There is Noisy Lunch: A Study of Noise in Evolutionary Optimization

Problems

Juan J. Merelo

, Federico Liberatore

, Antonio Fern

andez Ares

, Rub

en Garc

ıa

, Zeineb Chelly

Carlos Cotta

, Nuria Rico

, Antonio M. Mora

and Pablo Garc

ıa-S

anchez

Dept. ATC, University of Granada, Granada, Spain

Escuela de Doctorado, University of Granada, Granada, Spain

Lab. RODCP, Institut Sup

erieur de Gestion, Granada, Tunisia

Dept. LCC, University of M

alaga, M

alaga, Spain

Dept. EIO, University of Granada, Granada, Spain

Dept. TSTC, University of Granada, Granada, Spain

Keywords:

Games, Evolutionary Optimization, Noise, Uncertainty, Noisy Fitness.

Abstract:

Noise or uncertainty appear in many optimization processes when there is not a single measure of optimality

or ﬁtness but a random variable representing it. These kind of problems have been known for a long time,

but there has been no investigation of the statistical distribution those random variables follow, assuming in

most cases that it is distributed normally and, thus, it can be modelled via an additive or multiplicative noise

on top of a non-noisy ﬁtness. In this paper we will look at several uncertain optimization problems that

have been addressed by means of Evolutionary Algorithms and prove that there is no single statistical model

the evaluations of the ﬁtness functions follow, being different not only from one problem to the next, but in

different phases of the optimization in a single problem.

1 INTRODUCTION

Optimization methods usually need a crisp and ﬁxed

value to work correctly. This value, usually called

cost or ﬁtness, informs the procedure on how good is

the solution and is used to select particular solution

over others. This does not imply that these methods

need a single ﬂoating point number; since they are

based on comparisons, it is usually enough that the

values can be partially ordered. Multiobjective opti-

mization, for instance, just need to know when com-

paring two solutions whether one or the other is the

best or there can be no comparison between them. In

either case, the answer to the question “Is this solu-

tion better than the other?” needs to be either a crisp

‘Yes’ or ‘No’, or simply “Impossible to know”.

In many cases, however, the ﬁtness or cost of a so-

lution cannot be described by a crisp value. In those

cases where there is uncertainty in the measure, that

is, in most real world, physical cases, such as the one

described in (Chiaberge et al., 1994), where a con-

trol system was optimized , or in the procedure used

to evaluate the solution, for instance, when using a

stochastic procedure to make that measure, the best

way to describe a solution will be a random variable,

not a single, or even a vector, value. In our research

we have found this happens in many different opti-

mization problems:

• When optimizing the layout of a web-page using

Simulated Annealing (SA) (Pe

nalver and Merelo,

1998). Since SA is a stochastic procedure, the ﬁt-

ness obtained by a solution will be a random vari-

able.

• When training any kind of neural network, such as

those in (Chiaberge et al., 1994; Merelo-Guerv

et al., 2001); in the second case we dealt with a

physical installation, introducing another kind of

randomness. Since training a neural network is a

stochastic procedure, the error rate obtained after

every training run will also follow a statistical dis-

tribution.

• When evolving game bots (autonomous agents)

(Mora et al., 2010). In this case, the uncertainty

arises from the problem itself; in games, several

factors such as the initial positions of the players

Merelo, J., Liberatore, F., Ares, A., García, R., Chelly, Z., Cotta, C., Rico, N., Mora, A. and García-Sánchez, P..

There is Noisy Lunch: A Study of Noise in Evolutionary Optimization Problems.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 261-268

ISBN: 978-989-758-157-1

261

or the opponent’s behavior add certain stochastic

component so that ﬁnal score will also be uncer-

tain or noisy; in some cases, too, the bot itself

will rely on probabilities to generate its behavior

(Fern

andez-Ares et al., 2014b), in which case two

different runs with exactly the same initial condi-

tions and opponent will also yield different scores.

In all these cases it cannot be said that there is

noise added to a crisp ﬁtness. The ﬁtness itself is a

statistical variable whose value arises from a stochas-

tic process, evaluation or training, however, we have

not seen an exhaustive research of the behavior of ﬁt-

ness as a random variable.

