AN INVESTIGATION INTO THE USE OF SWARM INTELLIGENCE

FOR AN EVOLUTIONARY ALGORITHM OPTIMISATION

The Optimisation Performance of Differential Evolution Algorithm Coupled with

Stochastic Diffusion Search

Mohammad Majid al-Rifaie, John Mark Bishop and Tim Blackwell

Department of Computing, Goldsmiths College, University of London, London SE14 6NW, U.K.

Keywords:

SDS, DE, EC, SI, Metaheuristics.

Abstract:

The integration of Swarm Intelligence (SI) algorithms and Evolutionary algorithms (EAs) might be one of

the future approaches in the Evolutionary Computation (EC). This work narrates the early research on using

Stochastic Diffusion Search (SDS) – a swarm intelligence algorithm – to empower the Differential Evolution

(DE) – an evolutionary algorithm – over a set of optimisation problems. The results reported herein suggest

that the powerful resource allocation mechanism deployed inSDS has the potential to improve the optimisation

capability of the classical evolutionary algorithm used in this experiment. Different performance measures and

statistical analyses were utilised to monitor the behaviour of the ﬁnal coupled algorithm.

1 INTRODUCTION

In the literature, nature inspired swarm intelligence

algorithms and biologically inspired evolutionary al-

gorithms are typically evaluated using benchmarks

that are often small in terms of their objective func-

tion computational costs (Whitley et al., 1996); this

is often not the case in real-world applications. This

paper is an attempt to pave the way for more effec-

tively optimising computationally expensive objec-

tive functions, by deploying the SDS diffusion mech-

anism to more efﬁciently allocate DE resources via

information-sharingbetween the members of the pop-

ulation. The use of SDS as an efﬁcient resource allo-

cation algorithm was ﬁrst explored in (Nasuto, 1999)

and these results provided motivation to investigate

the application of the information diffusion mecha-

nism originally deployed in SDS

1

with DE.

In this paper, the swarm intelligence algorithm

and the evolutionary algorithm are ﬁrst introduced,

followed by the coupling strategy. The results are re-

ported afterwards and the performance of the coupled

algorithm will be discussed.

1

The ‘information diffusion’ and ‘randomised partial

objective function evaluation’ processes enable SDS to

more efﬁciently optimise problems with costly [discrete]

objective functions; see Stochastic Diffusion Search Sec-

tion for an introduction to the SDS metaheuristic.

2 STOCHASTIC DIFFUSION

SEARCH

This section introduces SDS (Bishop, 1989), a

multi-agent global search and optimisation algorithm,

which is based on simple interaction of agents (in-

spired by one species of ants, Leptothorax acervo-

rum, where a ‘tandem calling’ mechanism (one-to-

one communication) is used, where the forager ant

which ﬁnds the food location, recruits a single ant

upon its return to the nest, and therefore the location

of the food is physically publicised). A high-level

description of SDS is presented in the form of a so-

cial metaphor demonstrating the procedures through

which SDS allocates resources.

SDS introduced a new probabilistic approach

for solving best-ﬁt pattern recognition and matching

problems. SDS, as a multi-agent population-based

global search and optimisation algorithm, is a dis-

tributed mode of computation utilising interaction be-

tween simple agents. Unlike many nature inspired

search algorithms, SDS has a strong mathematical

framework, which describes the behaviour of the al-

gorithm by investigating its resource allocation, con-

vergence to global optimum, robustness and minimal

convergence criteria and linear time complexity. In

order to introduce SDS, a social metaphor the Mining

Game (al-Rifaie and Bishop, 2010) is used.

553

al-Rifaie M., Bishop J. and Blackwell T..

AN INVESTIGATION INTO THE USE OF SWARM INTELLIGENCE FOR AN EVOLUTIONARY ALGORITHM OPTIMISATION - The Optimisation

Performance of Differential Evolution Algorithm Coupled with Stochastic Diffusion Search.

