comp305. part i. artificial neural networks.. topic 3. learning rules of the artificial neural...
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COMP305. Part I.
Artificial neural networks.
Topic 3.
Learning Rules
of the Artificial Neural Networks.
Hebb’s rule (1949).
• Hebb conjectured that a particular type of use-dependent
modification of the connection strength of synapses might underlie learning in the nervous system.
Hebb’s rule (1949).• Hebb introduced a neurophysiological postulate :
“…When an axon of cell A is near enough to excite a cell B and repeatedly and persistently tales part in firing it, some growth process or metabolic change takes place in one or both cells, such that A’s efficiency as one of the cells firing B, is increased.”
Hebb’s rule (1949).
The simplest formalisation of Hebb’s rule is to increase weight of connection at every next instant in the way:
(1)where (2)
,1 kji
kji
kji www
kj
ki
kji XCaw
Hebb’s rule (1949). (1)where (2)here
wjik is the weight of connection at instant k,
wjik+1 is the weight of connection at the following instant k+1,
wjik is increment by which the weight of connection is enlarged,
C is positive coefficient which determines learning rate,
aik is input value from the presynaptic neuron at instant k,
Xjk is output of the postsynaptic neuron at the same instant k.
,1 kji
kji
kji www
kj
ki
kji XCaw
Hebb’s rule (1949). (1)where (2)
• Thus, the weight of connection changes at the next instant only if both preceding input via this connection and the resulting output simultaneously are not equal to 0.
,1 kji
kji
kji www
kj
ki
kji XCaw
Hebb’s rule (1949). (1)where (2)
• Equation (2) emphasises the correlation nature of a Hebbian synapse. It is sometimes referred to as the activity product rule.
,1 kji
kji
kji www
kj
ki
kji XCaw
Hebb’s rule (1949). (1)where (2)
• Hebb’s original learning rule (2) referred exclusively to excitatory synapses, and has the unfortunate property that it can only increase synaptic weights, thus washing out the distinctive performance of different neurons in a network, as the connections drive into saturation..
,1 kji
kji
kji www
kj
ki
kji XCaw
Hebb’s rule (1949). (1)where (2)
• However, when the Hebbian rule is augmented by a normalisation rule, e.g. keeping constant the total strength of synapses upon a given neuron, it tends to “sharpen” a neuron’s predisposition “without a teacher”, causing its firing to become better and better correlated with a cluster of stimulus patterns.
,1 kji
kji
kji www
kj
ki
kji XCaw
Normalised Hebb’s rule. (1)where (2)
normalisation: (3)
• Hebb’s rule plays an important role in studies of ANN algorithms much “younger” than the rule itself, such as unsupervised learning or self-organisation.
,1 kji
kji
kji www
kj
ki
kji XCaw
constk w
Normalised Hebb in practice. Input unit
No
1
2
3
4
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
1 1 1 1
t=0 C=1
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
1 1 1 1
t=0 C=1
.241111 2222
204
203
202
201
200
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
1 1 1 1
t=0 C=1
2
11
2
11
2
11
2
11
2
11
2
11
2
11
2
11
1;2
040
04
030
03
020
02
010
01
0
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
0.5 0.5 0.5 0.5
t=0 C=1
5.012
11
5.012
11
5.012
11
5.012
11
1;2
040
04
030
03
020
02
010
01
0
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
0.5 0.5 0.5 0.5
a01 a0
2 a03 a0
4
1 0 1 0
t=0 C=1
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
0.5 0.5 0.5 0.5
a01 a0
2 a03 a0
4
1 0 1 0
t=0 C=1
1
15.005.015.005.01
0
04
04
03
03
02
02
01
01
4
1
000
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
0.5 0.5 0.5 0.5
a01 a0
2 a03 a0
4
1 0 1 0
t=0 C=1
1
15.005.015.005.01
0
04
04
03
03
02
02
01
01
4
1
000
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w01 w0
2 w03 w0
4
0.5 0.5 0.5 0.5
a01 a0
2 a03 a0
4
1 0 1 0
t=0 C=1
.5.005.0,0101
;5.115.0,1111
;5.005.0,0101
;5.115.0,1111
04
04
14
04
03
03
13
03
02
02
12
02
01
01
11
01
001000
wwww
wwww
wwww
wwww
wwwXCaw iiiii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
1.5 0.5 1.5 0.5
a01 a0
2 a03 a0
4
1 0 1 0t=1 C=1
.5.005.0
;5.115.0
;5.005.0
;5.115.0
04
04
14
03
03
13
02
02
12
01
01
11
001
www
www
www
www
www iii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
1.5 0.5 1.5 0.5
t=1 C=1
24.255.05.15.05.1 2222
214
213
212
211
211
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
1.5 0.5 1.5 0.5
t=1 C=1
22.05.05
11
67.05.15
11
22.05.05
11
67.05.15
11
1;2
141
14
131
13
121
12
111
11
1
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1
22.05.05
11
67.05.15
11
22.05.05
11
67.05.15
11
1;2
141
14
131
13
121
12
111
11
1
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1w0
1 w02 w0
3 w04
0.5 0.5 0.5 0.5
28.0max
28.0
17.0
28.0
17.0
max
01
04
14
03
13
02
12
01
11
1 01.0?
