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Activation functions
• Unioplar
• Bipolar
Activation functions
Example
Suppose a feedforward neural network with n inputs,
m hidden units (tanh activation), and l output units (linear
activation). vji is the weight from input i to hidden unit j. wkj is
the weight from hidden unit j to output unit k.
If the error is we can find partial derivatives
(backpropagation) and apply gradient descent.
Hebbian Learning Rule
• The learning signal is equal simply to the neuron’s output (Hebb
1949). We have :
( )
w of the weight vector becomes
w ( )
t
i
i
t
i i
r f w x
The increment
cf w x x
=
∆
∆ =
single weight w is adapted using the following
w ( )
This can be briefly written as
w , fo rj=1, 2, 3.......n
ij
t
ij i j
ij i j
The
cf w x x
co x
∆ =
∆ =
Hebbian Learning Rule
• This rule represents a purely feed forward, unsupervised
learning.
• This rule states that if the crossproduct of the output and
the input or correlation term is positive, this results in an
increase of weight, otherwise the weight decreases.
Perceptron Learning Rule
• The learning signal is the difference between the desired
and the actual neuron’s response (Rosenblatt 1958).
Thus, learning is supervised and the learning signal is
equal to :
• And di is the desired response.
• Weight adjustments are obtained as follows :
where sgn( x)t
i i i ir d o o w= − =
[ sgn( x)]xt
i i iw c d w∆ = −
[ sgn( x)] for j=1,2....nt
ij i i jw c d w x∆ = −
Perceptron Learning Rule
• This rule is applicable only for binary neuron response,
and the above relationships express the rule for the bipolar
binary case.
• Here, the weights are adjusted if and only if is
incorrect.
• Since the desired response is either +1 or -1, the weight
adjustment reduces to
• Where a + sign is applicable when di = 1 and
• The weight adjustment is zero when the desired and the
actual responses agree.
io
2 xiw c∆ = ±
sgn( x) 1t
iw = −
Delta learning Rule
• The delta learning rule only valid for continuous activation
function and in the supervised training mode
• The learning signal for this mode is called delta and is
defined as follows
•
the term f1
(wt
ix) is the derivative of the activation function
f(net) computed for
net = wt
ix
[ ( x)] ( x)t t
i i ir d f w f w′= −
f(net i)
Continuous
perception
oi
di
di-oir
c
x
∆wi
x1
X2
.
.
.
.xj
xn
Delta learning rule
Delta learning Rule
• Learning rule can derived from the condition of least
squared error between oi and di
• Calculating the gradient vector with respect to wi of the
squared error defined as
• which is equivalent to
21
( )
2
i iE d o= −
21
[ ( x)]
2
t
i iE d f w= −
Delta learning Rule
• We obtain the error gradient vector value
∀∇E= -(di-oi) f1
(wt
ix)x
• The components of the gradient vector are
• since the minimization of the error requires the weight
changes to be in the negative gradient direction,we take
∀∆wi= -η∇E where η is a positive constant
Delta learning Rule
• We then obtain
∀∆wi = η(di-oi) f1
(neti)x
• or, for the single weight the adjustment becomes
∀∆wij = η(di-oi) f1
(neti)xj, for j=1,2,…,n
• note that weight adjustments computed based on
minimization of the squared error
Delta learning Rule
• Considering the use of the general learning rule and
plugging in the learning signal the weighting adjustment
becomes
∀∆wi = c(di-oi) f1
(neti)x
Widrow-Hoff learning Rule
• The Windrow-Hoff learning rule is applicable for the
supervised training of neural networks
• It is independent of the activation function of neurons
used since it maximizes the squared error between the
desired output value di and the neuron’s activation
value
neti = wi
t
x
Widrow-Hoff learning Rule
• The learning signal for this rule is defined as
follows r = di - wi
t
x
• the weight vector increment under this learning
rule is
or, for the single weight, the adjustment is
j = 1, 2 ….n
• this rule can be considered a special case of the
delta learning rule .
t
i i iw =c (d - w x) xV
t
ij i i jw =c (d - w x) xV
Widrow-Hoff learning Rule
• assuming that f(wi
t
x)= wi
t
x, or the activation function is
simply the identity function
f(net)=net, f ’
(net)=1.
