Evolution Strategies and Covariance Matrix Adaptation

Investigating New Shrinkage Techniques

Silja Meyer-Nieberg and Erik Kropat

Department of Computer Science, Universit

at der Bundeswehr M

unchen,

Werner-Heisenberg Weg 37, 85577 Neubiberg, Germany

Keywords:

Evolution Strategies, Covariance Matrix, Adaptation, Shrinkage Estimators.

Abstract:

This paper discusses the covariance matrix adaptation in evolution strategies, a central and essential mecha-

nism for the search process. Basing the estimation of the covariance matrix on small samples w.r.t. the search

space dimension is known to be problematic. However, this situation is common in optimization raising the

question, whether the performance of the evolutionary algorithms could be improved. In statistics, several ap-

proaches have been developed recently to improve the quality of the maximum-likelihood estimate. However,

they are seldom applied in evolutionary computation. Here, we focus on linear shrinkage which requires rela-

tively little additional effort. Several approaches and shrinkage targets are integrated into evolution strategies

and analyzed in a series of experiments.

1 INTRODUCTION

Evolution strategies (ESs) belong to the class of evo-

lutionary algorithms. Their performance compared to

other black-box optimization techniques is good as it

has been observed in practice and in several competi-

tions as e.g. (Hansen et al., 2010). Since they oper-

ate mainly in continuous search spaces, their central

search operator is mutation which is typically real-

ized as a multivariate normal distribution with mean

m and covariance matrix σ

C. In order to progress

fast and reliably towards the optimal point, the ex-

tent as well as the directions of the mutations must

be adapted so that the distribution of the random vari-

ables is suitable to ﬁtness landscape of the particu-

lar function. Techniques for controlling the mutation

process have therefore received a lot of attention in

research on evolution strategies. The focus lies on

the control of covariance matrix. The covariance ma-

trix adaptation techniques developed typically make

use of a variant of the sample covariance. However,

the estimation problem is ill-posed: Due to efﬁciency,

the sample size is relatively small w.r.t. the search

space dimension. This leads to a well-known problem

in statistics: The covariance matrix estimate may dif-

fer considerably from the underlying true covariance

(Stein, 1956; Stein, 1975). Taking a closer look at

the adaptation equations, reveals that nearly all adap-

tation techniques introduce correction or regulariza-

tion techniques by falling back to the previous covari-

ance matrix and/or by strengthening certain promis-

ing directions. These procedures exhibit similarities

to shrinkage estimation in statistical estimation the-

ory leading to the research question of the present pa-

per: If evolution strategies perform a kind of implicit

shrinkage, can they proﬁt from the introduction of ex-

plicit shrinkage operators?

The current analysis extends the work carried out

in (Meyer-Nieberg and Kropat, 2014; Meyer-Nieberg

and Kropat, 2015a) and augments the investigation

conducted in (Meyer-Nieberg and Kropat, 2015b;

Meyer-Nieberg and Kropat, 2015c) for the case of

thresholding estimators. (Meyer-Nieberg and Kropat,

2014; Meyer-Nieberg and Kropat, 2015a) presented

the ﬁrst approaches to apply Ledoit-Wolf shrinkage

estimators in evolution strategies. There, the shrink-

age estimators were combined with an approach stem-

ming from a maximum entropy covariance selection

principle. A literature review resulted in only two pa-

pers aside from our previous approaches: An applica-

tion in the case of Gaussian estimation of distribution

algorithms albeit with quite a different goal (Dong

and Yao, 2007). There, the learning of the covariance

matrix during the run lead to non positive deﬁnite ma-

trices. For this reason, a shrinkage procedure was ap-

plied to “repair” the covariance matrix towards the re-

quired structure. The authors used a similar approach

as in (Ledoit and Wolf, 2004b) but made the shrink-

age intensity adaptable. More recently Kramer con-

sidered Ledoit-Wolf-estimator based on (Ledoit and

Meyer-Nieberg, S. and Kropat, E.

Evolution Strategies and Covariance Matrix Adaptation - Investigating New Shrinkage Techniques.

DOI: 10.5220/0005703201050116

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 2, pages 105-116

ISBN: 978-989-758-172-4

105

Wolf, 2004a) for an evolution strategy which does not

follow a population-based approach but uses a vari-

ant of the (1+1)-ES with covariance matrix adaptation

for which past search points are taken into account

(Kramer, 2015).