ii
ti
ti
ww
ww
ww
ww
ww
ww
Continue…
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1
a11 a1
2 a13 a1
4
1 0 1 0
1
34.122.0067.0122.0067.01
1
14
14
13
13
12
12
11
11
4
1
111
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1
a11 a1
2 a13 a1
4
1 0 1 0
1
34.122.0067.0122.0067.01
1
14
14
13
13
12
12
11
11
4
1
111
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w11 w1
2 w13 w1
4
0.67 0.22 0.67 0.22
t=1 C=1
a11 a1
2 a13 a1
4
1 0 1 0
.22.0022.0,0101
;67.1167.0,1111
;22.0022.0,0101
;67.1167.0,1111
14
14
24
14
13
13
23
13
12
12
22
12
11
11
21
11
112111
wwww
wwww
wwww
wwww
wwwXCaw iiiii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
1.67 0.22 1.67 0.22
t=2 C=1
.22.0022.0
;67.1167.0
;22.0022.0
;67.1167.0
14
14
24
13
13
23
12
12
22
11
11
21
112
www
www
www
www
www iii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
1.67 0.22 1.67 0.22
t=2 C=1
38.222.067.122.067.1 2222
224
223
222
221
222
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
1.67 0.22 1.67 0.22
t=2 C=1
09.022.038.2
11
70.067.138.2
11
09.022.038.2
11
70.067.138.2
11
1;38.2
242
24
232
23
222
22
212
21
2
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1
09.022.038.2
11
70.067.138.2
11
09.022.038.2
11
70.067.138.2
11
1;38.2
242
24
232
23
222
22
212
21
2
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1w1
1 w12 w1
3 w14
0.67 0.22 0.67 0.22
13.0max
13.0
03.0
13.0
03.0
max
12
14
24
13
23
12
22
11
21
1 01.0?
ii
ti
ti
ww
ww
ww
ww
ww
ww
Continue…
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1
a21 a2
2 a23 a2
4
1 0 1 0
1
40.109.0070.0109.0070.01
2
24
24
23
23
22
22
21
21
4
1
222
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1
a21 a2
2 a23 a2
4
1 0 1 0
1
40.109.0070.0109.0070.01
2
24
24
23
23
22
22
21
21
4
1
222
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w21 w2
2 w23 w2
4
0.70 0.09 0.70 0.09
t=2 C=1
a21 a2
2 a23 a2
4
1 0 1 0
.09.0009.0,0101
;70.1170.0,1111
;09.0009.0,0101
;70.1170.0,1111
24
24
34
24
23
23
33
23
22
22
32
22
21
21
31
21
223222
wwww
wwww
wwww
wwww
wwwXCaw iiiii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
1.70 0.09 1.70 0.09
t=3 C=1a2
1 a22 a2
3 a24
1 0 1 0
.09.0009.0
;70.1170.0
;09.0009.0
;70.1170.0
24
24
34
23
23
33
22
22
32
21
21
31
223
www
www
www
www
www iii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
1.70 0.09 1.70 0.09
t=3 C=1
41.209.070.109.070.1 2222
234
233
232
231
233
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
1.70 0.09 1.70 0.09
t=3 C=1
04.009.041.2
11
71.070.141.2
11
04.009.041.2
11
71.070.141.2
11
1;41.2
343
34
333
33
323
32
313
31
3
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
0.71 0.04 0.71 0.04
t=3 C=1
04.009.041.2
11
71.070.141.2
11
04.009.041.2
11
71.070.141.2
11
1;41.2
343
34
333
33
323
32
313
31
3
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
0.71 0.04 0.71 0.04
t=3 C=1w2
1 w22 w2
3 w24
0.70 0.09 0.70 0.09
05.0max
05.0
01.0
05.0
01.0
max
23
24
34
23
33
22
32
21
31
1 01.0?
