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11
1
Backpropagation
11
2
Multilayer Perceptron
R – S1 – S2 – S3 Network
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3
Example
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4
Elementary Decision Boundaries
First Subnetwork
First Boundary:a1
1hardlim 1– 0 p 0.5+ =
Second Boundary:
a21
hardlim 0 1– p 0.75+ =
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5
Elementary Decision Boundaries
Third Boundary:
Fourth Boundary:
Second Subnetwork
a31
hardlim 1 0 p 1.5– =
a41
hardlim 0 1 p 0.25– =
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6
Total Network
W1
1– 0
0 1–1 0
0 1
= b1
0.50.75
1.5–0.25–
=
W2 1 1 0 0
0 0 1 1= b2 1.5–
1.5–=
W3
1 1= b30.5–=
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Function Approximation Example
f1
n 1
1 en–
+-----------------=
f2
n n=
w1 11
10= w 2 11
10= b11
10–= b21
10=
w1 12
1= w1 22
1= b2
0=
Nominal Parameter Values
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Nominal Response
-2 -1 0 1 2-1
0
1
2
3
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Parameter Variations
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
1– w 1 12
1
1– w1 22
1
0 b21
20
1– b2
1
11
10
Multilayer Network
am 1+
fm 1+
Wm 1+
amb
m 1++ = m 0 2 M 1– =
a0
p=
a aM=
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Performance Index
p1 t1{ , } p2 t2{ , } pQ tQ{ , }
Training Set
F x E e2 = E t a– 2 =
Mean Square Error
F x E eTe = E t a–
Tt a– =
Vector Case
F̂ x t k a k – T t k a k – eT k e k = =
Approximate Mean Square Error (Single Sample)
w i jm
k 1+ wi jm
k F̂
w i jm
------------–= bim
k 1+ bim
k F̂
bim
---------–=
Approximate Steepest Descent
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Chain Rule
f n w dwd
-----------------------f n d
nd--------------
n w dwd
---------------=
f n n cos= n e2w
= f n w e2w cos=
f n w dwd
-----------------------f n d
nd--------------
n w dwd
--------------- n sin– 2e2w e
2w sin– 2e2w = = =
Example
Application to Gradient Calculation
F̂
w i jm
------------
F̂
nim
---------ni
m
wi jm
------------= F̂
bim
---------F̂
nim
---------ni
m
bim
---------=
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Gradient Calculation
nim
wi jm
ajm 1–
j 1=
Sm 1–
bim
+=
nim
wi jm
------------ a jm 1–
=ni
m
bim
--------- 1=
sim F̂
nim
---------
Sensitivity
F̂
w i jm
------------ sim
ajm 1–
=F̂
bim
--------- si
m=
Gradient
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Steepest Descent
wi jm
k 1+ wi jm
k sim
a jm 1–
–= bim
k 1+ bim
k sim
–=
Wm
k 1+ Wm
k sma
m 1–
T–= bm
k 1+ bmk sm–=
sm F̂
nm
----------
F̂
n1m
---------
F̂
n2m
---------
F̂
nS
mm
-----------
=
Next Step: Compute the Sensitivities (Backpropagation)
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Jacobian Matrix
nm 1+
nm
-----------------
n1m 1+
n1m
----------------
n1m 1+
n2m
----------------
n1m 1+
nS
mm
----------------
n2m 1+
n1m
----------------
n2m 1+
n2m
----------------
n2m 1+
nS
mm
----------------
n
Sm 1+m 1+
n1m
----------------
nSm 1+m 1+
n2m
----------------
nSm 1+m 1+
nS
mm
----------------
nim 1+
n jm
----------------
wi lm 1+
alm
l 1=
Sm
bim 1+
+
n jm
----------------------------------------------------------- wi jm 1+ a j
m
n jm
---------= =
nim 1+
n jm
---------------- wi jm 1+ f m n j
m
njm
--------------------- wi jm 1+
fÝm
n jm = =
fÝm
n jm
f m n jm
njm
---------------------=
nm 1+
nm----------------- Wm 1+ FÝ
mnm = FÝ
mn
m
fÝm
n1m 0 0
0 fÝm
n2m 0
0 0 fÝm
nS
mm
=
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Backpropagation (Sensitivities)
sm F̂
nm---------- n
m 1+
nm
-----------------
T
F̂
nm 1+
----------------- FÝ
mnm Wm 1+
T F̂
nm 1+
-----------------= = =
sm
FÝmn
m( ) W
m 1+
Ts
m 1+=
The sensitivities are computed by starting at the last layer, andthen propagating backwards through the network to the first layer.
