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Major Defence 2014Major Defence 2014
Swarm Intelligence
Ants, Birds, Wolves and .... Bees

Major Defence 2014Major Defence 2014 2
Particle Swarm Optimisation (PSO)
PSO is an optimisation procedure based on the
social behaviour of groups of organisations (for
example the flocking of birds and the schooling of
fish)
Individual solutions in a population are viewed as
“particles” that evolve or change their positions
with time
Each particle modifies its position in search space
according to its own experience and also that of a
neighbouring particle by remembering that best
position visited by itself and its neighbours
(combining local and global search methods)

PSO update equations

Inspiration
Based on the hunting behavior of
canis lupis
Grey wolves
are considered as apex predators,
meaning that they are at the top
of the food chain. Grey wolves
mostly prefer to live in a pack. The
group size is 5–12 on average. Of
particular interest is that they
have a very strict social dominant
hierarchy.

Concept
●
The leaders are a male and a female, called alphas.
The alpha is mostly responsible for making decisions
about hunting, sleeping place, time to wake, and so on.
The alpha’s decisions are dictated to the pack. -
Interestingly,
●
the alpha is not necessarily the strongest member of the
pack but the best in terms of managing the pack.
●
The second level in the hierarchy of grey wolves is beta.
The betas are subordinate wolves that help the alpha in
decision-making or other pack activities.
●
and he/she is probably the best candidate to be the
alpha in caseone of the alpha wolves passes away or
becomes very old (i.e. non global optima follower). The
beta wolf should respect the alpha, but commands the
other lower-level wolves as well. It plays the role of an
advisor to the alpha and discipliner for the pack. The
beta reinforces the alpha’s commands throughout the
pack and gives feedback to the alpha.
●
Omega responsible for keeping structure
●
Delta are ranomd sample points akin to scout bees

Algorithm
●
Initialize the grey wolf population Xi (i = 1, 2, ..., n)
●
Initialize a, A, and C
●
Calculate the fitness of each search agent
●
X_{alpha} =the best search agent
●
X_{beta} =the second best search agent
●
X_{delta} =the third best search agent
●
while (t < Max number of iterations)
●
for each search agent
●
Update the position of the current search agent by
above equations
●
end for
●
Update a, A, and C C
●
Calculate the fitness of all search agents
●
Update X_alpha, X_beta, and X_delta
●
t=t+1
●
end while
●
return X_alpha

Algorithm
●
Circling the prey – local search
●
Where components of vec{a} are linearly
decreased from 2 to 0 over the course of iterations
and r_1, r_2 are random vectors in [0,1].
●
A grey wolf in the position of (X,Y) can update its
position according to the position of the prey
(X*,Y*). Different places around the best agent can
be reached with respect to the current position by
adjusting the value of vec{A} and vec{C} vectors.
For instance, (X*-X,Y*) can be reached by setting
vec{a}=(1,0) and vec{C}=(1,1). Note that the
random vectors r_1 and r_2 allow wolves to reach
any position between the two particular points. So a
grey wolf can update its position inside the space
around the prey in any random location by the
above-mentioned equations.
●
The same concept can be extended to a search
space with n dimensions, and the grey wolves will
move in hyper-cubes (or hyper-spheres) around the
best solution obtained so far.

(X*,Y*,Z*)
(X,Y*-Y,Z*-Z)
(X*-X,Y,Z*-Z)
(X*,Y*-Y,Z*-Z)
(X*-X,Y*-Y,Z-Z*)
(X*-X,Y*,Z*-Z)
(X,Y*,Z)
(X,Y*-Y,Z)
(X,Y*,Z*)
(X,Y,Z*)
(X*,Y*,Z*-Z) (X,Y*,Z*-Z)
(X*,Y,Z*-Z) (X,Y,Z*-Z)

Algorithm
●
On some indication of convergence we start an
exploitation mode. Mathematically the value of
vec{decrease}. Note that the fluctuation range
of vec{A} is also decreased by vec{a}. In other
words vec{A} is a random value in the interval
[-2a,2a] where a is decreased from 2 to 0 over the
course of iterations. When random values of
vec{A} are in [-1,1], the next position of a search
agent can be in any position between its current
position and the position of the prey.
●
With the operators proposed so far, the GWO
algorithm allows its search agents to update their
position based on the location of the alpha, beta,
and delta; and attack towards the prey. However,
the GWO algorithm is prone to stagnation in local
solutions with these operators. Hence the
alogithm's performance is sub optimal in areas of
high variance. The circling mechanism itself is not
sufficient, however it interesting component to
local search