Here, we focus on an approach in the eigenspace

of the covariance matrix. Several shrinkage targets

are analyzed and compared with each other and with

the original ES version. The paper is structured as fol-

lows: First, a brief introduction into evolution strate-

gies with covariance matrix adaptation is provided.

Afterwards, we focus on the problem of estimat-

ing high-dimensional covariance matrices. Several

shrinkage targets are introduced and their integration

into evolution strategies is described. The strategies

are assessed and compared to the original ES version

in the experimental section. The paper ends with the

conclusions and an outlook regarding open research

points.

2 EVOLUTION STRATEGIES

Let f ∶R

→R be a continuous function that allows

only the evaluation of the function itself but not the

derivation of higher order information. In this con-

text, metaheuristics as evolution strategies and similar

approaches can be applied. Evolution strategies (ESs)

are stochastic optimization methods that usually use

a sample or population of search points also called

candidate solutions. They distinguish between a pop-

ulation of µ parents and λ offspring. In many applica-

tions in continuous search spaces, the parent popula-

tion is discarded after the offspring have been created.

Therefore, λ >µ is required. Evolution strategies use a

multivariate normal distribution with mean m

(g)

and

covariance matrix



(g)



(g)

to obtain new search

points. The mean is taken as the (weighted) centroid

of the parent population whereas the covariance ma-

trix is updated by following one of the established

techniques. Sampling λ times from the normal dis-

tribution, results in the offspring population

= m

(g)

+σ

(g)

N(0, C

(g)

), l =1, . . . , λ. (1)

Afterwards, the new search points are evaluated using

the function f to be optimized. The µ best of the λ

offspring are selected for the following parent popu-

lation.

As stated previously, the parameters of the nor-

mal distribution must be adapted in order to allow

progress towards the optimal point. The mean is ob-

tained as the centroid of the parent population. In

contrast, the covariance matrix requires more effort.

Methods for adapting the scale factor σ or the full

covariance matrix have received a lot of attention in

work on evolution strategies (see (Meyer-Nieberg and

Beyer, 2007)). The investigation in this paper centers

on the covariance matrix self-adaptation evolution

strategy (CMSA-ES) (Beyer and Sendhoff, 2008).

2.1 Covariance Matrix Update

The covariance matrix (σ

(g)

)

(g)

can be interpreted

as the product of a general scaling factor σ

(g)

(or step-

size or mutation strength) and direction matrix C

(g)

Previous research in evolution strategies has shown

that both should be treated separately. The adapta-

tion therefore takes two distinct processes into ac-

count: the ﬁrst for the step-size, the second for the

matrix itself. Following established practice in evolu-

tion strategies, the matrix C

(g)

will be referred to as

the covariance matrix in the remainder of the paper.

Once the new sample has been created, the pa-

rameters of the distribution must be updated. First,

the µ best offspring provide the sample which is used

to estimate the covariance matrix. Considering only

the better candidate solutions shall introduce a bias

towards good search regions. It is uncessary to re-

estimate the mean m

(g)

, thus, the number of degrees

of freedom remains µ. Let x

m∶λ

denote the mth best of

the λ offspring w.r.t. the ﬁtness and let

(g+1)

m∶λ

∶=

(g)

x

(g+1)

m∶λ

−m

(g)

, (2)

stand for the normalized offspring. The covariance

update is then determined as

(g+1)

∶= (1 −

(g)



m=1

(g+1)

m∶λ

(g+1)

m∶λ

)

(3)

combining the old covariance and the population co-

variance and with the weights usually reading w

1µ (Beyer and Sendhoff, 2008). The parameter c

described in more detail later in the paper.