sM
sM 1–
s2
s1
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Initialization (Last Layer)
siM F̂
niM
----------t a–
Tt a–
niM
---------------------------------------
tj a j– 2
j 1=
SM
niM
----------------------------------- 2 ti ai– –ai
niM
----------= = = =
sM
2FÝMn
M( ) t a– –=
ai
niM
----------ai
M
niM
----------f
Mn i
M
niM
----------------------- fÝM
n iM = = =
siM
2 ti ai– – fÝM
n iM =
11
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Summary
am 1+
fm 1+
Wm 1+
amb
m 1++ = m 0 2 M 1– =
a0
p=
a aM=
sM
2FÝMn
M( ) t a– –=
sm
FÝmn
m( ) W
m 1+
Ts
m 1+= m M 1– 2 1 =
Wm
k 1+ Wm
k sma
m 1–
T–= b
mk 1+ b
mk s
m–=
Forward Propagation
Backpropagation
Weight Update
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Example: Function Approximation
g p 14---p
sin+=
1-2-1Network
+
-
t
a
ep
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Network
1-2-1Network
ap
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Initial Conditions
W10 0.27–
0.41–= b1
0 0.48–
0.13–= W2
0 0.09 0.17–= b20 0.48=
Network ResponseSine Wave
-2 -1 0 1 2-1
0
1
2
3
11
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Forward Propagation
a0
p 1= =
a1 f1 W1a0 b1+ l ogsig 0.27–
0.41–1
0.48–
0.13–+
logsig 0.75–
0.54–
= = =
a1
1
1 e0.75+--------------------
1
1 e0.54+--------------------
0.321
0.368= =
a2
f2 W2a1 b2
+ purelin 0.09 0.17–0.321
0.3680.48+( ) 0.446= = =
e t a– 1 4---p
sin+
a2– 1 4---1
sin+
0.446– 1.261= = = =
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Transfer Function Derivatives
fÝ1
n nd
d 1
1 en–
+----------------- e
n–
1 en–
+ 2
------------------------ 11
1 en–
+-----------------–
1
1 en–
+-----------------
1 a1
– a1 = = = =
fÝ2
n nd
dn 1= =
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Backpropagation
s2
2FÝ2n
2( ) t a– – 2 fÝ
2n
2 1.261 – 2 1 1.261 – 2.522–= = = =
s 1 FÝ1n1
( ) W2 Ts 2 1 a1
1– a1
1 0
0 1 a21
– a21
0.09
0.17–2.522–= =
s1 1 0.321– 0.321 0
0 1 0.368– 0.368 0.090.17–
2.522–=
s 1 0.218 0
0 0.233
0.227–
0.429
0.0495–
0.0997= =
11
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Weight Update
W21 W2
0 s2 a1 T
– 0.09 0.17– 0.1 2.522– 0.321 0.368–= =
0.1=
W21 0.171 0.0772–=
b21 b2
0 s2– 0.48 0.1 2.522–– 0.732= = =
W11 W1
0 s 1 a0 T
– 0.27–
0.41–0.1 0.0495–
0.09971– 0.265–
0.420–= = =
b11 b1
0 s1– 0.48–
0.13–0.1 0.0495–
0.0997– 0.475–
0.140–= = =
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Choice of Architecture
g p 1i 4----- p sin+=
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
1-3-1 Network
i = 1 i = 2
i = 4 i = 8
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Choice of Network Architecture
g p 16 4
------ p sin+=
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
1-5-1
1-2-1 1-3-1
1-4-1
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Convergence
g p 1 p sin+=
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
1
23
4
5
0
1
2
34
5
0
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Generalization
p1 t1{ , } p2 t2{ , } pQ tQ{ , }
g p 14---p
sin+= p 2– 1.6– 1.2– 1.6 2 =
-2 -1 0 1 2-1
0
1
2
3
-2 -1 0 1 2-1
0
1
2
3
1-2-1 1-9-1
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