Algorithm
●
Hunting:
●
Grey wolves have the ability to recognize the
location of prey and encircle them. The hunt
is usually guided by the alpha. The beta and
delta might also participate in hunting
occasionally. However, in an abstract search
space we have no idea about the location of
the optimum (prey). In order to
mathematically simulate the hunting behavior
of grey wolves, we suppose that the alpha
(best candidate solution) beta, and delta
have better knowledge about the potential
location of prey. Therefore, we save the first
three best solutions obtained so far and
oblige the other search agents (including the
omegas) to update their positions according
to the position of the best search agent

Algorithm
●
A alternating divergence and convergence is require to
prevent premature convergence. To measure
convergence/divergence criteria we use vec{c} and
vec{A}
●
Note that C>1 and convergence is indicated by A<1
●
The C vector can be also considered as the effect of
obstacles to approaching prey in nature. Generally
speaking, the obstacles in nature appear in the hunting
paths of wolves and in fact prevent them from quickly
and conveniently approaching prey. This is exactly
what the vector C does. Depending on the position of a
wolf, it can randomly give the prey a weight and make
it harder and farther to reach for wolves, or vice versa.
●
To sum up, the search process starts with creating a
random population of grey wolves (candidate solutions)
in the GWO algorithm. Over the course of iterations,
alpha, beta, and delta wolves estimate the probable
position of the prey. Each candidate solution updates
its distance from the prey. The parameter a is
decreased from 2 to 0 in order to emphasize
exploration and exploitation, respectively. Candidate
solutions tend to diverge from the prey when
|vec{A}|>1 and converge towards the prey when
|vec{A}|<1. Finally, the GWO algorithm is terminated
by the satisfaction of an end criterion.

Critique
➔
Works, but not a real hierarchy
➔
Since average is eqully weighted across all
named samples. Nothing seperates an
alpha from a beta
➔
Highly senistive.(Problem dependent) to
hieararchy depth

Bees in Nature
1- Bee colonies can span huge distances. They
are able to accurately search large spaces.

Bees in Nature
Flower patches with plentiful amounts of nectar or
pollen that can be collected with less effort should
be visited by more bees, whereas patches with
less nectar or pollen should receive fewer bees.
This is equivalent to saying areas of high fitness
have a high density of updates.

Bees in Nature
2- Scout bees search randomly from food source
to food source

Bees in Nature
3- The bees who return to the hive, evaluate the
different patches depending on certain quality
threshold (measured as a combination of some
elements, such as sugar content)

Bees in Nature
4- After depositing the nector or pollen the “dance
floor” to perform a “waggle dance”

Bees in Nature
5- Bees communicate through this waggle dance
which contains the following information:
1. The direction of flower patches (angle
between the sun and the patch)
2. The distance from the hive (duration
of the dance)
3. The quality rating (fitness) (frequency
of the dance)

Bees in Nature
These information helps the colony to send its
bees precisely
6- Follower bees go after the dancer bee to the
patch to gather food efficiently and quickly

Bees in Nature
7- The same patch will be advertised in the
waggle dance again when returning to the hive is it
still good enough as a food source (depending on
the food level) and more bees will be recruited to
that source
8- More bees visit flower patches with a (fitness)
amount of food

Bees in Nature
Depending on the fitness, a solution or food
source may be abandoned or may become
popular

Artificial Bee Colony Algorithm (ABC)
Initialization Phase
Repeat
Employed Bees Phase
Onlooker Bees Phase
Scout Bees Phase
Memorize the best solution achieved so far
Until(Cycle=Maximum Cycle Number)

Phases
●
Employed bees:
– Perform a local search to improve the existing solution
– When a solution point cannot be improved further the bee is
reassigned as a scout bee – after a certain threshold limit
●
Onlooker bees:
– Are probabalistically added to the neighborhoods of existing sample
points
– Thus better sources/samples attract more bees in their
neighborhood (positive feedback)
– using the expression given in equation :

Phases
●
Scout Phase
―
Purely random
―
Population fixed i.e. number of scouts are fixed -
―
Most unemployed bees are allocated to the onlooker worker
base

Swarm Criteria – What makes a smart
swarm?
The swarm should be able to do simple space and time computations (the proximity
principle).
The swarm should be able to respond to quality factors in the environment (the quality
principle).
The swarm should not commit its activities along excessively narrow channels (the principle
of diverse response).
The swarm should not change its mode of behavior upon every fluctuation of the environment
(the stability principle).
The swarm must be able to change behavior mode when needed (the adaptability principle).