2.2 Step-size Adaptation

The CMSA-ES applies self-adaptation in order to

tune the mutation strength σ

(g)

. Self-Adaptation has

been developed by Rechenberg (Rechenberg, 1973)

and Schwefel (Schwefel, 1981). It takes place at the

level of the individuals meaning that each population

member operates with its distinct set. The strategy pa-

rameters are adapted by including them in the genome

and subjecting them to evolution. That is, they un-

dergo recombination and mutation processes. After-

wards, they are used in the mutation of the search

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

106

Require: λ, µ, C

(0)

, m

(0)

, σ

(0)

, τ, c

1: g =0

2: while termination criteria not met do

3: for l =1 to λ do

4: σ

=σ

(g)

exp(τN(0, 1))

5: x

(g)

+σ



N(0, C

(g)

)

6: f

= f (x

)

7: end for

8: Select (x

1∶λ

, σ

1∶λ

), . . . , (x

µ∶λ

, σ

µ∶λ

)according to

their ﬁtness f

9: m

(g+1)

∑

m=1

m∶λ

10: σ

(g+1)

∑

m=1

m∶λ

11: z

m;λ

−m

(g)

for m =1, . . . , µ

12: C

∑

i=1

13: C

(g+1)

=(1 −

(g)

(g+1)

14: g =g +1

15: end while

Figure 1: The main steps of a CMSA-ES. Normally, the

weights w

are set to w

= 1/µ for m = 1, . . . , µ.

space position. The inﬂuence on the selection is in-

direct: Self-adaptation is based on the assumption

of a stochastic linkage between good objective val-

ues and appropriately tuned strategy parameters: Self-

adaptation is mainly used to adapt the step-size or

a diagonal covariance matrix. Here, the mutation

strength is considered. Its mutation process is realized

with the help of the log-normal distribution following

(g)

= σ

(g)

exp(τN(0, 1)). (4)

The parameter τ, the learning rate, should scale with

1

√

2N (Meyer-Nieberg and Beyer, 2005). Self-

adaptation with recombination has been shown as “ro-

bust” against noise (Beyer and Meyer-Nieberg, 2006).

In the case of ES with recombination, the variable

(g)

in (4) is the result of the recombination of the

mutation strengths. Here, the same recombination

type may be used as for the objective values, that is,

(g+1)

∑

m∶λ

with σ

m∶λ

standing for the muta-

tion strength associated with the mth best individual.

Figure 1 summarizes the main steps of the covariance

matrix self-adaptation ES (CMSA-ES).

3 COVARIANCE ESTIMATION:

SHRINKAGE

Estimating the covariance matrix in high-dimensional

search spaces, requires an appropriate sample size.

Using the population covariance matrix necessitates

µ ≫N for obtaining a high quality estimator. If this is

not the case, estimate and “true” covariance may not

aggree well. Among others, the eigen structure may

be signiﬁcantly distorted, see e.g. (Ledoit and Wolf,

2004b). However, this is the case in evolution strate-

gies. Typical recommendations for the population siz-

ing are to use an offspring population size λ of either

λ =O(log(N))or λ =O(N)and setting µ =cλwith

c ∈(0, 0.5). Thus, either µN →c or even µN →0

for N →∞ holds, showing that the textbook estima-

tion equation is not applicable in high-dimensional

settings.

As stated above, the estimation of high-

dimensional covariance matrices has received a lot

of attention, see e.g. (Chen et al., 2012). Several

approaches can be found in the literature, see e.g.

(Pourahmadi, 2013; Tong et al., 2014). This paper

focuses on linear shrinkage estimators that can be

computed comparatively efﬁciently and thus do not

burden the algorithm strongly. Other classes, as e.g.

thresholding operators for sparse covariance matrix

estimation, are currently considered in separate anal-

yses.

Based on (Stein, 1956; Ledoit and Wolf, 2004b),

linear shrinkage approaches consider an estimate of

the form

est

(ρ) = ρF +(1 −ρ)C

(5)

with F the target to correct the estimate provided by

the sample covariance C

. The parameter ρ ∈(0, 1)is

called the shrinkage intensity. Equation (5) is used to

shrink the eigenvalues of C

towards the eigenvalues

of F. The intensity ρ should be chosen to minimize

ES

est

(ρ)−Σ

 (6)

with ⋅

denoting the squared Frobenius norm with

A

TrAA

, (7)

see (Ledoit and Wolf, 2004b). Note the factor 1N is

additionally introduced in (Ledoit and Wolf, 2004b)

to normalize the norm w.r.t. the dimension.

Based on (6) and taking into account that the true

covariance is unknown in practice, Ledoit and Wolf

were able to obtain an optimal shrinkage intensity for

the target F =Tr(C

)N I for general probability dis-

tributions.

