Sensitivity Analysis of Checkpointing Strategies for Multimemetic Algorithms on Unstable Complex Networks

Abstract

The use of volatile decentralized computational platforms such as, e.g., peer-to-peer networks, is becoming an increasingly popular option to gain access to vast computing resources. Making an effective use of these resources requires algorithms adapted to such a changing environment, being resilient to resource volatility. We consider the use of a variant of evolutionary algorithms endowed with a classical fault-tolerance technique, namely the creation of checkpoints in a safe external storage. We analyze the sensitivity of this approach on different kind of networks (scale-free and small-world) and under different volatility scenarios. We observe that while this strategy is robust under low volatility conditions, in cases of severe volatility performance degrades sharply unless a high checkpoint frequency is used. This suggest that other fault-tolerance strategies are required in these situations.

Full text

Sensitivity Analysis of Checkpointing Strategies
for Multimemetic Algorithms on Unstable
Complex Networks
Rafael Nogueras and Carlos Cotta
Dept. Lenguajes y Ciencias de la Computaci´on, Universidad de M´alaga,
ETSI Inform´atica, Campus de Teatinos, 29071 M´alaga, Spain
ccottap@lcc.uma.es
Abstract. The use of volatile decentralized computational platforms
such as, e.g., peer-to-peer networks, is becoming an increasingly popular
option to gain access to vast computing resources. Making an effective
use of these resources requires algorithms adapted to such a changing
environment, being resilient to resource volatility. We consider the use
of a variant of evolutionary algorithms endowed with a classical fault-
tolerance technique, namely the creation of checkpoints in a safe external
storage. We analyze the sensitivity of this approach on different kind of
networks (scale-free and small-world) and under different volatility sce-
narios. We observe that while this strategy is robust under low volatil-
ity conditions, in cases of severe volatility performance degrades sharply
unless a high checkpoint frequency is used. This suggest that other fault-
tolerance strategies are required in these situations.
1 Introduction
Distributed computing platforms have been used for running population-based
metaheuristics for decades now. This is a direct consequence of the flexibility
and adaptability of these techniques whose functioning is intrinsically parallel.
Hence they can be naturally deployed on networked computers, cf. [1]. Numerous
research works have focused on different design aspects of these techniques and
how they affect performance in distributed environments – see, e.g., [2, 6, 25].
Exploiting efficiently distributed computing resources has become one of the
signature weapons of these techniques and is a major factor for boosting their
performance. In this sense, it is worth noting how technological advances are
reshaping both the underlying computational substrate and the very needs to
be addressed in computational terms. Regarding the latter, the problems and
their data are becoming increasingly larger and complex [22]. The term Big
Data [28] is nowadays a hot buzzword used to denote such large collections of
data, very much requiring vast computational power in order to harnessed.
While traditional supercomputing techniques (namely, dedicated systems
hosting a large array of processors and colossal memory banks) are certainly
one of the lines of attack to Big Data problems, the preponderance of computing

2 R. Nogueras and C. Cotta
resources permanently connected to the Internet has led to the emergence of
other computational environments such as peer-to-peer (P2P) networks [14] and
volunteer computing networks [23]. These are bound to play a key role in this
kind of endeavors since they allow the orchestration of enormous decentralized
collections of computational nodes. This comes at a cost though: these computa-
tional resources are unstable (they are typically contributed by volunteers during
their idle time) and this must be taken into account when deploying applications
on this kind of environments. Focusing specifically on applications (population-
based metaheuristics in our case) running natively on these environment (i.e.,
being aware of its dynamicity and dealing with it directly), they can either use
some fault-management policy for corrective purposes [12] or can self-adapt their
behavior/parameterization to cope with it. We aim our attention at the former
approach in this work. More precisely we analyze the performance of strategies
based on creating restoration checkpoints [16]. This is done within the context
of multimemetic algorithms [11], namely memetic algorithms which self-adapt
the local search procedure, cf. [21]. These are described next.
2 Fault-Tolerant Model in an Island-based Multimemetic
Algorithm
As stated before we consider the use of multimemetic algorithms (MMAs) on
a unstable computational scenario. Our MMA is organized as an island-based
algorithm [24, 29], that is, it has a population distributed over a collection of n
islands. Each of these islands comprises a panmictic (i.e., unstructured) subpopu-
lation and runs a basic steady-state MMA procedure. This procedure follows the
standard pattern of memetic algorithms, namely, selection, recombination, mu-
tation and local search [15] but has a distinctive feature: local search is not done
using a predefined strategy but using search patterns (memes) embedded in each
individual and evolving alongside the latter (note the connection with the con-
cept of memetic computing [20]). Inspired by the model by Smith [26,27], these
memes are expressed as variable-length pattern-based rewriting rules A→B
(i.e., find Ain the genome and change it into B; both A, B ∈Σ∪ {#}where Σ
is the alphabet used for encoding solutions and # is a wildcard). They evolve
via mutation and are transferred from parent to offspring via local selection. We
refer to [18] for further details.
The islands are distributed over a network of nodes and perform migration
asynchronously (randomly picking an individual from an island and transferring
it to another one, where it replaces the worst individual [17]). Two factors de-
fine this computational scenario: the interconnection topology and the dynamic
model of the network. Regarding the topology, we consider to possibilities:
–Scale-free networks (SF): these are characterized by the existence of a power-
law distribution in node degrees, and are often observed in many natural
processes. We use the Barab´asi-Albert (BA) model [3] to generate this kind of
SF networks. This model uses preferential attachment [4] to grow a network