• This rule is sometimes called the LMS (Least mean
square)learning rule.
• weights are initialized at any values in this method.
Correlation Learning Rule
• By substituting r = di into the general learning
rule we obtain the correlation learning rule.
• The adjustments for the weight vector and the
single weights respectively, are
∆wi=cdix
∆ wij =cdixj for j=1,2,….n
Winner_take_All Learning Rule
• Winner_take_All Learning Rule is used for learning
statistical properties of input.
• The learning is based on the premise that one of the
neurons in the layer, say the m’th
, has the max. response
due to input x,as shown in.
• This neuron is declared the winner.As a result of this
winning event, the weight vector wm
Figure 2.25
Winning
neuron
X1
.
.
Xj
.
.
.
Xn
W11
W1j
W1n
Wm1
Wmj
Wmn
Wp1
Wpj
Wpn
o1
op
on
Winner_take_All Learning Rule
• Wm=[wm1 wm2 …. Wmn]t
• containing weights highlighted in the figure is the only
one adjusted in the given unsupervised learning step
• Its increment is computed as follows
∆wm=α(x-wm)
• or,the individual weight adjustment becomes
∆wmj=
α(xj-wmj) for j=1,2, …n
Winner_take_All Learning Rule
• Where ∝>0 is a small learning constant,typically
decreasing as learning progresses
• the winner selection is based on the following criterion of
max activation among all p neurons participating in a
competition:
wm
t
x = max(wi
t
x) i=1,2, … n
Outstar Learning Rule
• The weight adjustments in this rule are computed as
follows ∆wj
=β (d-wj)
• or, the individual adjustments are
∆wmj =β (dm-wmj) for m=1,2,..p
• note that in contrast to any learning rule discussed so
far, the adjusted weights are fanning out of the j’th node
in this learning
Outstar Learning Rule
method and the weight vector is defined accordingly as
wj=[w1j w2j … wpj]t
X1
.
.
Xj
.
.
.
Xn
W11
W1j
W1n
Wm1
Wmj
Wmn
Wp1
Wpj
Wpn
o1
op
on
d1
dm
dp
β
β
β
∆wij
∆wmj
∆wpj

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Nural network ER. Abhishek k. upadhyay Learning rules

  • 1.
  • 2.
  • 5. Example Suppose a feedforward neural network with n inputs, m hidden units (tanh activation), and l output units (linear activation). vji is the weight from input i to hidden unit j. wkj is the weight from hidden unit j to output unit k. If the error is we can find partial derivatives (backpropagation) and apply gradient descent.
  • 6. Hebbian Learning Rule • The learning signal is equal simply to the neuron’s output (Hebb 1949). We have : ( ) w of the weight vector becomes w ( ) t i i t i i r f w x The increment cf w x x = ∆ ∆ = single weight w is adapted using the following w ( ) This can be briefly written as w , fo rj=1, 2, 3.......n ij t ij i j ij i j The cf w x x co x ∆ = ∆ =
  • 7. Hebbian Learning Rule • This rule represents a purely feed forward, unsupervised learning. • This rule states that if the crossproduct of the output and the input or correlation term is positive, this results in an increase of weight, otherwise the weight decreases.