Several other approaches can be identiﬁed in liter-

ature. One the one hand, different targets can be con-

sidered, see e.g. (Sch

affer and Strimmer, 2005; Fisher

and Sun, 2011; Ledoit and Wolf, 2003; Touloumis,

2015). Sch

afer and Strimmer analyze among others

diagonal matrices with equal and unequal variance

or special correlation models (Sch

affer and Strimmer,

2005). Fisher and Sun also allow for several targets

Evolution Strategies and Covariance Matrix Adaptation - Investigating New Shrinkage Techniques

107

(Fisher and Sun, 2011) assuming a multivariate nor-

mal distribution. Toulumis relaxed the normality as-

sumption, considered several targets, and provided a

new non-parametric family of shrinkage estimators

(Touloumis, 2015). Other authors introduced differ-

ent estimators, see e.g. (Chen et al., 2010) or (Chen

et al., 2012)). Recently, Ledoit and Wolf extended

their work to include non-linear shrinkage estimators

(Ledoit and Wolf, 2012; Ledoit and Wolf, 2014). A

problem arises concerning the complexity of the ap-

proaches. Especially the non-linear estimates require

solving an associated optimization problem. Since the

estimation has to be performed in every generation of

the ES, only computationally simple approaches can

be taken into account. Therefore, the paper focuses on

linear shrinkage with shrinkage targets and intensities

taken from (Ledoit and Wolf, 2004a; Ledoit and Wolf,

2004b; Fisher and Sun, 2011; Touloumis, 2015).

Interestingly, Equation (3) of the ES algorithm

represents a special case of shrinkage with the old co-

variance matrix as the target. The shrinkage intensity

is determined by

=1 +

N(N +1)

2µ

(8)

as ρ =1 −1c

. The paper investigates whether an

additional shrinkage could improve the performance.

Transferring shrinkage estimators to ESs must take

the situation in which the estimation occurs into ac-

count since it differs from the assumptions in sta-

tistical literature. First, the covariance matrix Σ =

g−1

that was used to create the offspring is known.

Second, the sample is based on truncation selec-

tion. Therefore, the variables cannot be assumed

to be identically and independently distributed (iid).

Only if there were no selection pressure, the sample

, . . . , x

would represent normally distributed ran-

dom variables. In this context, it is interesting to note

that during the discussion in (Hansen, 2006) with re-

spect to the setting of the CMA-ES parameters it is

argued to choose the parameters so that the distribu-

tion of the random variables remains unchanged if no

selection pressure were present. This paper uses a

similar argument to justify the usage of the shrinkage

intensities obtained for assuming iid random or even

normally distributed random variables. Since we are

aware of the fact that the situation may differ vastly

from the analysis assumptions of the statistical litera-

ture, other settings will be considered in future work.

Before continuing, it is worthwhile to take a closer

look at the covariance matrix update (3). As stated, it

can be interpreted as a shrinkage equation. Its effects

become more clear, if we move into the eigenspace

of C

(g)

. Since the covariance matrix is a positive

deﬁnite matrix, we can carry out a spectral compo-

sition of C

(g)

with C

(g)

ΛM. The modal matrix

M =(v

, . . . , v

) contains the eigenvectors of C

(g)

whereas Λ =diag(λ

, . . . , λ

)represents the diagonal

matrix with the corresponding eigenvalues λ

, . . . , λ

The representation C

(g+1)

of C

(g+1)

in the eigenspace

of C

(g)

then reads

(g+1)

= ρΛ +(1 −ρ)C

= diag(C

)+ρΛ −diag(C

)+

(1 −ρ)L

 (9)

with C

M. The matrix L

denotes the matrix

with the entries of C

below the diagonal, whereas

comprises the elements above. In other words, the

covariance matrix update decreases the off-diagonal

elements of the transformed population covariance.

In the case of the diagonal entries, two cases may

appear: if c

<λ

, the new entry is in the inter-

val [c

, λ

] and thus the estimate increases towards

, otherwise it is shrunk towards λ

. Thus, in the

eigenspace of C

(g)

, Equation (9) behaves similar to

shrinkage with a diagonal matrix as target matrix and

therefore in original space it shrinks the eigenvalues

of the population matrix towards those of the tar-

get. In contrast to shrinkage, the target matrix is not

computed via the sample but with the old covariance

(which is not obtainable in the general case).