DOI: 10.5220/0003723005530558

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FEC-2011), pages 553-558

ISBN: 978-989-8425-83-6

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

2.1 The Mining Game

This metaphor provides a simple high-level descrip-

tion of the behaviour of agents in SDS, where moun-

tain range is divided into hills and each hill is divided

into regions:

A group of miners learn that there is gold to be

found on the hills of a mountain range but have

no information regarding its distribution. To maxi-

mize their collective wealth, the maximum number

of miners should dig at the hill which has the rich-

est seams of gold (this information is not available

a-priori). In order to solve this problem, the min-

ers decide to employ a simple Stochastic Diffusion

Search.

• At the start of the mining process each miner is

randomly allocated a hill to mine (his hill hy-

pothesis, h).

• Every day each miner is allocated a randomly

selected region, on the hill to mine.

At the end of each day, the probability that a miner

is happy is proportional to the amount of gold he

has found. Every evening, the miners congregate

and each miner who is not happy selects another

miner at random for communication. If the chosen

miner is happy, he shares the location of his hill and

thus both now maintain it as their hypothesis, h; if

not, the unhappy miner selects a new hill hypothe-

sis to mine at random.

As this process is isomorphic to SDS, miners will nat-

urally self-organise to congregate over hill(s) of the

mountain with high concentration of gold.

In the context of SDS, agents take the role of

miners; active agents being ‘happy miners’, inactive

agents being ‘unhappy miners and the agent’s hypoth-

esis being the miner’s ‘hill-hypothesis’.

2.2 SDS Architecture

The SDS algorithm commences a search or optimi-

sation by initialising its population (e.g. miners, in

the mining game metaphor). In any SDS search, each

agent maintains a hypothesis, h, deﬁning a possible

problem solution. In the mining game analogy, agent

hypothesis identiﬁes a hill. After initialisation two

phases are followed (for high-level SDS description

see Algorithm 1):

• Test Phase (e.g. testing gold availability)

• Diffusion Phase (e.g. congregation and exchang-

ing of information)

In the test phase, SDS checks whether the agent

hypothesis is successful or not by performing a partial

hypothesis evaluation which returns a boolean value.

Later in the iteration, contingent on the precise re-

cruitment strategy employed, successful hypotheses

Algorithm 1: SDS Algorithm.

Initialising agents ()

While (stopping condition is not met)

Testing hypotheses ()

Diffusion hypotheses ()

End

diffuse across the population and in this way informa-

tion on potentially good solutions spreads throughout

the entire population of agents.

In the Test phase, each agent performs partial

function evaluation, pFE, which is some function of

the agent’s hypothesis; pFE = f(h). In the mining

game the partial function evaluation entails mining a

random selected region on the hill, which is deﬁned

by the agent’s hypothesis (instead of mining all re-

gions on that hill).

In the Diffusion phase, each agent recruits another

agent for interaction and potential communication of

hypothesis. In the mining game metaphor, diffusion

is performed by communicating a hill hypothesis.

2.3 Partial Function Evaluation

The commonly used benchmarks for evaluating the

performance of swarm intelligence algorithms are

typically small in terms of their objective functions

computational costs, which is often not the case in

real-world applications. Examples of costly evalua-

tion functions are seismic data interpretation (Whit-

ley et al., 1996), selection of sites for the transmis-

sion infrastructure of wireless communication net-

works and radio wave propagation calculations of one

site (Whitaker and Hurley, 2002) etc.

Many ﬁtness functions are decomposable to com-

ponents that can be evaluated separately. In partial

evaluation of the ﬁtness function in SDS, the evalua-

tion of one or more of the components may provide

partial information to guide the subsequent optimisa-

tion process.

3 DIFFERENTIAL EVOLUTION

DE, one of the most successful Evolutionary Algo-

rithms (EAs), is a simple global numberical optimiser

over continuous search spaces which was ﬁrst intro-

duced by Storn and Price (Storn and Price, 1995).

DE is a population based stochastic algorithm,

proposed to search for an optimum value in the feasi-

ble solution space. The parameter vectors of the pop-

ulation are deﬁned as follows:

x

g

i

=

h

x

g

i,1

, x

g

i,2

, ..., x

g

i,D

i

, i = 1, 2, ..., NP (1)

FEC 2011 - Special Session on Future of Evolutionary Computation

554

where g is the current generation, D is the dimension

of the problem space and NP is the population size.