That is why, after some initial study of noise in

a particular game in (Merelo et al., 2014), where our

ﬁndings indicated that, in some cases, noise followed

a Gamma, that is a skewed normal distribution and

proposing a solution to this using Wilcoxon compar-

ison as a selection operator, we dug into data dis-

covering that, even if the distribution in that partic-

ular case was always a gamma, the parameters of the

distribution were different, which meant that the ran-

dom variable behaved in different ways depending on

the particular individual, the state of evolution and, of

course, the particular problem.

This initial conclusion disagrees with the usual as-

sumptions in optimization in uncertain environments,

where it is frequent to assume that the noise is nor-

mally distributed and with a ﬁxed sigma (Arnold,

2001). For instance, in the Black Box Optimization

Benchmarks (Hansen et al., 2009) the uncertainty was

simulated by adding noise centered in 0 and with a

Cauchy that is, a centered, sharp bell shaped distri-

bution, with different widths. Either multiplicative or

additive noise has been used in different occasions.

That is why in this paper we have collected data

from three different problems, which will be pre-

sented later on in this paper and tried to ﬁnd a model

for the ﬁtness using statistical tools. Our aim is to

eventually ﬁnd a model that is as general as possible

and that is able to account for most sources of uncer-

tainty; failing that, to try and ﬁnd selection operators

that are able to work with random variables in a nat-

ural way. However, this is not the focus of this paper

and, if it is eventually needed, is left as future work.

The rest of the paper is organized as follows. Next

we present the state of the art in evolutionary algo-

rithms in uncertain environments, to be followed by

a short presentation of the three problems with uncer-

tainty whose measures will be used in this paper in

Section 3. Results will be presented in Section 4, fol-

lowed by our conclusions.

2 STATE OF THE ART

The most recent and comprehensive review of the

state of the art in evolutionary algorithms in uncertain

environments was done by (Jin and Branke, 2005), al-

though recent papers such as (Qian et al., 2013; Bhat-

tacharya et al., 2014) and (Qian et al., 2014) include

brief updates. In that ﬁrst survey the authors state that

uncertainty is categorized into noise, robustness is-

sues, ﬁtness approximation, and time-varying ﬁtness

functions, and then, different options for dealing with

it are proposed. In principle, the approach presented

in this paper was designed to deal with the ﬁrst kind

of uncertainty, noise or uncertainty in ﬁtness evalua-

tion, although it could be argued that there is uncer-

tainty in the true ﬁtness as stated in the third category;

however, we do not think that is the case and, in gen-

eral, that third issue refers to the case in which ex-

pensive ﬁtness functions are substituted by surrogate

functions which carry a certain amount of error. They

suggest several methods, based either on using aver-

aging or using a selection threshold over which one

or other individual is selected. But since then, several

other solutions have been proposed.

For scientists not concerned on solving the prob-

lem of noise, but on a straightforward solution of the

optimization problem without modiﬁcation of exist-

ing tools and methodologies, an usual approach is

just to disregard the fact that the ﬁtness is noisy and

use whatever value is returned by a single evaluation

or after re-evaluation each generation. This was the

option in our previous research in games although

one evaluation in some of those works consists, in

fact, in an average of several evaluations, in differ-

ent maps or considering different opponents, for in-

stance. (Mora et al., 2010; Mora et al., 2012; Liber-

atore et al., 2015) and evolution of neural networks

(Castillo et al., 1999; Merelo-Guerv

os et al., 2001)

and leads, if the population is large enough, to an

implicit averaging as mentioned in (Jin and Branke,

2005).

In fact, selection used in evolutionary algorithms

is also stochastic, so noise in ﬁtness evaluation will

have the same effect as randomness in selection or a

higher mutation rate, which might make the evolu-

tion process easier and not harder in some particular

cases (Qian et al., 2013). In fact, Miller and Goldberg

proved that an inﬁnite population would not be af-

fected by noise (Miller and Goldberg, 1996) and Jun-

Hua and Ming studied the effect of noise in conver-

gence rates (Jun-hua and Ming, 2013), proving that

an elitist genetic algorithm ﬁnds at least one solution,

although with a lowered convergence rate.