Applying shrinkage requires among others the

choice of an appropriate target. Most shrinkage ap-

proaches consider regular structures as e.g. the scaled

unity matrix, diagonal matrices, or matrices with con-

stant correlations as shrinkage targets. However, a

shrinkage towards a regular structure concerning the

coordinate system does not appear as an optimal

choice concerning the optimization of arbitrary func-

tions.

4 SHRINKAGE ESTIMATION

EVOLUTION STRATEGIES

Since we cannot assume that the covariance matrix

adaptation would proﬁt from a shrinkage towards reg-

ular structures in every application case, we do not

perform the shrinkage in the original search space.

Instead, we consider the eigenspace of a positive def-

inite matrix. A similar approach was introduced in

(Thomaz et al., 2004) and used as the foundation of

(Meyer-Nieberg and Kropat, 2014; Meyer-Nieberg

and Kropat, 2015a). In (Thomaz et al., 2004) the

authors were faced with the task to obtain a reliable

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

108

covariance matrix. To this end, a sample covariance

matrix S

was combined with a pooled variance ma-

trix C

– similar to (3)

mix

(ξ)=ξC

+(1 −ξ)S

(10)

with the parameter ξ to be determined. To proceed,

the authors switched to the eigenspace of the non-

weighted mixture matrix where they followed a max-

imal entropy approach to determine an improved esti-

mate of the covariance matrix. Furthermore, (Hansen,

2008) suggested that changing the coordinate sys-

tem may result in an improved performance. There-

fore, Hansen introduced an adaptive encoding for the

CMA-ES. It is based on the spectral decomposition of

the covariance matrix. New search points are created

in the eigenspace of the covariance matrix.

Similar to (Hansen, 2008), we suggest the ES will

proﬁt from a change of the coordinate system. How-

ever, the covariance matrix adaptation and estimation

which in (Hansen, 2008) occurs in the original space

will be performed in the transformed space.

This paper considers a combination of a shrink-

age estimator and the basis transformation for a

use in the CMSA-ES. First results were obtained in

(Meyer-Nieberg and Kropat, 2014; Meyer-Nieberg

and Kropat, 2015a). Here, the work is extended

by considering several mixture matrices, targets, and

choices for the shrinkage intensity. The resulting

shrinkage CMSA-ES (Shr-CMSA-ES) approaches

follow the same general principle:

• First, the coordinate system is transformed,

• followed by a shrinkage towards a particular

shrinkage target.

In order to conduct the search space transformation, a

positive-deﬁnite matrix is required. This paper con-

siders the following choices for the transformation

matrix which arise as combinations of the population

covariance matrix C

and C

(g)

mix

= C

(g)

, (11)

g+1

= (1 −c

(g)

, (12)

= C

(g)

. (13)

The variants (11) - (13) are based on different assump-

tions: The ﬁrst (11) follows (Thomaz et al., 2004).

The inﬂuence of the old covariance and the population

covariance are balanced. Structural changes caused

by C

will be dampened but will inﬂuence the re-

sult more strongly than in the case of (12) and (13).

The second (13) considers the covariance mixture that

would have been used in the original CMSA-ES. De-

pending on the size of c

, which in turn is a function

of µ and N, see (8), the inﬂuence of the population co-

variance matrix may be stronger or lesser. The third

considers the eigenspace of the old covariance matrix

and reduces therefore the inﬂuence of the new esti-

mate. Equations (11) - (13) are used to change the

coordinate system. Assuming that for example (13) is

used, the following steps are performed

• spectral decomposition: M, D ←spectral(S

(g)

• determination of C

∶= M

and C

∶=

(g)

• shrinkage resulting in C

shr

• retransformation C

shr

• covariance adaptation

(g+1)

=(1 −

(g)

The representations of the covariance matrices in the

eigenspace are given as C

∶=M

and C

∶=

(g)

with S standing for one of the variants

(11) - (13). Once the change is performed, different

targets can be taken into account. In this paper, we

consider the matrices

= vI, (14)

with v =Tr(C

)N (Ledoit and Wolf, 2004b),

= diag(C

) (15)

the diagonal entries of C

(Fisher and Sun, 2011;

Touloumis, 2015), the constant correlation model

with matrix F

the entries of which read

i j

= 

if i = j

¯r

√

j j

if i ≠ j

(16)

and ¯r =2((N −1)N)