In the ﬁrst generation, (when g = 0), the i

th

vector’s

j

th

component could be initialised as:

x

0

i, j

= x

min, j

+ r(x

max, j

− x

min, j

) (2)

where r is a random number drawn from a uniform

distribution on the unit intervalU (0, 1), and x

min

, x

max

are the lower and upper bounds of the j

th

dimen-

sion, respectively. The evolutionary process (muta-

tion, corssover and selection) starts after the initiali-

sation of the population.

3.1 Mutation

At each generation g, the mutation operation is ap-

plied to each member of the population x

g

i

(target vec-

tor) resulting in the corresponding vector v

g

i

(mutant

vector). Among the most frequently used mutation

approaches are the following:

• DE/rand/1

v

g

i

= x

g

r

1

+ F

x

g

r

2

− x

g

r

3

(3)

• DE/target-to-best/1

v

g

i

= x

g

i

+ F

x

g

best

− x

g

i

+ F

x

g

r

1

− x

g

r

2

(4)

• DE/best/1

v

g

i

= x

g

best

+ F

x

g

r

1

− x

g

r

2

(5)

• DE/best/2

v

g

i

= x

g

best

+ F

x

g

r

1

− x

g

r

2

+ F

x

g

r

2

− x

g

r

3

(6)

• DE/rand/2

v

g

i

= x

g

r

1

+ F

x

g

r

2

− x

g

r

3

+ F

x

g

r

4

− x

g

r

5

(7)

where r

1

, r

2

, r

3

, r

4

are different from i and are distinct

random integers drawn from the range [1, NP]; In gen-

eration g, the vector with the best ﬁtness value is x

g

best

and F is a positive control parameter for constricting

the difference vectors.

3.2 Crossover

Crossover operation, improves population diversity

through exchanging some components of v

g

i

(mutant

vector) with x

g

i

(target vector) to generate u

g

i

(trial vec-

tor). This process is led as follows:

u

g

i, j

=

v

g

i, j

, if r ≤ CR or j = r

d

x

g

i, j

, otherwise

(8)

where r is a uniformly distributed random number

drawn from the unit interval U (0, 1), r

d

is randomly

generated integer from the range [1, D]; this value

guaranteesthat at least one componentof the trial vec-

tor is different from the targetvector. The valueofCR,

which is another control parameter, specifes the level

of inheritance from v

g

i

(mutant vector).

3.3 Selection

The selection operation decides whether x

g

i

(target

vector) or u

g

i

(trial vector) would be able to pass to

the next generation (g + 1). In case of a minimisa-

tion problem, the vector with a smaller ﬁtness value

is admitted to the next generation:

x

g+1

i

=

u

g

i

, if f

u

g

i

≤ f

x

g

i

x

g

i

, otherwise

(9)

where f (x) is the ﬁtness function.

DE is known to be relatively good in compar-

ison with other EAs and PSOs at avoiding prema-

ture convergence. However, in order to reduce the

risk of premature convergence in DE and to pre-

serve population diversity, several methods have been

proposed, among which are: multi-population ap-

proaches (Brest et al., 2009); providing extra knowl-

edge about the problem space (Weber et al., 2010);

information storage about previously explored areas

(Zhang and Sanderson, 2009) and utilising adapting

and control parameters to ensure population diversity

(Zaharie, 2003).

4 COUPLING SDS AND DE

The initial motivating thesis justifying the coupling

of SDS and DE is the partial function evaluation de-

ployed in SDS, which may mitigate the high com-

putational overheads entailed when deploying a DE

algorithm onto a problem with a costly ﬁtness func-

tion. However, before commenting on and explor-

ing this area – which remains an ongoing research –

an initial set of experiments aimed to investigate the

scenario where the optimisation process is initialised

by n number of function evaluations (FEs) performed

within the SDS test-diffusion cycle, in order to allo-

cate the resources (agent) to the promising areas of

the search space and then passing on the agents’ po-

sitions to DE to resume the optimisation process, in

most cases as a local search.

The goal of this process is to verify whether the in-

formation diffusion and dispensation mechanisms de-

ployed in SDS may on its own improve DE behaviour.

