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Ming-Feng Yeh 1
CHAPTER 11CHAPTER 11
Back-Back-PropagationPropagation
Ming-Feng Yeh 2
ObjectivesObjectives
A generalization of the LMS algorithm, called backpropagation, can be used to train multilayer networks.
Backpropagation is an approximate steepest descent algorithm, in which the performance index is mean square error.
In order to calculate the derivatives, we need to use the chain rule of calculus.
Ming-Feng Yeh 3
MotivationMotivation
The perceptron learning and the LMS algorithm were designed to train single-layer perceptron-like networks.They are only able to solve linearly separable classification problems.Parallel Distributed ProcessingThe multilayer perceptron, trained by the backpropagation algorithm, is currently the most widely used neural network.
Ming-Feng Yeh 4
Three-Layer NetworkThree-Layer Network
321 SSSR Number of neurons in each layer:
Ming-Feng Yeh 5
Pattern Classification: Pattern Classification: XOR gateXOR gate
The limitations of the single-layer perceptron (Minsky & Papert, 1969)
0,
0
011 tp
1,
1
022 tp
1,
0
133 tp
0,
1
144 tp
1P
2P 4P
3P
Ming-Feng Yeh 6
Two-Layer XOR NetworkTwo-Layer XOR Network
Two-layer, 2-2-1 network
11w
12w
1P
4P
AND
11
1
11n
12n
1
5.1
11a
12a
21n 2
1a
1p
2p
22
11
5.1
Individual Decisions
Ming-Feng Yeh 7
Solved Problem P11.1Solved Problem P11.1
Design a multilayer network to distinguish these categories.
T1 1111 p
T2 1111 p
T3 1111 p
T4 1111 p
Class I Class II01 bWp02 bWp
03 bWp04 bWp
There is no hyperplane that can separate these two categories.
Ming-Feng Yeh 8
Solution of Problem P11.1Solution of Problem P11.1
11
1
11n
12n
1
11a
12a
21n 2
1a
1p
2p
1
1
2
2
2
2
3p
4p
AND
OR
Ming-Feng Yeh 9
Function ApproximationFunction Approximation
Two-layer, 1-2-1 networknnf
enf
n
)( ,
1
1)( 21
.10,10,10,10 12
11
12
11 bbww
.0,1,1 221
21 bww
Ming-Feng Yeh 10
Function ApproximationFunction Approximation
The centers of the steps occur where the net input to a neuron in the first layer is zero.
The steepness of each step can be adjusted by changing the network weights.
110100
110)10(012
12
12
12
12
11
11
11
11
11
wbpbpwn
wbpbpwn
Ming-Feng Yeh 11
Effect of Parameter ChangesEffect of Parameter Changes
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1
0
1
2
3
12b
20 15 10 5 0
Ming-Feng Yeh 12
Effect of Parameter ChangesEffect of Parameter Changes
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1
0
1
2
3
21w
1.0
0.5
0.0
-0.5
-1.0
Ming-Feng Yeh 13
Effect of Parameter ChangesEffect of Parameter Changes
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1
0
1
2
3
21w
1.0
0.5
0.0
-0.5
-1.0
Ming-Feng Yeh 14
Effect of Parameter ChangesEffect of Parameter Changes
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1
0
1
2
3
2b
1.0
0.5
0.0
-0.5
-1.0
Ming-Feng Yeh 15
Function ApproximationFunction Approximation
Two-layer networks, with sigmoid transfer functions in the hidden layer and linear transfer functions in the output layer, can approximate virtually any function of interest to any degree accuracy, provided sufficiently many hidden units are available.
Ming-Feng Yeh 16
Backpropagation AlgorithmBackpropagation Algorithm
For multilayer networks the outputs of one layer becomes the input to the following layer.
1,...,2,1,0 ),( 1111 Mmmmmmm baWfaMaapa ,0
Ming-Feng Yeh 17
Performance IndexPerformance Index
Training Set:
Mean Square Error:
Vector Case:
Approximate Mean Square Error:
Approximate Steepest Descent Algorithm
p1 t1{ , } p2 t2{ , } pQ tQ{ , }
F x E e2 = E t a– 2 =
F x E eTe = E t a–
Tt a– =
F̂ x t k a k – T t k a k – eTk e k = =
w i jm
k 1+ wi jm
k F̂
w i jm
------------–= bim
k 1+ bim
k F̂
bim
---------–=
Ming-Feng Yeh 18
Chain RuleChain Rule
If f(n) = en and n = 2w, so that f(n(w)) = e2w.