Analysis of Checkpointing for MMAs on Unstable Complex Networks 3
Fig. 1: Example networks for n= 32. The left figure is a SF network (m= 2)
and the right one is a SW network (M= 61).
by adding a new node at a time. A parameter mdetermines the number of
links each new node gets.
–Small-world networks (SW): these are characterized by very small average
distances between nodes (often O(log n) where nis the number of nodes).
We use a variant of the Barmpoutis-Murray (BM) model [5] to create ultra-
SW networks. This model takes as a parameter the total number of nodes
nand the total number of links Mand uses a backtracking procedure to
successively build the largest clique that leaves enough links available to
connect the rest of the network. In our variant, each of these cliques are
then connected using random vertices in the first clique created so as to
make the resulting network more resilient.
Fig. 1 shows an example of both kinds of network with the same number of
nodes and links.
As to the dynamics of the network, it is characterized by the availability
patterns of computing nodes. We use the model in [16]: all nnodes are initially
available and then their permanence in the system follows a Weibull distribu-
tion. This distribution is characterized by a shape parameter ηthat determines
whether failure probability increases with time (η > 1), decreases with time
(η < 1) or is time-independent (η= 1), and a scale parameter βdetermining
the mean lifetime for a given shape. Each node has an independent dynamics
and will contribute to the so-called churn phenomenon, namely the collective
effect on the network of computing nodes independently entering and leaving
it over time. Churn can have different effects on a distributed population-based
metaheuristic, the most obvious being that the current incumbent solution can
be lost [9]. Needless to say, the progress of the search will be also affected by the
disappearance of whole subpopulations. To tackle this in the context of corrective
fault-management policies, we consider two strategies [16]:
–rand: when a node becomes available again, it is initialized from scratch much
like in the initialization of the algorithm.

4 R. Nogueras and C. Cotta
–checkpoint: the algorithm uses some external safe storage in order to create
restoration checkpoints, namely periodical backups of the populate state that
are used to recover the last state of the population when a node becomes
available again.
It is clear that rand is a simpler strategy that has also the potential advantage
of reintroducing diversity in the search process. On the other hand, checkpoint
has the advantage of not wasting the previous progress of the search, being
more amenable to keep its momentum. The negative side of this latter strategy
is the requirement of this external safe storage and the associated overhead
(particularly if security and privacy concerns are important [13]) introduced by
the periodical backups. The latter effect can be somehow ameliorated by tuning
the period λ(measured in number of iterations) between checkpoints. The effect
of this parameter is studied next.
3 Experiments
We have done experiments using a distributed MMA with n= 32 islands. Each
island has a population size of µ= 16 individuals. Meme lengths evolve within
lmin = 3 and lmax = 9, mutating their length with probability pr= 1/9 following
[18]. We use crossover probability pX= 1.0 (one-point crossover), mutation
probability pM= 1/` (bit-flip mutation), where `is the genotype length, and
migration probability pmig = 1/80. In order to generate the network topology
we use m= 2 in the BA model of SF networks, and the corresponding value of
M=nm −m(m+ 1)/2 in the BM model of SW networks so that the number of
links is the same in both cases. As to node dynamics, we use the shape parameter
η= 1.5 (and hence the probability of failure increases with time), and scale
parameters β=−1/log p(k) for p(k)=1−(kn)−1,k∈ {1,2,5,10,20}. By doing
this, the mean availability stint per node is about 90%·kn iterations. We therefore
obtain scenarios ranging from rather low (k= 20) churn up to extremely high
(k= 1) churn. To analyze the sensitivity of the checkpoint strategy we consider
values λ∈ {µ, 10µ, 100µ}where µis the island population size. For comparison
purposes we also consider in the experimentation the use of the rand strategy.
We have considered four test functions, namely Deb’s trap (TRAP) function [7],
Watson et al.’s Hierarchical-if-and-only-if (HIFF) and Hierarchical-Exclusive-OR
(HXOR) functions [30] and Goldberg et al.’s Massively Multimodal Deceptive
Problem (MMDP) [8]. We perform 25 simulations running for a total number
of 50 000 evaluations for each value of λ, churn scenario, problem and network
topology.
Fig. 2 shows the results obtained in terms of deviation with respect to the
optimal solution (averaged for the four problems) as a function of the churn rate,
separately for each λvalue and for each network topology. First of all, it is clear
that performance degrades for increasing churn rate. This fact notwithstanding,
we can observe that variants using checkpoint reactivation perform notably bet-
ter than random reactivation. This confirms previous research on this kind of
strategies [16] and validates its usefulness on different network topologies. Note