But real populations are ﬁnite, so the usual ap-

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

262

proach to dealing with ﬁtness with a degree of ran-

domness is to increase the population size to a value

bigger than would be needed in a non-noisy environ-

ment. In fact, it has been recently proved that us-

ing sex, that is, crossover, is able to deal successfully

with noise (Friedrich et al., 2015), while an evolution-

ary algorithm based mainly on mutation, such as the

µ+1 EA, or evolutionary programming, would suffer

a considerable degradation of performance. However,

crossover is part of the standard kit of evolutionary

algorithms, so using it and increasing the population

size has the advantage that no special provision or

change in the implementation has to be made, just dif-

ferent values of the standard parameters.

Another more theoretically sound way is using a

statistical central tendency indicator, which is usu-

ally the average; which happens to be equal to the

median in the case of the random variable following

the normal distribution. This strategy is called ex-

plicit averaging by Jin and Branke and is used, for

instance, in (Jun-hua and Ming, 2013). Averaging

decreases the variance of ﬁtness but the problem is

that it is not clear in advance what would be the sam-

ple size used for averaging (Aizawa and Wah, 1994).

We have used it in some cases but in a different way:

not re-evaluating individuals every additional genera-

tion and computing the average but computing the ﬁt-

ness using the average of several evaluations, usually

ﬁve or more (Mora et al., 2012). Most authors use

several measures of ﬁtness for each new individual

(Costa et al., 2013), although other averaging strate-

gies have also been proposed, like averaging over the

neighbourhood of the individual or using resampling,

that is, more measures of ﬁtness in a number which is

decided heuristically (Liu et al., 2014). This assumes

that there is, effectively, an average of the ﬁtness val-

ues which is true for Gaussian random noise and other

distributions such as Gamma or Cauchy, but not nec-

essarily for all distributions.

To the best of our knowledge, other measures like

the median which might be more adequate for certain

noise models, but which is the same for the normal

distribution usually attributed to noise, have not been

tested; the median always exists, while the average

might not exist for non-centrally distributed variables.

Besides, most models keep the number of evaluations

ﬁxed and independent of its value, which might result

in bad individuals being evaluated many times before

being discarded; some authors have proposed resam-

pling, (Rada-Vilela et al., 2014; Rakshit et al., 2014),

which will effectively increase the number of evalu-

ations and thus slow down the search. In any case,

using average is also a small change to the algorithm

framework, requiring only using as new ﬁtness func-

tion the average of several evaluations.

These two approaches that are focused on the eval-

uation process might be complemented with changes

to the selection process. For instance, using a thresh-

old (Rudolph, 2001; Rakshit et al., 2014) that is re-

lated to the noise characteristics to avoid making com-

parisons of individuals that might, in fact, be very

similar or statistically the same; this is usually called

threshold selection and can be applied either to ex-

plicit or implicit averaging ﬁtness functions. The al-

gorithms used for solution, themselves, can be also

tested, with some authors proposing, instead of tak-

ing more measures, testing different solvers (Cauwet

et al., 2014), some of which might be more af-

fected by noise than others. However, recent papers

have proved that sampling might be ineffective (Qian

et al., 2014) in some types of evolutionary algorithms,

adding running time without an additional beneﬁt in

terms of performance. This is one lead we will use in

the current paper.

Any of these approaches do have the problem of

statistical representation of the true ﬁtness, even more

so if there is not such a thing, but several measures

that represent, as a set the ﬁtness of an individual.

This is what we are going to use in this paper, where

we present a method that uses resampling via an in-

dividual memory and use either explicit averaging or

statistical tests like the non-parametric Wilcoxon test.

First we will examine and try to ﬁnd the shape of the

noise that actually appears in some games and other

optimization problems; then we will check in this pa-

per what is the inﬂuence on the quality of results of

these two strategies and which one, if any, is the best

when working in noisy environments.

