x w

The optimization problem is to find weights w and b that maximizes the above log-likelihood

How to do it efficiently ?

Gradient Ascent Compute the gradient

Increase weights w and threshold b in the gradient direction

21 1

21 1

log ( | )

log ( | )

where is learning rate.

N mi i ii i

N mi i ii i

w w p y x s ww

c c p y x s wc

21 1 1

21 1 1

log ( | ) (1 ( | ))

log ( | ) (1 ( | ))

N m Ni i i i i i ii i i

N m Ni i i i i ii i i

p y x s w sw x y p y xw

p y x s w y p y xb

Problem with Gradient Ascent Difficult to find the appropriate step

size Small slow convergence Large oscillation or “bubbling”

Convergence conditions Robbins-Monroe conditions

Along with “regular” objective function will ensure convergence

20 0

, t tt t

Newton Method Utilizing the second order derivative Expand the objective function to the second order around x0

The minimum point is Newton method for optimization

Guarantee to converge when the objective function is convex

0 0

20 0 0( ) ( ) ( ) ( )

2'( ) , ''( )

x x x x

bf x f x a x x x x

a f x b f x

0 /x x a b

'( )

''( )

old

old

new old x x

x x

f xx x

f x

Multivariate Newton Method Object function comprises of multiple variables

Example: logistic regression model

Text categorization: thousands of words thousands of variables Multivariate Newton Method

Multivariate function:

First order derivative a vector

Second order derivative Hessian matrix Hessian matrix is mxm matrix Each element in Hessian matrix is defined as:

21 1

1( ) log

1 exp ( )N m

reg train ii il D s w

y b x w

21 2

,( , ,..., )m

i ji j

f x x x

x x

H

1 2( ) ( , ,..., )mf x f x x x

1 1, ,...,

m

f f f f

x x x x

Multivariate Newton Method Updating equation:

Hessian matrix for logistic regression model

Can be expensive to compute Example: text categorization with 10,000 words Hessian matrix is of size 10,000 x 10,000 100 million entries Even worse, we have compute the inverse of Hessian matrix H-1

1 ( )new old f xx x

x

H

1( | )(1 ( | ))

n Ti i i i m mi

p y x p y x x x s H I

Quasi-Newton Method Approximate the Hessian matrix H-1 with another B matrix:

B is update iteratively (BFGS):

Utilizing derivatives of previous iterations

( )new old f xx x

x

B

1

1 1,

T Tk k k k k k

k k T Tk k k k k

k k k k k k

p p y y

p p y p

p x x y g g

B BB B

B

Limited-Memory Quasi-Newton Quasi-Newton

Avoid computing the inverse of Hessian matrix But, it still requires computing the B matrix large storage

Limited-Memory Quasi-Newton (L-BFGS) Even avoid explicitly computing B matrix

B can be expressed as a product of vectors Only keep the most recently vectors of (3~20)

1

1 1,

T Tk k k k k k

k k T Tk k k k k

k k k k k k

p p y y

p p y p

p x x y g g

B BB B

B

{ , }k kp y

{ , }k kp y

Efficiency

Num

ber of Variable

Standard Newton method: O(n3)

Small

Medium Quasi Newton method (BFGS): O(n2)

Limited-memory Quasi Newton method (L-BFGS): O(n)

Large

Con

verg

ence

Rat

e

V-Fast

Fast

R-Fast

Empirical Study: Learning Conditional Exponential Model

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 350 4.8

81 1.13

Lex 1545 114.21

176 20.02

Summary 3321 190.22

69 8.52

Shallow 14527 85962.53

421 2420.30

Limited-memory Quasi-Newton method

Gradient ascent

Free Software http://www.ece.northwestern.edu/~nocedal/so

ftware.html L-BFGS L-BFGSB

Linear Conjugate Gradient Method Consider optimizing the quadratic function

Conjugate vectors The set of vector {p1, p2, …, pl} is said to be conjugate with respect to

a matrix A if

Important property The quadratic function can be optimized by simply optimizing the

function along individual direction in the conjugate set. Optimal solution:

k is the minimizer along the kth conjugate direction

* arg min2

TT

x

x xx b x

A

0, for any Ti jp p i j A

1 1 2 2 ... l lx p p p

Example Minimize the following function

Matrix A

Conjugate direction

Optimization First direction, x1 = x2=x:

Second direction, x1 =- x2=x:

Solution: x1 = x2=1

-4

-2

0

2

4

-3

-2

-1

0

1

2

3-10

0

10

20

302 21 2 1 2 1 2 1 2( , )f x x x x x x x x

1 0.5

0.5 1A

1 21 1

,1 1

p p

21 2 1( , ) 2 Minimizer 1f x x x x

21 2 2( , ) 2 Minimizer 0f x x x

How to Efficiently Find a Set of Conjugate Directions Iterative procedure

Given conjugate directions {p1,p2,…, pk-1}

Set pk as follows:

Theorem: The direction generated in the above step is conjugate to all previous directions {p1,p2,…, pk-1}, i.e.,

Note: compute the k direction pk only requires the previous direction pk-1

11 1

1 1

( ), , where ,

k

Tk k

k k k k k k k k k kTx xk k

r p f xp r p r x x p

xp p

A

A

, for any [1, 2,..., 1]Tk ip p i k A

Nonlinear Conjugate Gradient Even though conjugate gradient is derived for a

quadratic objective function, it can be applied directly to other nonlinear functions

Several variants: Fletcher-Reeves conjugate gradient (FR-CG) Polak-Ribiere conjugate gradient (PR-CG)

More robust than FR-CG

Compared to Newton method No need for computing the Hessian matrix No need for storing the Hessian matrix