ii
ti
ti
ww
ww
ww
ww
ww
ww
Continue…
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
0.71 0.04 0.71 0.04
t=3 C=1
a31 a3
2 a33 a3
4
1 0 1 0
1
42.104.0071.0104.0071.01
3
34
34
33
33
32
32
31
31
4
1
333
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
0.71 0.04 0.71 0.04
t=3 C=1
a31 a3
2 a33 a3
4
1 0 1 0
1
42.104.0071.0104.0071.01
3
34
34
33
33
32
32
31
31
4
1
333
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
0.71 0.04 0.71 0.04
t=3 C=1
a31 a3
2 a33 a3
4
1 0 1 0
.04.0004.0,0101
;71.1171.0,1111
;04.0004.0,0101
;71.1171.0,1111
34
34
44
34
33
33
43
33
32
32
42
32
31
31
41
31
334333
wwww
wwww
wwww
wwww
wwwXCaw iiiii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w31 w3
2 w33 w3
4
1.71 0.04 1.71 0.04
t=3 C=1a3
1 a32 a3
3 a34
1 0 1 0
.04.0004.0
;71.1171.0
;04.0004.0
;71.1171.0
34
34
44
33
33
43
32
32
42
31
31
41
334
www
www
www
www
www iii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
1.71 0.04 1.71 0.04
t=4 C=1
42.204.071.104.071.1 2222
244
243
242
241
244
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
1.71 0.04 1.71 0.04
t=4 C=1
02.004.042.2
11
71.071.142.2
11
02.004.042.2
11
71.071.142.2
11
1;42.2
444
44
434
43
424
42
414
41
4
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
0.71 0.02 0.71 0.02
t=4 C=1
02.004.042.2
11
71.071.142.2
11
02.004.042.2
11
71.071.142.2
11
1;42.2
444
44
434
43
424
42
414
41
4
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
0.71 0.02 0.71 0.02
t=4 C=1w3
1 w32 w3
3 w34
0.71 0.04 0.71 0.04
02.0max
02.0
00.0
02.0
00.0
max
34
34
44
33
43
32
42
31
41
1 01.0?
ii
ti
ti
ww
ww
ww
ww
ww
ww
Continue…
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
0.71 0.02 0.71 0.02
t=4 C=1
a41 a4
2 a43 a4
4
1 0 1 0
1
42.102.0071.0102.0071.01
4
44
44
43
43
42
42
41
41
4
1
444
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
0.71 0.02 0.71 0.02
t=4 C=1
a41 a4
2 a43 a4
4
1 0 1 0
1
42.102.0071.0102.0071.01
4
44
44
43
43
42
42
41
41
4
1
444
X
wawawawa
waSi
ii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w41 w4
2 w43 w4
4
0.71 0.02 0.71 0.02
t=4 C=1
a41 a4
2 a43 a4
4
1 0 1 0
.02.0002.0,0101
;71.1171.0,1111
;02.0002.0,0101
;71.1171.0,1111
44
44
54
44
43
43
53
43
42
42
52
42
41
41
51
41
445444
wwww
wwww
wwww
wwww
wwwXCaw iiiii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w51 w5
2 w53 w5
4
1.71 0.02 1.71 0.02
t=4 C=1a4
1 a42 a4
3 a44
1 0 1 0
.02.0002.0
;71.1171.0
;02.0002.0
;71.1171.0
44
44
54
43
43
53
42
42
52
41
41
51
445
www
www
www
www
www iii
Normalised Hebb in practice. Input unit
No
1
2
3
4
w51 w5
2 w53 w5
4
1.71 0.02 1.71 0.02
t=5 C=1
42.202.071.102.071.1 2222
254
253
252
251
255
wwww
wwi
i
Normalised Hebb in practice. Input unit
No
1
2
3
4
w51 w5
2 w53 w5
4
1.71 0.02 1.71 0.02
t=5 C=1
01.002.042.2
11
71.071.142.2
11
01.002.042.2
11
71.071.142.2
11
1;42.2
545
54
535
53
525
52
515
51
5
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w51 w5
2 w53 w5
4
0.71 0.01 0.71 0.01
t=5 C=1
01.002.042.2
11
71.071.142.2
11
01.002.042.2
11
71.071.142.2
11
1;42.2
545
54
535
53
525
52
515
51
5
oldnew
oldnew
oldnew
oldnew
oldtinew
ti
ww
w
ww
w
ww
w
ww
w
ww
ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w51 w5
2 w53 w5
4
0.71 0.01 0.71 0.01
t=5 C=1w4
1 w42 w4
3 w44
0.71 0.02 0.71 0.02
01.0max
01.0
00.0
01.0
00.0
max
45
44
54
43
53
42
52
41
51
1 01.0?
ii
ti
ti
ww
ww
ww
ww
ww
ww
STOP!!!!
Normalised Hebb in practice. Input unit
No
1
2
3
4
1max ti
ti ww
Normalised Hebb in practice. Input unit
No
1
2
3
4
w1 w2 w3 w4
0.71 0.01 0.71 0.01
Testa1 a2 a3 a4
1 1 0 0
0
72.001.0071.0001.0171.0144332211
4
1
X
wawawawa
waSi
ii
I do not know you…