3 PROBLEMS USED IN THIS

PAPER

The ﬁtness of three different problems, all of them re-

lated to computational intelligence in games, has been

used in this paper: generation of character backsto-

ries in artiﬁcial worlds, described in subsection 3.1,

optimization of bots for playing the real time strat-

egy game Planet Wars in 3.2, and optimization of

the ghost team in Ms. Pac-Man, which will be de-

scribed in subsection 3.3. These three problems have

been chosen for several reasons, the most important

of which is that we have been working on them and

thus have data available; another reason is that the ori-

gin of the uncertainty is different in the three cases.

In the case of MADE, ﬁtness is computed through a

simulation; in the case of Planet Wars, the bot them-

selves have a random component, with its represen-

There is Noisy Lunch: A Study of Noise in Evolutionary Optimization Problems

263

tation including probabilities of different courses of

action; and ﬁnally in Ms. Pac-Man, it is due to the

nature of the game itself. It is not a complete repre-

sentation of all causes of uncertainty, but the sample

is big enough so that we can generalize the results ob-

tained, which will be presented in the next section.

3.1 Creation of Character Backstories

MADE (Massive Drama Engine for non-player char-

acters) (Garc

ıa-Ortega et al., 2014) is a framework for

the automatic generation of virtual worlds that allow

the emergence of backstories for secondary charac-

ters that can later on be included in videogames. In

this context, an archetype is a well-known behaviour

present in the imaginary collective (for example, a

“hero” or a “villain”). Given a ﬁtness to model the ex-

istence of different N

archetypes for a virtual world,

MADE uses a genetic algorithm to optimize the pa-

rameter values of a Finite State Machine (FSM) that

model the agents of that world. For the evaluation,

a world is simulated using this parameter set, and

the log is analyzed to detect behaviours of the world

agents to match with the desired archetypes.

As the evolved parameters are the probabilities

to jump from one state to another in the FSM, each

ﬁtness evaluation is performed executing the virtual

world ﬁve times per individual, obtaining the average

ﬁtness. Selection is, therefore, performed comparing

this average ﬁtness, using a binary tournament in this

case. Fitness values range from 0 and N

and are cal-

culated taking into account the rate of occurrence of

the archetypes in the execution log.

3.2 Real Time Strategy: Planet Wars

Planet Wars (Fern

andez-Ares et al., 2011) is a simple

Real-Time strategy (RTS) game. RTS games are not

turn-based and their objective is to defeat the enemy

using resources available in the map to build units and

structures.

Computational intelligence methods have been

applied to Planet Wars since it provides a simpliﬁ-

cation of the elements of the RTS: one kind of units

(spaceships) and one kind of resources and structures

(planets). Spaceships are automatically generated in

the planets owned by the player and they are used to

conquer the enemy planets, as this is the objective of

the game.

In this paper we are using the results obtained

from the Genebot algorithm (Garc

ıa-S

anchez et al.,

2014). This algorithm optimizes the parameters of a

hand-coded FSM that indicates how many ships send

from each planet to attack or reinforce another planet

depending of some other values (such as the distance

between planets). The generated bot is not determin-

istic, as some of the jumps of the states are based in

probabilities. Fitness is calculated confronting ﬁve

times the bot obtained from the parameter set of the

FSM against a competitive hand-coded bot. The re-

sult of each match takes into account the ‘slope’ of

the number of player spaceships during the time of the

match. Positive results mean that the bot won, as the

slope will be positive, and vice versa. Theoretical val-

ues are in the range [−1, 1], although these values are

impossible to attain in the game. A value of -1 would

indicate that the player lost all their ships in the initial

time, while 1 would mean the contrary: it generated

all the spaceships and won in the initial time. The

ﬁtness of an individual is the sum of all ﬁve results,

and therefore being in the range [−5, 5]. This ﬁtness

has been explained in more detail in (Fern

andez-Ares

et al., 2014a).

3.3 Ghost Team Optimization

Ms. Pac-Man is a variant of the famous Pac-Man

game that extends its mechanics with features such

as the inclusion of a random event that reverses the

direction of the ghosts. This game is used in the Ms.

