This document discusses self-sampling strategies for multimemetic algorithms (MMAs) in unstable computational environments subject to churn. It proposes using probabilistic models to sample new individuals when populations need to be enlarged due to node failures. Experimental results show the bivariate model is superior for high churn, maintaining diversity and convergence better than random strategies. Future work aims to extend these self-sampling strategies to dynamic network topologies and more complex probabilistic models.

1. 6th International Work-Conference on the Interplay between Natural and Artificial Computation
Self-sampling Strategies for Multimemetic
Algorithms in Unstable Computational
Environments
Rafael Nogueras Carlos Cotta
Departamento de Lenguajes y Ciencias de la Computación
Universidad de Málaga, Spain
IWINAC 2015, Elche-Elx, 1-5 June 2015
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2. Introduction Model Description Experimental Analysis Conclusions
Parallel Computing & EAs
Use of parallel and distributed
models of EAs (GAs, MAs,
MMAs, etc.) to improve solution
quality and reduce computational
times.
The island model spatially
organizes populations into
partially isolated panmictic
demes.
island1
island2
island3
island4
migrants
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3. Introduction Model Description Experimental Analysis Conclusions
Emergent Paralell Environments
Two emergent computational environments are offering new
opportunities to EAs:
I P2P networks: Equally privileged computing nodes carry out a
distributed computation without need for central coordination.
I Desktop Grids: Distributed networks of heterogeneous systems
which typically contribute computing cycles while they are
inactive (volunteer computing platforms).
Churn
The combined effect of multiple computing nodes leaving and
entering the system along time.
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4. Introduction Model Description Experimental Analysis Conclusions
Scope
Dynamic population sizing has been proposed to deal with the
phenomenon of churn.
1. Enlarge populations to cope with loss of subpopulations.
2. Exchange individuals to balance subpopulation size.
Goal
Study EAs running on unstable computational environments with
scale-free topology and fault-tolerance mechanisms:
I Use of dynamic population sizes by means of probabilistic
models.
I Impact on performance and comparison with random
strategies.
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5. Introduction Model Description Experimental Analysis Conclusions
Network Topology
Scale-free networks are commonly observed in many natural
phenomena. They feature a power-law distribution in node degrees.
This kind of networks is often the result of processes driven by
preferential attachment.
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2 3
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5
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7
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10
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18 19
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10
0
10
1
10
−2
10
−1
10
0
10
1
α=−1.9346
degree
P(degree)
Cumulative distribution function
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6. Introduction Model Description Experimental Analysis Conclusions
Network Topology
Algorithm 1: Barabási-Albert Model
function BA-Model (↓ m, n : N) : Network
m0 ← min(n, m);
net ← CreateClique(m0);
δ[1 . . . m0] ← m0;
for i ← m0 + 1 to n do
net ← AddNode(net);
for j ← 1 to m do
k ← Pick(δ) // Sampling w/o replacement ∝ δ
δ[k] ← δ[k] + 1;
net ← AddLink(net, i, k);
end
δ[i] ← m;
end
return net
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7. Introduction Model Description Experimental Analysis Conclusions
Instability
Algorithms must be executed on platforms with multiple
computing elements (processors)...
...but distributed platforms are prone to errors.
0 100 200 300 400 500 600 700 800 900 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
survival
probability
k=1
k=2
k=5
k=10
k=20
We assume node availability
follows a Weibull distributiona:
p(t1 | t0) = e
−
h
t1
β
η
−
t0
β
ηi
If the shape parameter η 1 the
hazard rate increases with time.
a
J Grid Comput, doi: 10.1007/s10723-014-9315-6, 2015
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8. Introduction Model Description Experimental Analysis Conclusions
Coping with Instability
Computing in an unstable environment requires fault-tolerance.
Classical approaches are redundancy or checkpointing. Note that:
I These strategies require access to external safe storage and
possibly some central monitoring.
I As nodes go up and down, overall population size will
fluctuate.
An alternative strategy is used1:
I No central command required: decision making and
information exchange is done locally among neighboring
islands.
I Qualitative exchange of information among islands.
1
Comput Appl Math, doi:10.1016/j.cam.2015.03.047, 2015.
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9. Introduction Model Description Experimental Analysis Conclusions
Self-balancing
b
10 16
a
22 16 18 17
c
20 18
d
16 17
← ping
→ ping
→ ping
→ pong
← status?
→ h10, 4i
← push(6)
← pong
→ status?