These are the results that are primarily reported in this

paper.

AN INVESTIGATION INTO THE USE OF SWARM INTELLIGENCE FOR AN EVOLUTIONARY ALGORITHM

OPTIMISATION - The Optimisation Performance of Differential Evolution Algorithm Coupled with Stochastic Diffusion

Search

555

In this new architecture, a standard set of bench-

marks are used to evaluate the performance of the

coupled algorithm. The resource allocation (or re-

cruitment) and partial function evaluation sides of

SDS (see Section 2.3) are used to assist allocating re-

sources after partially evaluating the search space.

Each DE agent has three vectors (target, mutant

and trial vectors); and each SDS agent has one hy-

pothesis and one status. In the experiment reported

here (coupled algorithm), as stated before, SDS test-

diffusion cycle is run for n of FEs and then DE com-

mences with the optimisation, taking its target vectors

from SDS agents’ positions.

The behaviour of the coupled algorithm in its sim-

plest form is presented in Algorithm 2.

Algorithm 2: Coupled Algorithm.

Initialise Agents

x = initialInactiveErrorVector (e.g. 4)

y = initialActiveErrorVector (e.g. 1)

// x > y

n = SDS_FE_Allowed

//SDS cycle

While ( FE <= n )

{

// Decreasing the error vector over time

If ( FE < stoppingErrV_DecreasePoint )

iErrorV = x - (x*FE) / stoppingErrV_DecPoint

aErrorV = y - (y*FE) / stoppingErrV_DecPoint

End If

// stoppingErrV_DecPoint < SDS_FE_Allowed

// TEST PHASE

For ag = 1 to NP

r_ag = pick-random -agent ()

If ( F(ag) < F(r_ag) )

ag.setActivity (true)

Else

ag.setActivity (false)

End If

End For

// DIFFUSION PHASE

For ag = 1 to NP

If ( ag is not active )

r_ag = pick -random -agent()

If ( r_ag is active )

ag.setHypo (

Gaussian (r_ag.getHypo (),iErrorV))

Else

ag.setHypo ( randomHypo () )

End If

End If

Else

ag.setHypo(Gaussian(ag.getHypo(), aErrorV))

End for

}

// DE

While ( FE < FE_Allowed )

For ( Agent = 1 to NP )

Mutation : generate mutant vector

Crossover : generate trial vector

Selection : generate target vector

End For

Find Agent with best fitness value

End For

4.1 Test and Diffusion Phases in the

Coupled Algorithm

In the test-phase of a stochastic diffusion search, each

agent has to partially evaluate its hypothesis. The

guiding heuristic is that hypotheses that are promis-

ing are maintained and those that appear unpromising

are discarded.

In the context of the coupled SDS-DE algorithm,

it is clear that there are many different tests that could

be performed in order to determine the activity of

each agent.

A very simple test is illustrated in Algorithm 2.

Here, the test-phase is simply conducted by compar-

ing the ﬁtness of each agent’s ﬁtness against that of a

random agent; if the selecting agent has a better ﬁt-

ness value, it will become active, otherwise it will be

ﬂagged inactive. On average, this mechanism will en-

sure 50% of agents remain active from one iteration

to another.

In the Diffusion Phase, each inactive agent picks

another agent randomly, if the selected agent is active,

the selected agent communicates its hypothesis to the

inactive one; if the selected agent is also inactive, the

selecting agent generates a new hypothesis at random

from the search space.

As outlined in the pseudo-code of the coupled al-

gorithm (see Algorithm 2), after the initial n function

evaluations during which SDS test-diffusion cylce it-

erates, DE algorithm should run.

In the next section, the experiment setup is re-

ported and the results will follow.

5 EXPERIMENTAL SETUP

In this work, a number of experiments are carried

out and the performance of one variation of DE al-

gorithm (DE/best/1) is contrasted against the coupled

SDS-DE algorithm (sDE). The algorithms are tested

over a number of standard benchmarking functions,

preserving different dimensionality and modality (see

(al-Rifaie et al., 2011a; al-Rifaie et al., 2011b) for

more information on the benchmarks used). The ex-

periments are conducted with the population of 100

agents. The halting criterion for this experiment is

when the function evaluations reaches 300, 000.

