Transcript Chapter 7 - Montana State University

Introduction to Neural
Networks
John Paxton
Montana State University
Summer 2003
Chapter 7: A Sampler Of Other
Neural Nets
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Optimization Problems
Common Extensions
Adaptive Architectures
Neocognitron
I. Optimization Problems
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Travelling Salesperson Problem.
Map coloring.
Job shop scheduling.
RNA secondary structure.
Advantages of Neural Nets
• Can find near optimal solutions.
• Can handle weak (desirable, but not
required) constraints.
TSP Topology
• Each row has 1 unit that is on
• Each column has 1 unit that is on
1st
City A
City B
City C
2nd
3rd
Boltzmann Machine
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Hinton, Sejnowski (1983)
Can be modelled using Markov chains
Uses simulated annealing
Each row is fully interconnected
Each column is fully interconnected
Architecture
• ui,j connected to uk,j+1 with –di,k
• ui1 connected to ukn with -dik
b
U11
Un1
-p
U1n
Unn
Algorithm
1.
Initialize weights b, p
p>b
p > greatest distance between cities
Initialize temperature T
Initialize activations of units to
random binary values
Algorithm
2.
while stopping condition is false, do
steps 3 – 8
3.
do steps 4 – 7 n2 times (1 epoch)
4.
choose i and j randomly
1 <= i, j <= n
uij is candidate to change state
Algorithm
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6.
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Compute c = [1 – 2uij]b + S S ukm (-p)
where k <> i, m <> j
Compute probability to accept change
a = 1 / (1 + e(-c/T) )
Accept change if random number [0..1]
< a. If change, uij = 1 – uij
Adjust temperature T = .95T
Stopping Condition
• No state change for a specified number of
epochs.
• Temperature reaches a certain value.
Example
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T(0) = 20
½ units are on initially
b = 60
p = 70
10 cities, all distances less than 1
200 or fewer epochs to find stable
configuration in 100 random trials
Other Optimization Architectures
• Continuous Hopfield Net
• Gaussian Machine
• Cauchy Machine
– Adds noise to input in attempt to escape from
local minima
– Faster annealing schedule can be used as a
consequence
II. Extensions
• Modified Hebbian Learning
– Find parameters for optimal surface fit of
training patterns
Boltzmann Machine With Learning
• Add hidden units
• 2-1-2 net below could be used for simple
encoding/decoding (data compression)
y1
x1
z1
x2
y2
Simple Recurrent Net
• Learn sequential or time varying patterns
• Doesn’t necessarily have steady state
output
• input units
• context units
• hidden units
• output units
Architecture
c1
x1
z1
y1
xn
zp
ym
cp
Simple Recurrent Net
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f(ci(t)) = f(zi(t-1))
f(ci(0)) = 0.5
Can use backpropagation
Can learn string of characters
Example: Finite State Automaton
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4 xi
4 yi
2 zi
2 ci
A
BEGIN
END
B
Backpropagation In Time
• Rumelhart, Williams, Hinton (1986)
• Application: Simple shift register
1 (fixed)
x1
y1
x1
z1
x2
y2
x2
1 (fixed)
Backpropagation Training for Fully
Recurrent Nets
• Adapts backpropagation to arbitrary
connection patterns.
III. Adaptive Architectures
• Probabilistic Neural Net (Specht 1988)
• Cascade Correlation (Fahlman, Lebiere
1990)
Probabilistic Neural Net
• Builds its own architecture as training
progresses
• Chooses class A over class B if hAcAfA(x) >
hBcBfB(x)
• cA is the cost of classifying an example as
belonging to A when it belongs to B
• hA is the a priori probability of an example
belonging to class A
Probabilistic Neural Net
• fA(x) is the probability density function for
class A, fA(x) is learned by the net
• zA1: pattern unit, fA: summation unit
zA1
x1
zAj
fA
y
xn
zB1
zBk
fB
Cascade Correlation
• Builds own architecture while training
progresses
• Tries to overcome slow rate of
convergence by other neural nets
• Dynamically adds hidden units (as few as
possible)
• Trains one layer at a time
Cascade Correlation
• Stage 1
x0
y1
x1
y2
x2
Cascade Correlation
• Stage 2 (fix weights into z1)
x0
y1
x1
z1
x2
y2
Cascade Correlation
• Stage 3 (fix weights into z2)
x0
y1
x1
z1
z2
y2
x2
Algorithm
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Train stage 1. If error is not
acceptable, proceed.
2.
Train stage 2. If error is not
acceptable, proceed.
3.
Etc.
IV. Neocognitron
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Fukushima, Miyako, Ito (1983)
Many layers, hierarchical
Very spare and localized connections
Self organizing
Supervised learning, layer by layer
Recognizes handwritten 0, 1, 2, 3, … 9,
regardless of position and style
Architecture
Layer
# of Arrays
Size
Input
1
192
S1 / C1
12 / 8
192 / 112
S2 / C2
38 / 22
112 / 72
S3 / C3
32 / 30
72 / 72
S4 / C4
16 / 10
32 / 12
Architecture
• S layers respond to patterns
• C layers combine results, use larger field
of view
• For example S11 responds to
000
111
000
Training
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Progresses layer by layer
S1 connections to C1 are fixed
C1 connections to S2 are adaptable
A V2 layer is introduced between C1 and
S2, V2 is inhibatory
• C1 to V2 connections are fixed
• V2 to S2 connections are adaptable