Pac-Man vs Ghosts competition, where participants

can submit controllers for both Ms. Pac-Man and the

Ghost Team, the ﬁrst trying to maximize its score,

the second trying to minimize Ms. Pac-Man’s. The

framework used to test the methodology analyzed de-

ﬁnes the following restrictions for the Ghost Team:

• A ghost can never stop and if it is in a corridor it

must move forward.

• A ghost can choose its direction only at a junction.

• Every time a ghost is at a junction the controller

has to provide a direction from the set of feasible

directions.

• After 4000 game ticks, a level is considered com-

pleted and the game moves on to the next one.

Also, in this method, which was published in (Lib-

eratore et al., 2015), the ﬁtness of each individual is

computed as the maximum score obtained by eight

different Ms. Pac-Man controllers. Some of these

controllers were the best in past editions of the inter-

national competition, so they are very tough rivals for

the ghost team.

4 EXPERIMENTS AND RESULTS

With the problems presented above, data on ﬁtness

was collected by selecting a few random individu-

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

264

als in every generation and measuring its ﬁtness 100

times. Thus, every individual is represented by a

random variable with the 100 measures taken with

its ﬁtness. According to the usual assumptions, this

random variable should follow a normal distribution,

with probably different σ and centered on the true

ﬁtness value. In order to check that hypothesis, we

plotted the skewness, that is, asymmetry of the distri-

bution, and kurtosis, which is a parameter related to

the shape of the distribution. A symmetrical distribu-

tion like the normal distribution has a skewness and

kurtosis equal to 0; asymmetric distributions, such

as the Gamma that we had found in previous papers

(Merelo et al., 2014), has non-zero skewness and kur-

tosis which are related to their α and κ parameters, for

instance. Any random variable has skewness and kur-

tosis at any rate, and we have computed and plotted

them in the next ﬁgures, where skewness is plotted as

x axis against kurtosis in the y axis.

●

−1 0 1 2

Skewness

Kurtosis

Gen

●

Gen 64

Gen 128

Gen 256

Figure 1: Skewness and kurtosis for ﬁtness in several gen-

erations of the MADE problem. Different colors represent

diffent generations.

Figure 1 represents them for the MADE problem

for which we took measures for a variable amount of

individuals every generation, from 100 in generation

64 to around 50 in the latest generation. A curious

convergence, but without reaching, the normal distri-

bution is observed as generations proceed; in the ﬁrst

generations, values of skewness and kurtosis are quite

high and correspond to arbitrary distribution (Beta or

uniform), however, as the simulation proceeds, val-

ues approach zero. However, they do not converge

exactly to 0, meaning that, even if uncertainty can be

approached by a normal distribution, that approxima-

tion would only be correct for the latest generations

of the simulation. In general, individual ﬁtness will

follow an arbitrary distribution with a general shape

and asymmetry.

●

−1 0 1 2

Skewness

Kurtosis

Gen

●

Gen 1

Gen 50

Figure 2: Skewness and kurtosis for ﬁtness in several gen-

erations of the Planet Wars problem. Different colors repre-

sent different generations.

The shape of the graph for the Planet Wars prob-

lem, shown in Figure 2 for two different generations,

is different but has some similarities. The disper-

sion also decreases as evolution proceeds, with shape

becoming closer to normal distribution in generation

50. However, initial kurtosis is quite high and values

above 2 and below 0 are found even late in the evolu-

tion. Noise is, thus, noisy and does not conform to a

single shape, even less a normal one.

The graph for the ﬁnal problem, Ms. Pac-Man,

is different in several aspects, and is shown in Figure

3. First we have to take into account, as explained

in 3.3, that differently from the previous cases, the

ﬁtness for a ghost team is the maximum, not an av-

erage of several values. This causes a curious behav-

ior of ﬁtness: in the ﬁrst generation, several individ-

uals have crisp values; however, this is decreasingly

so, becoming more “random” as generations proceed,

that is, the set of values the ﬁtness has got starts to

have many different values while in the ﬁrst genera-

tions it had one or a few. That is why the behavior

shown in the graph is completely different: distribu-

tions get increasingly asymmetric and its shape more

different from a normal distribution and more like a

Beta distribution. Even if the trend is different from

the other two problems, the overall aspect is the same:

there is no single distribution that is able to describe

the shape of ﬁtness with an uncertainty component.