Analysis of Checkpointing for MMAs on Unstable Complex Networks 5
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 (%)
rand
λ=16
λ=160
λ=1600
(a)
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 (%)
rand
λ=16
λ=160
λ=1600
(b)
Fig. 2: Deviation from the optimal solution as a function of the churn rate for
(a) SF and (b) SW.
Table 1: Results of Holm Test (α= 0.05) using λ= 16 as control parameter.
istrategy z-statistic p-value α/i
1λ= 160 2.598e+00 4.687e–03 5.000e–02
2λ= 1600 7.015e+00 1.151e–12 2.500e–02
3rand 8.747e+00 1.097e–18 1.667e–02
however that there is a marked performance degradation when the checkpoint
frequency is increased. This degradation is shown to be statistically significant
according to Quade test (p-value ≈0) both globally and when SF and SW are
separately analyzed. Subsequently we used Holm test to do a post-hoc analysis.
The use of checkpoint with parameter λ=µis shown to be statistically superior
to the remaining techniques at α= 0.05 level – see Table 1. This result suggests
that less expensive strategies (in terms of requiring less frequent state snapshots)
are not capable of dealing with churn (this result also holds if a separate analysis
is conducted for SF and SW topologies). A more clear depiction of the behavior
of the MMAs is provided by Fig. 3 for low (k= 20), high (k= 5) and extremely
high (k= 1) churn. Note that in the most stable scenario the algorithm performs
robustly regardless of the frequency of the snapshots (although as seen in Fig. 3f
there is a noticeable difference in genetic diversity when the period λis large).
However, as churn increases the difference in fitness turns out to be remarkably
higher in favor of λ=µ. As seen in Fig. 3d and 3f, the MMA has convergence
problems in these scenarios when λis high. The less frequent snapshots cannot
keep the momentum of the search in such unstable environments.

6 R. Nogueras and C. Cotta
12345
x 104
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20
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32
evaluations
best fitness
λ= 16
λ= 160
λ= 1600
(a)
12345
x 104
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evaluations
best fitness
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x 104
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(f)
Fig. 3: Best fitness (top) and entropy (bottom) for TRAP with SF topology.
From left to right: k= 1, k= 5 and k= 20.
4 Conclusion
Any algorithm directly deployed on a unstable computational environment must
be resilient to the volatility of its substrate. Metaheuristics are no exception and,
while they are intrinsically resilient to some extent [10], they must be augmented
with adequate policies in order to cope with the loss of information associated
to computing nodes that become inactive. A classical fault-tolerance technique
for this purpose is the creation of periodical backups of the state of these nodes
in order to recover from failures. We have performed a sensitivity analysis of
this strategy in the context of island-based multimemetic algorithms. It turns
out that this approach can be affordable in scenarios with low churn rates. In
such a situation, checkpoints need not be frequent for the algorithm to perform
adequately. However, scenarios featuring large churn rates require much more
frequent backups in order to cope with node volatility. The additional overhead
of such backups together with the need for having access to persistent external
storage makes this approach less appealing in such situations, suggesting other
approaches –autonomous, self-adaptive and purely local– can be more appropri-
ate. Work is already in progress in this direction [19].

Analysis of Checkpointing for MMAs on Unstable Complex Networks 7
Acknowledgements This work is partially supported by the MINECO project
EphemeCH (TIN2014-56494-C4-1-P), by the Junta de Andaluc´ıa project DNEME-
SIS (P10-TIC-6083) and by the Universidad de M´alaga, Campus de Excelencia
Internacional Andaluc´ıa Tech.
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