Part 3: Autonomous Agents
11/10/04
Sufficient Conditions for
Instability (Case 1)
Conditions for Stability
Stability of entire pattern :
x m = sgn x m + 1n x k cos km km
Suppose x im = 1. Then unstable if :
(1) + 1n x ik cos km > 0
km
Stability of a single bit :
x im = sgn x im + 1n x ik cos km km
11/10/04
1
n
2
Sufficient Conditions for
Stability
Suppose x im = +1. Then unstable if :
(+1) + x
cos km > 1
11/10/04
Sufficient Conditions for
Instability (Case 2)
k
i
k
i
km
1
1
n
x
1
n
x
k
i
cos km 1
km
cos km < 0
km
1
n
x
k
i
The crosstalk with the sought pattern must be
sufficiently small
cos km < 1
km
11/10/04
3
11/10/04
4
1
Part 3: Autonomous Agents
11/10/04
Single Bit Stability Analysis
Capacity of Hopfield Memory
• For simplicity, suppose xk are random
• Then xk xm are sums of n random ±1
• Depends on the patterns imprinted
• If orthogonal, pmax = n
– but every state is stable trivial basins
• So pmax < n
• Let load parameter = p / n
binomial distribution Gaussian
in range –n, …, +n
with mean µ = 0
and variance 2 = n
• Probability sum > t:
1
2
t 1 erf 2n [See “Review of Gaussian (Normal) Distributions” on course website]
11/10/04
5
equations
Approximation of Probability
Let crosstalk Cim =
1
n
x (x
k
i
km
k
11/10/04
6
Probability of Bit Instability
n Pr{nCim > n} = 12 1 erf 2np xm )
We want Pr{Cim > 1} = Pr{nCim > n}
p
=
n
Note : nC = x x x
m
i
k
i
k
j
1
2
[1 erf (
n 2p
)]
m
j
k=1 j=1
km
A sum of n( p 1) np random ± 1
Variance 2 = np
11/10/04
7
11/10/04 (fig. from Hertz & al. Intr. Theory Neur. Comp.)
8
2
Part 3: Autonomous Agents
11/10/04
Tabulated Probability of
Single-Bit Instability
–
–
–
–
–
–
Perror
11/10/04
Spurious Attractors
• Mixture states:
0.1%
0.105
0.36%
0.138
1%
0.185
5%
0.37
10%
0.61
• Spin-glass states:
– not correlated with any finite number of imprinted patterns
– occur beyond overload because weights effectively random
(table from Hertz & al. Intr. Theory Neur. Comp.)
9
11/10/04
10
Fraction of Unstable Imprints
(n = 100)
Basins of Mixture States
x k1
sums or differences of odd numbers of retrieval states
number increases combinatorially with p
shallower, smaller basins
basins of mixtures swamp basins of retrieval states overload
useful as combinatorial generalizations?
self-coupling generates spurious attractors
x k3
x mix
x k2
11/10/04
x imix = sgn( x ik1 + x ik2 + x ik3 )
11
11/10/04
(fig from Bar-Yam)
12
3
Part 3: Autonomous Agents
11/10/04
Number of Stable Imprints
(n = 100)
11/10/04
(fig from Bar-Yam)
Number of Imprints with Basins
of Indicated Size (n = 100)
13
11/10/04
(fig from Bar-Yam)
14
Summary of Capacity Results
• Absolute limit: pmax < cn = 0.138 n
• If a small number of errors in each pattern
permitted: pmax n
• If all or most patterns must be recalled
perfectly: pmax n / log n
• Recall: all this analysis is based on random
patterns
• Unrealistic, but sometimes can be arranged
11/10/04
Stochastic Neural Networks
(in particular, the stochastic Hopfield network)
15
11/10/04
16
4
Part 3: Autonomous Agents
11/10/04
Trapping in Local Minimum
11/10/04
Escape from Local Minimum
17
11/10/04
18
Escape from Local Minimum
11/10/04
19
5