← h20, 3i
→ request(2)
← push(2)
← pong
→ status?
← h16, 5i
→ push(1) b a
17 21
← ping
[timeout]
Compensate the loss of islands and balance population sizes2.
2
Comput Appl Math, doi:10.1016/j.cam.2015.03.047, 2015.
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10. Introduction Model Description Experimental Analysis Conclusions
Self-sampling Strategies
Self-balancing only captures the quantitative aspect of resizing:
I New solutions are randomly constructed from scratch.
I This method introduces diversity but does not keep up the
momentum of the search.
Improvement by using smart strategies:
1. Probabilistic model to estimate the population of each island
to be enlarged.
2. New individuals are generated by sampling from previous
model.
3. Diversity is still introduced since new individuals can be
different.
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11. Introduction Model Description Experimental Analysis Conclusions
Self-sampling Strategies
Model Definition I
We consider two alternatives:
I Univariate model (UMDA) → the joint distribution is the
product of independent distributions:
p(~
x = hv1, · · · , vni) =
n
Y
j=1
p(xj = vj )
where
p(xj = vj ) =
1
µ
µ
X
i=1
δ(popij , vj )
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12. Introduction Model Description Experimental Analysis Conclusions
Self-sampling Strategies
Model Definition II
I Bivariate model (COMIT) → relations among pairs of
variables are assumed:
p(~
x = hv1, · · · , vni) = p(xj1 = vj1 )
n
Y
i=2
p(xji
= vji
| xja(i)
= vja(i)
)
where j1 · · · jn is a permutation of the indices 1 · · · n built as
follows:
• j1 is the variable with the lowest entropy H(Xk ),
• a(i) i is the permutation index of the variable which xji
depends on. It is chosen as the variable that minimizes
H(Xk | Xjs
, s i).
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13. Introduction Model Description Experimental Analysis Conclusions
Benchmark and Settings
Parameters for island-based model:
I nι = 32 islands and µ = 16 individuals (at the beginning).
I m = 2 (BA model).
Node deactivation/reactivation:
I shape parameter η = 1.5.
I scale parameters β = −1/ log(p) for p = 1 − (knι)−1,
k ∈ {1, 2, 5, 10, 20, ∞}.
Problems used:
I Deb’s trap function (concatenating 32 four-bit traps).
I HIFF function (using 128 bits).
I MMDP (using 24 six-bit blocks).
25 runs @ 50,000 evaluations are performed for each problem and
algorithm.
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14. Introduction Model Description Experimental Analysis Conclusions
Numerical Results
Approximation to the Optimum
Deviation from the optimum as a function of the churn rate.
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
80
90
100
1/k
deviation
from
optimum
(%)
noB
LBQ
rand
LBQ
umda
LBQ
comit
0.5 1 1.5 2 2.5 3 3.5 4 4.5
rank
LBQcomit
LBQ
umda
LBQrand
noB
Performance degrades with increasing churn rates but not in the
same way for the different strategies.
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15. Introduction Model Description Experimental Analysis Conclusions
Numerical Results
Evolution of Best Fitness
Evolution of best fitness on the TRAP function for different churn
rates. (Left) UMDA and (Right) COMIT.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
4
14
16
18
20
22
24
26
28
30
32
evaluations
best
fitness
K = 1
K = 2
K = 5
K = 10
K = 20
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
4
14
16
18
20
22
24
26
28
30
32
evaluations
best
fitness
K = 1
K = 2
K = 5
K = 10
K = 20
LBQcomit is clearly superior in the most severe scenarios (k = 1
and k = 2).
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16. Introduction Model Description Experimental Analysis Conclusions
Numerical Results
Evolution of Genetic Diversity
Population entropy is an indicator of algorithmic convergence.
(Left) UMDA and (Right) COMIT.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
evaluations
entropy
K = 1
K = 2
K = 5
K = 10
K = 20
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
evaluations
entropy
K = 1
K = 2
K = 5
K = 10
K = 20
LBQumda faces convergence problems as churn increases.
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17. Introduction Model Description Experimental Analysis Conclusions
Conclusions
Resilience is a key feature on unstable computational environments.
Self-sampling strategies based on probabilistic models to enlarge
populations can improve the performance of the MMA, especially
with severe churn.
Bivariate model seems superior when churn is high.
Future work:
I extend to dynamically-rewired network topologies,
I consider more complex probabilistic models (multivariate).
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18. Introduction Model Description Experimental Analysis Conclusions
Thank You!
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