There are 30 independent runs for each bench-

mark function and the results are averaged over these

independent trials.

The stopping condition for decreasing the error

vectors is reaching 80, 000 FEs. DE is run after

100, 000 FEs until the temination criterion which is

300, 000 FEs. These values were selected merely to

FEC 2011 - Special Session on Future of Evolutionary Computation

556

Table 1: Accuracy Details.

DE sDE sDispDE

SDS-DE SDS (Disp) DE

f

1

2.80E-78±2.65E-78 1.35E-37±1.06E-37 3.36E-54±2.01E-54

f

2

6.31E-02±1.55E-02 8.15E-01±2.00E-01 7.58E+00±1.55E+00

f

3

3.45E+01±8.04E+00 3.45E+01±4.52E+00 2.65E+01±4.08E+00

f

4

4.59E+02±1.31E+02 8.55E+02±2.44E+02 6.17E+00±1.10E+00

f

5

1.75E+02±8.18E+00 5.69E+01±1.80E+00 2.48E+01±1.26E+00

f

6

1.87E+01±8.84E-01 2.29E+00±6.48E-02 7.52E-01±1.30E-01

f

7

5.79E-02±1.77E-02 1.02E+00±4.68E-01 1.18E-02±2.99E-03

f

8

1.34E+01±2.94E+00 3.80E-02±2.20E-02 1.69E-01±8.07E-02

f

9

1.62E+00±3.56E-01 9.36E-02±2.50E-02 3.33E-02±1.48E-02

f

10

4.90E-01±7.42E-02 1.04E-16±2.06E-17 1.18E-16±2.06E-17

f

11

1.57E+02±4.21E+01 0.00E+00±0.00E+00 5.92E-17±2.80E-17

f

12

5.05E+00±7.38E-17 1.06E-08±2.37E-09 2.28E+00±4.90E-01

f

13

5.27E+00±0.00E+00 2.64E-07±4.22E-08 1.76E+00±4.63E-01

f

14

5.36E+00±9.99E-17 2.84E-07±5.17E-08 2.85E+00±5.37E-01

Accuracy (±standard error) is shown with two decimal places after 30 trials

of 300,000 FEs. For each benchmark, the best algorithm(s) which is signif-

icantly better (see Table 2) than the others is highlighted. In cases where

more than one algorithm is highlighted in a row, the highlighted algorithms

do not signiﬁcantly outperform each other.

provide a brief initial exploration of the behaviour of

the new coupled algorithm; no claim is made for their

optimality.

6 EXPERIMENTAL RESULTS

Table 1 shows the performance of the coupled algo-

rithm (sDE) alongside DE algorithm. For each bench-

mark and algorithm, the table shows the accuracy

measure. The overal reliability of each algorithm is

also reported.

The focus of this paper is not ﬁnding the best stop-

ping point for decreasing the error vectors or the val-

ues of the initial error vectors (for this set of bench-

marks), but rather investigate the effect of SDS algo-

rithm on the performance of DE algorithm.

As Table 2 shows, over all benchmarks, other than

f

7

, DE algorithm does not signiﬁcantly outperform

the coupled algorithm. On the other hand, in most

cases (f

5−6

and f

8−14

), the coupled algorithm signiﬁ-

cantly outperforms the classical DE algorithm.

More results and analyses are presented in the next

section.

7 DISCUSSION

The resource allocation process underlying SDS of-

fers three closely coupled mechanisms to the algo-

rithm’s search component to speed its convergence to

global optima:

• ‘efﬁcient, non-greedy information sharing’ in-

stantiated via positive feedback of potentially

good hypotheses between agents;

• dispensation mechanism – SDS-led random-

restarts – deployed as part of the diffusion phase;

• random ‘partial hypothesis evaluation’, whereby

a complex, computationally expensive objective

function is broken down into ‘k independent

partial-functions’, each one of which, when eval-

uated, offers partial information on the absolute

quality of current algorithm search parameters. It

is this mechanism of iterated selection of a ran-

dom partial function that ensures SDS does not

prematurely converge on local minimum.