Approximate mean square error:
dw
wdn
dn
ndf
dw
wndf )()())((
2)()())((
nedw
wdn
dn
ndf
dw
wndf
)()()]()([)]()([)(ˆ TT kkkkkkF eeatatx
mji
mi
mi
mjim
ji
mji
mji w
n
n
Fkw
w
Fkwkw
,,
,,,
ˆ)(
ˆ)()1(
mi
mi
mi
mim
i
mi
mi b
n
n
Fkb
b
Fkbkb
ˆ
)(ˆ
)()1(
Ming-Feng Yeh 19
Sensitivity & GradientSensitivity & Gradient
The net input to the ith neurons of layer m:
The sensitivity of to changes in the ith element of the net input at layer m:
Gradient:
mi
S
j
mj
mji
mi bawn
m
1
1
1, 1 ,1
,
mi
mim
jmji
mi
b
na
w
n
F̂mi
mi nFs ˆ
1
,,
ˆˆ
m
jmim
ji
mi
mi
mji
asw
n
n
F
w
F
mi
mim
i
mi
mi
mi
ssb
n
n
F
b
F
1ˆˆ
Ming-Feng Yeh 20
Steepest Descent AlgorithmSteepest Descent Algorithm
The steepest descent algorithm for the approximate mean square error:
Matrix form:
1,
,,, )(
ˆ)()1(
mj
mi
mjim
ji
mi
mi
mji
mji askw
w
n
n
Fkwkw
mi
mim
i
mi
mi
mi
mi skb
b
n
n
Fkbkb
)(ˆ
)()1(
Wm
k 1+ Wm
k sm
am 1–
T
–=
bmk 1+ bm
k sm–=
sm F
nm
----------
F
n1m
---------
F
n2m
---------
F
nS
mm
-----------
=
Ming-Feng Yeh 21
BP the SensitivityBP the Sensitivity
Backpropagation: a recurrence relationship in which the sensitivity at layer m is computed from the sensitivity at layer m+1. Jacobian matrix:
.
1
2
1
1
1
12
2
12
1
12
11
2
11
1
11
1
111
m
s
m
sm
m
sm
m
s
m
s
m
m
m
m
m
m
s
m
m
m
m
m
m
m
m
mmm
m
m
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
Ming-Feng Yeh 22
Matrix RepressionMatrix Repression
The i,j element of Jacobian matrix
).(
)(
1,
1,
1,
1
11,1
mj
mmji
mj
mj
mm
jimj
mjm
jimj
s
l
mi
ml
mli
mj
mi
nfw
n
nfw
n
aw
n
baw
n
n
m
,)(11
mmmm
m
nFWn
n
.
)(00
0)(0
00)(
)( 2
1
m
s
m
mm
mm
mm
mnf
nf
nf
nF
Ming-Feng Yeh 23
Recurrence RelationRecurrence Relation
The recurrence relation for the sensitivity
The sensitivities are propagated backward through the network from the last layer to the first layer.