Variation - the cause
●
Employed bees that cannot locally improve their sources are
re-assigned.
●
The reassignment is independent of where the particle used to be.
●
Thus leading to large jumps. And hence abrupt changes in the
fitness/error.
●
Leads to large improvements very quickly. But...
●
Leads to high variations in the perfomance,
●
Centalizes the algorithm -> something sarm intelligence tries to
prevent

Lifespan
●
Allows a popuation to improve itself quickly (since information can be
lost or deleted between generations)
●
However,
– No control over information lost
– No information gained

Where genetic Algs Come in
●
Framework built around the generational concept
●
Allows information transfer between generations:
●
Genetic operators like crossover can be modified to favor specific
types of information transferr
●
e.g. diemensional bias
●
This reduces the variation without icreasing centralization

Genetic Operators
●
Cross over:
– Single point
– Multi point
– Arithmetic
– Heuristic
– One-way
●
Crossover probability parameter should be reduced as the algorithm
progresses
– Since after initial area allocation is done; conserving generational
information loses priority
●
Mutation

Support: Experiments and Results

Proposed Algorithmic Improvement
Initialization Phase
Repeat
●
Context Sensitive parameters
Employed Bees Phase
●
Crossover Phase
Onlooker Bees Phase
Scout Bees Phase
Memorize the best solution achieved so far
●
Mutation Phase
Until(Cycle=Maximum Cycle Number)
•
The Crossover ensures that
information is carried from one
generation to the next
•
Crossover also dampens the
variations effects
•
Muatation allows more effective
local searching
•
Context senitive parameters allow
a more efficient algorithm
•
Why is the operator inserted in this
order? - the answer is not trivial.

GA – results of experiments
●
Several Xover operators were tried
e.g.
●
Single pt
●
Uniform – best results
●
We even made our own operator
●
Single dir Xover – leads to
a better general population
but hits thrashing problems

Effectiveness
●
Clearly leads to a more
montonica decrease in error
●
A smoother curve
●
Less varitaion –
uniformity and
consistancy
●
Behavior near higher iterations
show definite improvement

Effectiveness contd.
● Offers better performance in
higher dimensional problems
● The graph shows the the
performace of ABC vs GA across
30 runs
● Run paramters:
● Low mutation
● Low crossover
● For rastrigin(100)

PSO type falloff

Rosenbrock Greiwank

Experimental Results
Function ABC - mean(across30runs)
minimum error
ABCwith proposed modi-
fications - mean(across 30
runs) minimum error
ABC std ABC withproposed
modifications std
Sphere
n
i=1
x
2
i 562.55 541.36 39.343 31.394
Griewank 1 + 1
4000
n
i=1
x
2
i
−
n
i=1
cos(
x i√
i
)
0.1497 0.1376 0.008757 0.0086948
Rastrigin An+
n
i=1
x
2
i
− Acos(2πx i ) 1348.84 1307.85 64.885 60.479
Rosenbrock
n
i=1
[(1 − x
2
i
)
2
− 100(x i+1 −
x
2
i
)
2
]
767751.64 729240.13 9.54E+004 8.24E+004
Ackley − ae
− b
i=D
i=1
x 2
i
D
−
e
i=D
i=1
cos (cx i )
D
+a +e
9.1524613695 9.0952396525 0.045036 0.03556
Multi
Dimensional
Easom
i=D
i=1
cos (x i )e
−
i=D
i=1
(x i
− π)
2
0 0 0 0
Schwefel 418.9829D −
D
1
x i sin( |x i |)
41389.9780380048 41371.3919573781 195.8 156.84
Zakharov
i=D
i=1
x
2
i
+
0.5
i=D
i=1
ix i
2
+
0.5
i=D
i=1
ix i
4
7.85E+002 7.91E+002 2.51E+003 3.95E+003