∑

N−1

i=1

∑

j=i+1

i j



√

j j

(Ledoit

and Wolf, 2004a). The shrinkage intensities are taken

from the corresponding publications. For (14) the pa-

rameter is based on (Ledoit and Wolf, 2004b), for (15)

it is follows (Fisher and Sun, 2011) and (Touloumis,

2015) while it is taken from (Ledoit and Wolf, 2004a))

in the case of (16). The question remains whether the

additional term consisting of the old covariance ma-

trix in (3) remains necessary or whether the ES may

operate solely with shrinkage. Our preliminary in-

vestigations indicate that the latter strategies perform

worse than the original CMSA-ES but more detailed

investigations will be carried out.

5 EXPERIMENTAL ANALYSIS

Experiments were performed to investigate the

shrinkage estimators introduced. First, the question

of ﬁnding a suitable transformation was addressed. To

Evolution Strategies and Covariance Matrix Adaptation - Investigating New Shrinkage Techniques

109

this end, a comparison of the effects of (11) - (13) was

conducted for a combination of (15) and the shrinkage

intensity from (Touloumis, 2015). The experiments

showed that using (12) provided the best results for

this combination. Therefore, the remaining discus-

sion in this paper is restricted to ESs using the new

covariance matrix. However, as we will see below,

the increased variability provided by (11) should be

considered together with (16) or (14) in further exper-

iments. The analysis considers ES-algorithms which

use shrinkage estimators as deﬁned in (14) to (16).

Aside from the CMSA-ES, we denote the strategies

as follows

1. CI-ES: a CMSA-ES using (14) as shrinkage tar-

get,

2. CC-ES: the CMSA-ES with the constant correla-

tion model (16),

3. FS-ES: the CMSA-ES which uses (15) and fol-

lows (Fisher and Sun, 2011) to determine the

shrinkage intensity,

4. Tou-ES: a CMSA-ES based on (15) which uses

the shrinkage intensity of (Touloumis, 2015).

The approaches were coded in MATLAB. In the case

of the CI-ES and the CC-ES we used the estimation

source code provided by the authors on their web-

page

. The implementation of the Tou-ES follows

closely the R package

5.1 Experimental Set-up

The parameters for the experiments read as follows.

Each experiment uses 15 repeats. The initial pop-

ulation is drawn uniformly from [−4, 4]

, whereas

the mutation strength is chosen from [0.25, 1]. The

search space dimensions were set to N =5, 10, 20,

and 40. The maximal number of ﬁtness evaluations

is set to FE

max

=2 ×10

N. All evolution strategies

use λ =log(3N)+8 offspring and µ =λ4 par-

ents. A run terminates prematurely if the difference

between the best value so far and the optimal ﬁtness

value f

best

−f

opt

is below a predeﬁned precision set

to 10

−8

. Furthermore, we introduce a restart mecha-

nism into the ESs so that the search is re-initialised

when the search has stagnated for 10 +30Nλgen-

erations. Stagnation is determined by measuring the

best function values in a generation. If the difference

between minimal and maximal values of the sample

lies below 10

−8

for the given the time-interval, the ES

does not make signiﬁcant movements anymore and

the search is started anew.

http://www.econ.uzh.ch/faculty/wolf/publications.html

http://cran.r-project.org/web/packages/ShrinkCovMat

The experiments are conducted with the help of

the black box optimization benchmarking (BBOB)

software framework and the test suite, see (Hansen

et al., 2012). The framework allows the analysis of

algorithms and provides means to generate tables and

ﬁgures of the results.

This paper considers 24 noise-less functions

(Finck et al., 2010). They consist of four subgroups:

separable functions (function ids 1-5), functions with

low/moderate conditioning (ids 6-9), functions with

high conditioning (ids 10-14), and two groups of mul-

timodal functions (ids 15-24).

The experiments use the expected running time

(ERT) as performance measure. The ERT is deﬁned

as the expected value of the function evaluations ( f -

evaluations) the algorithm needs to reach the target

value with the required precision for the ﬁrst time, see

(Hansen et al., 2012). In this paper, the estimate

ERT =

#(FEs(f

best

≥ f

target

))

#succ

(17)

is used, that is, the ﬁtness evaluations FEs(f

best

≥

target

) of each run until the ﬁtness of the best indi-

vidual is smaller than the target value are summed up

and divided by the number of successful runs.

