Matakuliah
Tahun
Versi
: H0434/Jaringan Syaraf Tiruan
: 2005
:1
Pertemuan 10
PERFORMANCE SURFACES
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menjelaskan pengertian tentang
performance learning.
2
Outline Materi
• Permukaan kinerja.
• Titik Optimal
3
Taylor Series Expansion
F ( x ) = F ( x* ) +
d
F( x )
dx
x = x*
( x – x* )
2
1 d
+ --F( x)
2 d x2
2
( x – x* ) + 
x = x*
n
1 d
+ ----F( x)
n! d x n
( x – x* ) + 
n
x = x*
4
Example
–x
F( x ) = e
Taylor series of F(x) about x* = 0 :
F (x ) = e
–x
–0
= e
–0
1 –0
2 1 –0
3
– e ( x – 0 ) + ---e ( x – 0 ) – -- e ( x – 0 ) + 
2
6
1 2 1 3
F ( x ) = 1 – x + -- x – --- x + 
2
6
Taylor series approximations:
F( x )  F0 ( x ) = 1
F( x)  F1 ( x) = 1 – x
1 2
F ( x )  F 2 ( x ) = 1 – x + --- x
2
5
Plot of Approximations
6
5
4
F2 ( x )
3
2
1
F1 ( x )
F0 ( x )
0
-2
-1
0
1
2
6
Vector Case
F( x) = F( x1 x 2   x n )


F ( x ) = F ( x* ) +
F(x )
F(x)
( x 1 – x 1* ) +
( x 2 – x 2* )
*
*
 x1
 x2
x=x
x=x
2
2
1 

*
*
++
F( x)
x
–
x
+
-F
x
–
x
x
(
)
(
)
(
)
n
n
1
1
2  x2
 xn
x = x*
x = x*
1
2
1 
+ --F(x)
( x 1 – x 1* ) ( x 2 – x 2* ) + 
*
2  x 1 x 2
x=x
7
Matrix Form
T
*
F(x) = F(x ) +  F(x)
x
= x*
( x – x* )
T 2
1
*
+ --- ( x – x )  F ( x )
( x – x* ) + 
*
2
x=x

F(x)
 xn
2
2
 x1
F(x)
2

F(x)
2 F ( x ) =  x 2  x 1
2
2
2

F(x)

 x 2 x n


F(x ) 
F(x)
 x 1 x 2
 x 1 x n

2
 x2
F(x)



F(x)
F ( x ) =  x 2



F(x)
 x1
Hessian
2
2


F(x)
F(x ) 
 x n x 1
 x n x 2
2

Gradient

2
2
 xn
F(x)
8
Directional Derivatives
First derivative (slope) of F(x) along xi axis:  F( x)   xi
(ith element of gradient)
2
2
Second derivative (curvature) of F(x) along xi axis:  F(x )  x i
(i,i element of Hessian)
T
First derivative (slope) of F(x) along vector p:
p
F ( x )
----------------------p
T
Second derivative (curvature) of F(x) along vector p:
p
2 F ( x ) p
-----------------------------2
p
9
Example
2
2
F(x ) = x 1 + 2x 1 x2 + 2 x2
x* =
 F ( x)
x = x*
=
0.5
0
p =

F( x )
 x1

F( x )
 x2
=
1
–1
2x 1 + 2x 2
2x 1 + 4x 2
x = x*
1
1
–
1
T
1
0
p
F ( x )
----------------------- = ------------------------ = ------- = 0
p
2
1
–1
x = x*
= 1
1
10
Plots
Directional
Derivatives
2
20
15
1
1.4
10
1.3
x2
5
1.0
0
0.5
0
2
1
2
1
0
x2
0.0
-1
0
-1
-1
-2
-2
x1
-2
-2
-1
0
1
2
x1
11
Minima
Strong Minimum
The point x* is a strong minimum of F(x) if a scalar d > 0 exists,
such that F(x*) < F(x* + Dx) for all Dx such that d > ||Dx|| > 0.
Global Minimum
The point x* is a unique global minimum of F(x) if
F(x*) < F(x* + Dx) for all Dx ° 0.
Weak Minimum
The point x* is a weak minimum of F(x) if it is not a strong
minimum, and a scalar d > 0 exists, such that F(x*) Š F(x* + Dx)
for all Dx such that d > ||Dx|| > 0.
12
Scalar Example
4
2 1
F ( x ) = 3x – 7x – --- x + 6
2
8
Strong Maximum
6
4
2
Strong Minimum
Global Minimum
0
-2
-1
0
1
2
13
Vector Example
4
F(x ) = ( x2 – x1 ) + 8x 1 x2 – x1 + x2 + 3
2
2
2
F( x) = ( x 1 – 1.5x 1 x2 + 2 x2 )x1
2
2
1.5
1
1
0.5
0
0
-0.5
-1
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-2
-2
2
12
-1
0
1
2
8
6
8
4
4
2
0
2
0
2
1
2
1
0
0
-1
-1
-2
-2
1
2
1
0
0
-1
-1
-2
-2
14