There is Noisy Lunch: A Study of Noise in Evolutionary Optimization Problems

265

●

100

−2.5 0.0 2.5 5.0 7.5 10.0

Skewness

Kurtosis

Gen

●

Gen 1

Gen 25

Gen 50

Figure 3: Skewness and kurtosis for ﬁtness in several gen-

erations of the Ms. Pac-Man problem. Different colors rep-

resent different generations.

5 CONCLUSIONS

In this paper we set out to study the statistical distri-

bution followed by the ﬁtness of single individuals in

several problems in the area of games in which we

have worked. These problems: MADE, Planet Wars,

and Ms. Pac-Man, have different natures and ways

to compute the ﬁtness, but all of them have in com-

mon that ﬁtness is not a ﬁxed number but a random

variable. We have set out to do to prove the hypoth-

esis that not only noise does not follow the normal,

or Gaussian, distribution or other centrally-distributed

models such as Cauchy, which have been used repeat-

edly in the literature, but that it does not follow a sin-

gle distribution even considering a single problem.

The study presented here proves that hypothesis.

The best way to describe statistical variables is using

two parameters: kurtosis and skewness. These two

parameters have been computed and plotted for each

one of the problems, proving that not only distribu-

tions are asymmetrical and not bell-shaped, but that

its shape changes within a single problem and in dif-

ferent stages of the computation. In some case, like

MADE, it seems clear that due to the fact that aver-

ages are used as a representative for selection, those

individuals whose ﬁtness is closer to a central shape

are oversampled and thus selected preferably, with

almost-central individuals in the latest stages being a

consequence of this fact. In other cases, when ﬁtness

is computed in a different way or selection takes an-

other form, the effect is exactly the opposite. At any

rate, using averages, after the study done in this paper,

is discouraged since in many cases and almost always

in the early stages of the evolution, ﬁtness, being a

random variable, does not pass a centrality test and

it might not even have an average. A better way of

comparing any ﬁtness with uncertainty would be, as

proposed by the authors, using non-parametric tests

such as the Wilcoxon test that impose a partial order

on the individuals (Merelo et al., 2014); this partial

order can be used, in several different ways, for selec-

tion.

The fact that there is no single model representing

the distribution of ﬁtness also implies that it is an error

to use centrally distributed random variables added to

a crisp ﬁtness to test operators and algorithms that

operate in uncertainty. Either real values should be

used, such as the ones proposed above, or a distribu-

tion with varying shape and symmetry such as Beta

should be used. However, in this case we should take

into account that “true” or “crisp” ﬁtness does not re-

ally exist, so any modelization of uncertain ﬁtness that

uses noise added to crisp ﬁtness is, in the more gen-

eral case, wrong, although it might obviously be true

in some cases. If the ﬁtness evaluation is expensive

and tests want to be performed for some new oper-

ators, the best way to model uncertainty would be to

use different models applied to every individuals, with

different skewness and kurtosis. However, this would

be only a ﬁrst-order approximation and it might favor

methods that use averages.

What remains to be done is to effectively apply

Wilcoxon-based comparisons to the problems above,

but since they are costly to evaluate, we will try to cre-

ate a benchmark for problems with uncertainty which

reﬂects in the best possible way how ﬁtness is orga-

nized in a wide array of problems. In order to do that

we will try to examine as many uncertain problems as

possible and deduce what would be the most general

model.

ACKNOWLEDGEMENTS

This work has been supported in part by projects

TIN2014-56494-C4-3-P (Spanish Ministry of Econ-

omy and Competitiveness), SPIP2014-01437 (Di-

recci

on General de Tr

aﬁco), PRY142/14 (Fundaci

ublica Andaluza Centro de Estudios Andaluces en la

IX Convocatoria de Proyectos de Investigaci

on), and

project V17-2015 of the Microprojects program 2015

from CEI BioTIC Granada.

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

266

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