Proposed Modifications to the ABC:
Analysis
●
The Proposed algorithm show significantly lower variation in runs and
a steadier monotonic decrease in error
●
The probability of premature convergence is mch lower
– Due to underlying ABC structure
– Control of information flow between generation
●
Proposed modifications are Local
●
Enables a high degree of distributiveness and parallizability
―
Relatively Cheap O(1) – O(n) [depending on option opted for]
―
Minimizes variance

Models
●
No known accurate models for algorithm
●
Original model very complex uses tensors... really doesn't give too much insight
– Tensor construct grows expoentially ... not computationaly feasible
●
2 phase model
– Local phase
– Non local phase
●
Pumping model
– Uses modified voroni diagram (may not use euclidean distance metric)
– Scales with number of extreama

Pumping model
●
Create a cover of the space
●
Cover must centre aound
extremas
●
Each set has a fixed search
time associated with it
●
Can be based on saddle
points/extreamas, contours,
voronoi diagram with any metric
●
The search time depends on
●
Area of the cover
●
Magnitude of extrema (relative
depth of minima and relative
height for maxima)

Pumping Model 2: update equations

Visualization
Projection into subspace leads to loss information;
we need to minimize it
Users preferr:
●
Linear projection easier to corelate
●
Limited or no change in basis
This implies:
●
Use linear embedding (eigen vector based,
LDA, etc) no non linear (kohonen maps, etc)
●
Cannot use PCA, etc at each instant of time
instead rotate the axis by certain amount
●
Amount calculated is theta. Have found an
equation forit
●
How to rotate is still a big question
Fraction of population at maximum extrema

An Intelligent System:
from Birds, Bees, Genes and
Wolves to...
The swarm
Context sensitive parameters
Intra generation information passing
Inter generationimformation passing
Hirarchies
global local
Finite lifespan
PSO
GA
ABC
Multiple sample points
GWO

What does this remind you off?
humans

Future work
Context sensitive parameters: we have just scratached the surface
Layered models
– Hierarchy based GWO is not a true hierarchy based
Semi localized models – control degree of localization
Memtic search – type1,2,3

Bibliography
[1]Hagan, Martin T., Howard B. Demuth, and Mark H. Beale. Neural network design. Boston: Pws Pub., 1996.
[2]Brownlee, Jason. Clever algorithms: nature-inspired programming recipes. Jason Brownlee, 2011.
[3]Kollar, Daphne, and Nir Friedman. Probabilistic graphical models: principles and techniques. The MIT Press, 2009.
[4]Pham, D. T., et al. "The bees algorithm–a novel tool for complex optimisation problems." Proceedings of the 2nd Virtual International Conference on
Intelligent Production Machines and Systems (IPROMS 2006). 2006.
[5]The article describes the Bees Algorithm. It first describes existing swarm/evolutionary approaches pointing out that these require a number of
parameters. It then details the behavior of bees in nature.
[6]Karaboga, Dervis, and Bahriye Akay. "A comparative study of artificial bee colony algorithm." Applied Mathematics and Computation 214.1 (2009):
108-132.

[7]Larrañaga, Pedro, et al. "A review on evolutionary algorithms in Bayesian network learning and inference tasks."
Information Sciences (2013).
[8]Zitzler, Eckart, Kalyanmoy Deb, and Lothar Thiele. "Comparison of multiobjective evolutionary algorithms: Empirical
results. (revised version)" Evolutionary computation 8.2 (2000): 173-195.
[9]Shah, Sameena, Ravi Kothari, and Suresh Chandra. "Trail formation in ants. A generalized Polya urn process." Swarm
Intelligence 4.2 (2010): 145-171.
[10]Shah, Sameena, et al. "Mathematical Modeling and Convergence Analysis of Trail Formation." AAAI. 2008.
[11]Mitchell, Melanie. "An introduction to genetic algorithms (complex adaptive systems)." (1998).

[12]Pham, D. T., et al. "The bees algorithm–a novel tool for complex optimisation problems." Proceedings of the 2nd Virtual International
Conference on Intelligent Production Machines and Systems (IPROMS 2006). 2006.
[13]Goldberg, David E., and John H. Holland. "Genetic algorithms and machine learning." Machine learning 3.2 (1988).

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