To further analyse the role of SDS in the cou-

pled algorithm, the Diffusion Phase of SDS algorithm

is modiﬁed (see Algorithm 3) to investigate the dis-

pensation effect caused by randomising a selection of

agent hypotheses (effectively instantiating the popu-

lation with SDS-led random-restarts). In other words,

after the SDS test-phase, the hypothesis of each inac-

tive agent is randomised.

As detailed in Table 1, although, information shar-

ing plays an important role in the performance of

the coupled algorithm, the signiﬁcance of dispensa-

tion mechanism (in randomly restarting some of the

agents) in improving the performance of the algo-

rithm cannot be discarded.

In some cases (f

4,5,7

), solely the dispensation

mechanism (sDispDE), which is facilitated by the

test-phase of the SDS algorithm, demonstrates a sig-

niﬁcantly better performance compared to the cou-

pled algorithm (see Table 1). However, in the several

cases, the coupled algorithms outperform the modi-

ﬁed algorithm: f

2,8

and f

10−14

, out of which f

2

and

f

12−14

are performing signiﬁcantly better (see Table

2).

Table 1 shows that among the highlighted algo-

rithms, out of 14 bechmarks, sDE exhibits the best

performance as it is among the most signiﬁcant in 9

cases; sDispDE and DE are among the best in 7 and

2 cases, respectively.

The results show the importance of coupling

the SDS-led restart mechanism (dispensation mecha-

nism) and the information sharing which are both de-

ployed in SDS algorithm.

The third SDS component feature, which is cur-

rently only implicitly exploited by the coupled algo-

rithm, is ‘randomised partial hypothesis evaluation’

(see (al-Rifaie et al., 2011b) for a detailed explana-

tion on the implicit deployment of this feature).

AN INVESTIGATION INTO THE USE OF SWARM INTELLIGENCE FOR AN EVOLUTIONARY ALGORITHM

OPTIMISATION - The Optimisation Performance of Differential Evolution Algorithm Coupled with Stochastic Diffusion

Search

557

Table 2: TukeyHSD Test Results for Accuracy.

DE - sDE DE - sDispDE sDE - sDispDE

f

1

– – –

f

2

– X–o X–o

f

3

– – –

f

4

– – o–X

f

5

o–X o–X o–X

f

6

o–X o–X –

f

7

X–o – o–X

f

8

o–X o–X –

f

9

o–X o–X –

f

10

o–X o–X –

f

11

o–X o–X –

f

12

o–X o–X X–o

f

13

o–X o–X X–o

f

14

o–X o–X X–o

Based on TukeyHSD Test, if the difference between each pair of algorithms

is signiﬁcant, the pairs are marked. X–o shows that the left algorithm is

signiﬁcantly better than the right one; and o–X shows that the right algorithm

is signiﬁcantly better than the one, on the left.

Algorithm 3: SDS Dispensation coupled with DE

(sDispDE).

// DIFFUSION PHASE

For ag = 1 to No_of_agents

If ( ag is not active )

ag.setHypo( randomHypo () )

Else

ag.setHypo(Gaussian (ag.getHypo (),aErrorV ))

End If

End For

7.1 Conclusions

This paper presents a brief overview about the poten-

tial of coupling of DE with SDS. Here, SDS is pri-

marily used as an efﬁcient resource allocation and

dispensation mechanism responsible for facilitating

communication between the agents at the early stages

of the optimisation. Results reported in this paper

have demonstrated that initial explorations with the

coupled sDE algorithm outperform the performance

of (one variation of) classical DE architecture. We

believe similar techniques (e.g. (Omran et al., 2011))

can be applied to other swarm intelligence and evo-

lutionary algorithms. As reported in (al-Rifaie et al.,

2011a; al-Rifaie et al., 2011b) SDS has been also suc-

cessfully integrated (vs. coupled) into PSO and DE

in a different framework. In ongoing research, fur-

ther theoretical work seeks to develop the core ideas

presented in this paper on problems with signiﬁcantly

more computationally expensive objective functions.

This reinforces the idea of the integration of SI

algorithms with EAs as a potential future approach in

Evolutionary Computation.

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