.))((
ˆ))((
ˆˆ
1T1
11
1
T1
mmmm
mTmm
mm
m
mm
sWnF
n
FWnF
n
F
n
n
n
Fs
.121 ssss MM
Ming-Feng Yeh 24
Backpropagation AlgorithmBackpropagation Algorithm
At the final layer:
.)(2
)()()(ˆ
1
2
T
Mi
iiiM
i
S
jjj
Mi
Mi
Mi n
aat
n
at
nn
Fs
M
atat
)()( M
iM
Mi
Mi
M
Mi
Mi
Mi
i nfn
nf
n
a
n
a
)()(2 Mi
Mii
Mi nfats
))((2 atnFs MMM
Ming-Feng Yeh 25
SummarySummary
The first step is to propagate the input forward through the network:
The second step is to propagate the sensitivities backward through the network: Output layer: Hidden layer:
The final step is to update the weights and biases:
Maa1,...,2,1,0 ),( 1111 Mmmmmmm baWfa
pa 0
))((2 atnFs MMM 1,2,...,1 ,))(( 1T1 Mmmmmmm sWnFs
T1)()()1( mmmm kk asWW mmm kk sbb )()1(
Ming-Feng Yeh 26
BP Neural NetworkBP Neural Network
m
jS mw,1
mjw ,1
mjiw ,
Layer m
1
j
mS
1
i
Layer m-1
1mS
mw 1,1
miw 1,
m
S mw1,1
m
SS mmw,1
m
S mw,1
m
Si mw,
1
k
MS
Layer MMa1
Mka
M
S Ma
Layer 1
1p
2p
Rp
11,1w
11,2w
1
1,1Sw
1
,1 RSw
1
2
1S
Ming-Feng Yeh 27
Ex: Function ApproximationEx: Function Approximation
g p 14---p sin+=
1-2-1Network
+
t
ep
Ming-Feng Yeh 28
Network ArchitectureNetwork Architecture
1-2-1Network
ap
Ming-Feng Yeh 29
Initial ValuesInitial ValuesW1
0 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
Initial Network Response:
p
2a
Ming-Feng Yeh 30
Forward PropagationForward Propagationa
0p 1= =
a1 f1 W1a0 b1+ l ogsig 0.27–
0.41–1
0.48–
0.13–+
logsig 0.75–
0.54–
= = =
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= = = =
a1
1
1 e0.75+--------------------
1
1 e0.54+--------------------
0.321
0.368= =
Initial input:Output of the 1st layer:
Output of the 2nd layer:
error:
Ming-Feng Yeh 31
Transfer Func. DerivativesTransfer Func. Derivatives
))(1(1
1
1
11
)1(1
1)(
11
21
aaee
e
e
edn
dnf
nn
n
n
n
1)()(2 ndn
dnf
Ming-Feng Yeh 32
BackpropagationBackpropagation
The second layer sensitivity:
The first layer sensitivity:
522.2261.112
)]([2))((2 22222
enfn atFs
0997.0
0495.0
522.217.0
09.0
368.0)368.01(0
0321.0)321.01(
))(1(0
0))(1())(( 2
22,1
21,1
12
12
11
1122111 ssWnFs
w
w
aa
aaT
Ming-Feng Yeh 33
Weight UpdateWeight Update
Learning rate 1.0
0772.0171.0
368.0321.0]522.2[1.017.009.0
)()0()1( 1222
TasWW
]732.0[]522.2[1.0]48.0[)0()1( 222 sbb
420.0
265.0]1[
0997.0
0495.01.0
41.0
27.0
)()0()1( 0111 TasWW
140.0
475.0
0997.0
0495.01.0
13.0
48.0)0()1( 111 sbb
Ming-Feng Yeh 34
Choice of Network StructureChoice of Network Structure
Multilayer networks can be used to approximate almost any function, if we have enough neurons in the hidden layers.
We cannot say, in general, how many layers or how many neurons are necessary for adequate performance.
Ming-Feng Yeh 35
Illustrated Example 1Illustrated Example 1
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 1i 2i
4i 8i
Ming-Feng Yeh 36
Illustrated Example 2Illustrated Example 2
g p 164
------ 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
22 p
Ming-Feng Yeh 37
ConvergenceConvergenceg p 1 p sin+=
-2 -1 0 1 2-1
0
1
2
3
1
23
4
5
0
-2 -1 0 1 2-1
0
1
2
3
1
2
34
5
0
22 p
Convergence to Global Min. Convergence to Local Min.The numbers to each curve indicate the sequence of iterations.
Ming-Feng Yeh 38
GeneralizationGeneralization
In most cases the multilayer network is trained with a finite number of examples of proper network behavior:
This training set is normally representative of a much larger class of possible input/output pairs.
Can the network successfully generalize what it has learned to the total population?
p1 t1{ , } p2 t2{ , } pQ tQ{ , }
Ming-Feng Yeh 39
Generalization ExampleGeneralization Exampleg p 1
4---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
For a network to be able to generalize, it should have fewer parameters than there are data points in the training set.
Generalize well Not generalize well