T&C LAB-AI
Robotics
Neural Network
Lecture 8
Jeong-Yean Yang
2020/12/10
1
T&C LAB-AI
H.W. Neural Network
0
2
T&C LAB-AI
Robotics
HW.1 Sigmoidal NN for Sin(x)
• 0<x<10 X=linspace(0,10,20) N=20
• Y = sin(X)
• Find the best NN result with Sigmoidal NN
– W1 and W2 = zeros or randn
– How many iterations are required?
–
3
Y= sin(x)
T&C LAB-AI
Robotics
HW 2 Why J shows this Phenomenon?
• During Learning process,
J shows sudden Jump.
Why?
4
T&C LAB-AI
Robotics
HW3. Using RBF for Y=Sin(x)
• 0<x<10 X=linspace(0,10,20) N=20
• Y = sin(X)
5
T&C LAB-AI
Robotics
HW 4. Tell me What the differences are
Between Sigmoidal and RBF NN
• Iteration, Convergence, alpha, Initial value,
• Anything is O.K.
6
HW.5. Why Sin(x) is Not Smooth?
Find the Answer and the RESULT
T&C LAB-AI
Robotics
HW. 6 Noisy Data With RBF NN
• n=100
• X=linspace(0,10,n)
• y= -0.1*pow(x-2,2)+randn(n)
• Try RBF with the above x and y.
• RBF becomes what?
7
T&C LAB-AI
Unbalanced Cost Function
1
8
T&C LAB-AI
Robotics
We used Squared Error
• Remind Differentiation for Gradient Descent Method
9
2
2
1
2
1
1
1
2
2
2
1
1
ˆ
||
||
2
2
,
i
i
i D
i D
J
e
y
y
J
J
W
W
J
W
W
W
J
W
W
W
• Question:
If we use Absolute Error, |e| , then What occurs?
T&C LAB-AI
Robotics
Absolute Error
• Absolute Error is not well used because of Differentiation.
– It is NOT continuous
10
2
1
2
'
'
i
i D
i
i
i D
J
e
J
e e
1
|
|
2
i
i D
J
e
'
0
1
'
'
0
2
0
0
i
i
i
i
i
e
e
J
e
e
e
T&C LAB-AI
Robotics
In Spite of All, Why We Concern |e|?
• Convergence
– At a solution, convergence rate is too slow.
11
2
1
2
i
i D
J
e
1
|
|
2
i
i D
J
e
A
B
'
'
w A
w B
J
J
Convergence Rate is NOT
constant
'
'
w A
w B
J
J
A
B
Convergence rate is constant
Sliding into a goal
Remind
sgn( )
T
K
s
Benito
Fernandez
J
e
J
e
T&C LAB-AI
Robotics
What Changes in OUR RBF Network?
12
2
2
2
1
1
2
2
T
k
k
T
J
e
e e
J
e
e
W
W
1
2
1
1
(
1)
2
2
2
2
1
(
1)
1
1
2
(
1) 1
[
]
[
]
[
]
[
]
T
T
T
T
n
n
h
T
T
T
n
h
n
h
Y
I W
J
e
Y
e
e
e
e
Y
I
W
W
W
W
J
Transpose
Vector
Y
I
e
Y
I e
W
2
2
|
|
( , , 0)
k
k
J
e
J
e
W
W
How we solve Matrix Row-Column Problem in this
|e| Network?
New equation is required!
T&C LAB-AI
Robotics
See Derivative of Error in RBF NN
• Error vector e, is replaced by [+, -, or 0] vector
13
2
2
2
1
1
2
2
T
k
k
T
J
e
e e
J
e
e
W
W
2
2
|
|
( , , 0)
k
k
J
e
J
e
W
W
2
2
2
1 (
1)
1
2
1
1
(
1)
2
[
]
[
]
T
T
h
T
T
n
n
h
J
e
Y
e
e
W
W
W
Y
I W
e
e
Y
I
W
2
2
2
1 (
1)
1
(
1)
1
( , 0)
( , 0)
1
1
[
]
0
...
T
T
h
T
n
h
n
J
e
Y
W
W
W
Y
I
T&C LAB-AI
Robotics
Derivatives of W1 in RBF NN
14
2
2
2
1,
1
1
2
2
2
k
k
T
T
k
k
k
k
k
k
k
k
J
e
J
eW
Z
eW
Y
Z
W
2
2
1,
1
|
|
1
1
1
1
2
2
0
0
...
...
k
k
T
T
k
k
k
k
k
k
k
k
J
e
J
W
Z
W
Y
Z
W
T&C LAB-AI
Robotics
Exampl) l8abs1.py
15
|
|
k
k
J
e
1
1
1
sgn( )
0
...
n
e
RBF-NN
2
1
2
k
k
J
e
T&C LAB-AI
Robotics
Example) l8abs1.py
• Alpha=0.1 Too many oscillations(chattering)
16
T&C LAB-AI
Robotics
Example) l8abs1.py with Small Alpha
• Alpha=0.01 Too many oscillation
17
T&C LAB-AI
Robotics
Alpha is the Key for Chattering
Remind Gradient Descent method
18
J
e
J
e
0.1
0.01
Big alpha moves faster and farther.
J
e
J
e
0.1
0.01
Large Chattering
Small chattering
T&C LAB-AI
Robotics
Alternative Strategy
for Small Chattering
19
J
e
0.1
J
e
0.1
When near J=0, differentiation is
NOT continuous.
We can use Hybrid method.
In Small error regions, we use
2
e
Insight from Sliding Mode Control( with Low pass filter)
2
| e |
: J
, J=2
'
| e |
: J
| |,
[1, 1, 0] '
e
ee
e
J
e
T&C LAB-AI
Robotics
Example) l8abs2.py
20
No Chattering
Low pass filter
Activates!
T&C LAB-AI
Robotics
Another Idea of |e|
• Unbalanced Error
21
J
e
J
e
What is it?
Case 1: if e>0, very generous.
if e<0, very tough.
Case 1
Case 2
1/a
-a
a
-1/a
T&C LAB-AI
Robotics
Example) l8abs3.py
• a= 3 or 1/3
22
J
e
Case 1
1/a
-a
J
e
Case 2
a
-1/a
T&C LAB-AI
Robotics
Can You Image the Result?
• Example) test a=3 and a=1/3 for case I
23
J
e
Case 1
1/a
-a
J
e
Case 2
a
-1/a
T&C LAB-AI
Robotics
RBF in Noisy Signal (l8abs4.py)
• RBF learning in the Noisy Signal, y= f(x)+ sin(20x)
• The results becomes, mean value
24
T&C LAB-AI
Robotics
Unbalanced Error with Noisy Signal
25
a=6.554
How we find the magic number, 6.554?