  • 8. Perceptron Learning Rule • The learning signal is the difference between the desired and the actual neuron’s response (Rosenblatt 1958). Thus, learning is supervised and the learning signal is equal to : • And di is the desired response. • Weight adjustments are obtained as follows : where sgn( x)t i i i ir d o o w= − = [ sgn( x)]xt i i iw c d w∆ = − [ sgn( x)] for j=1,2....nt ij i i jw c d w x∆ = −
  • 9. Perceptron Learning Rule • This rule is applicable only for binary neuron response, and the above relationships express the rule for the bipolar binary case. • Here, the weights are adjusted if and only if is incorrect. • Since the desired response is either +1 or -1, the weight adjustment reduces to • Where a + sign is applicable when di = 1 and • The weight adjustment is zero when the desired and the actual responses agree. io 2 xiw c∆ = ± sgn( x) 1t iw = −
  • 10. Delta learning Rule • The delta learning rule only valid for continuous activation function and in the supervised training mode • The learning signal for this mode is called delta and is defined as follows • the term f1 (wt ix) is the derivative of the activation function f(net) computed for net = wt ix [ ( x)] ( x)t t i i ir d f w f w′= −
  • 12. Delta learning Rule • Learning rule can derived from the condition of least squared error between oi and di • Calculating the gradient vector with respect to wi of the squared error defined as • which is equivalent to 21 ( ) 2 i iE d o= − 21 [ ( x)] 2 t i iE d f w= −
  • 13. Delta learning Rule • We obtain the error gradient vector value ∀∇E= -(di-oi) f1 (wt ix)x • The components of the gradient vector are • since the minimization of the error requires the weight changes to be in the negative gradient direction,we take ∀∆wi= -η∇E where η is a positive constant
  • 14. Delta learning Rule • We then obtain ∀∆wi = η(di-oi) f1 (neti)x • or, for the single weight the adjustment becomes ∀∆wij = η(di-oi) f1 (neti)xj, for j=1,2,…,n • note that weight adjustments computed based on minimization of the squared error
  • 15. Delta learning Rule • Considering the use of the general learning rule and plugging in the learning signal the weighting adjustment becomes ∀∆wi = c(di-oi) f1 (neti)x
  • 16. Widrow-Hoff learning Rule • The Windrow-Hoff learning rule is applicable for the supervised training of neural networks • It is independent of the activation function of neurons used since it maximizes the squared error between the desired output value di and the neuron’s activation value neti = wi t x
  • 17. Widrow-Hoff learning Rule • The learning signal for this rule is defined as follows r = di - wi t x • the weight vector increment under this learning rule is or, for the single weight, the adjustment is j = 1, 2 ….n • this rule can be considered a special case of the delta learning rule . t i i iw =c (d - w x) xV t ij i i jw =c (d - w x) xV
  • 18. Widrow-Hoff learning Rule • assuming that f(wi t x)= wi t x, or the activation function is simply the identity function f(net)=net, f ’ (net)=1. • This rule is sometimes called the LMS (Least mean square)learning rule. • weights are initialized at any values in this method.
  • 19. Correlation Learning Rule • By substituting r = di into the general learning rule we obtain the correlation learning rule. • The adjustments for the weight vector and the single weights respectively, are ∆wi=cdix ∆ wij =cdixj for j=1,2,….n
  • 20. Winner_take_All Learning Rule • Winner_take_All Learning Rule is used for learning statistical properties of input. • The learning is based on the premise that one of the neurons in the layer, say the m’th , has the max. response due to input x,as shown in. • This neuron is declared the winner.As a result of this winning event, the weight vector wm
  • 22. Winner_take_All Learning Rule • Wm=[wm1 wm2 …. Wmn]t • containing weights highlighted in the figure is the only one adjusted in the given unsupervised learning step • Its increment is computed as follows ∆wm=α(x-wm) • or,the individual weight adjustment becomes ∆wmj= α(xj-wmj) for j=1,2, …n
  • 23. Winner_take_All Learning Rule • Where ∝>0 is a small learning constant,typically decreasing as learning progresses • the winner selection is based on the following criterion of max activation among all p neurons participating in a competition: wm t x = max(wi t x) i=1,2, … n
  • 24. Outstar Learning Rule • The weight adjustments in this rule are computed as follows ∆wj =β (d-wj) • or, the individual adjustments are ∆wmj =β (dm-wmj) for m=1,2,..p • note that in contrast to any learning rule discussed so far, the adjusted weights are fanning out of the j’th node in this learning
  • 25. Outstar Learning Rule method and the weight vector is defined accordingly as wj=[w1j w2j … wpj]t