5.2 Results and Discussion

First of all, let us take a look at the behavior of the

strategies for two exemplary functions, the sphere,

f (x)=x

, and the discus, f (x)=10

∑

i=2

Figure 2 shows the ratio of the largest to the smallest

eigenvalue of the covariance matrix for the CMSA-

ES and for one of the shrinkage approaches, the CC-

ES. In the case of the sphere, the ﬁgures illustrate

that the largest and smallest eigenvalue develop dif-

ferently and diverge for N =10. Shrinkage causes the

problem to be less pronounced. In the case of the dis-

cus, different eigenvalues are expected. Both strate-

gies achieve this, the CC-ES shows again a lower rate

of increase. This may be a hint that the adaptation

process of the covariance matrix may be decelerated

by the additional shrinkage. Whether this lowers the

performance is investigated in the experiments with

the complete test suite.

The results of the experiments are summarized by

Tables 1 - 3. They provide the estimate of the ex-

pected running time (ERT) for several precision tar-

gets ranging from 10

to 10

−7

. Also shown is the

number of successful runs. Several functions repre-

sent challenges for the ESs considered. These com-

prise the Rastrigin functions (id 3, id 4, id 15, and

id 24) of the test suite, the step ellipsoidal function

with a condition number of 100 (id 7), and a multi-

modal function with a weak global structure based on

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

110

a) CMSA-ES, sphere, N = 10 b) CMSA-ES, sphere, N = 40

c) CC-ES, sphere, N = 10 d) CC-ES, sphere, N = 40

e) CMSA-ES, discus, N = 10 f) CMSA-ES, discus, N = 40

g) CC-ES, discus, N = 10 h) CC-ES, discus, N = 40

Figure 2: The development of the ratio of the largest to the smallest eigenvalue of the covariance on the sphere and the discus.

Shown are the results from 15runs per dimensionality.

Evolution Strategies and Covariance Matrix Adaptation - Investigating New Shrinkage Techniques

111

Table 1: Expected running time (ERT in number of function evaluations) divided by the respective best ERT measured during

BBOB-2009 in dimension 10. The ERT and in braces, as dispersion measure, the half difference between 90 and 10%-tile

of bootstrapped run lengths appear for each algorithm and target, the corresponding best ERT in the ﬁrst row. The different

target ∆f -values are shown in the top row. #succ is the number of trials that reached the (ﬁnal) target f

opt

+10

−8

. The median

number of conducted function evaluations is additionally given in italics, if the target in the last column was never reached.

10-D

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f1 22 23 23 23 23 23 23 15/15

CMSA 4.0(2) 8.6(3) 14(5) 19(3) 26(5) 38(6) 50(5) 15/15

CC-ES 3.7(2) 8.4(3) 13(3) 18(3) 23(3) 35(5) 45(3) 15/15

CI-ES 4.2(1) 9.3(4) 14(4) 19(5) 24(4) 34(6) 43(2) 15/15

Tou-ES4.2(2) 9.1(3) 15(4) 21(3) 27(4) 40(7) 54(11) 15/15

FS-ES 4.2(2) 10(2) 14(4) 19(2) 25(4) 37(5) 49(6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f2 187 190 191 191 193 194 195 15/15

CMSA 65(30) 85(23) 96(23) 105(22) 109(31) 113(27) 129(27) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES496(217) 850(584) 1517(1640)1835(1954)1833(1007)2248(1961)2757(2246) 5/15

FS-ES 62(28) 90(26) 102(37) 110(25) 117(22) 125(31) 129(22) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f5 20 20 20 20 20 20 20 15/15

CMSA 12(5) 17(6) 17(10) 17(9) 17(6) 17(11) 17(9) 15/15

CC-ES 18(13) 102(307) 104(305) 104(16) 104(16) 104(17) 104(601) 15/15

CI-ES 14(7) 19(12) 20(15) 20(10) 20(9) 20(13) 20(12) 15/15

Tou-ES12(11) 18(17) 19(8) 19(12) 19(4) 19(6) 19(32) 15/15

FS-ES 12(5) 16(7) 16(6) 16(7) 16(9) 16(8) 16(6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f6 412 623 826 1039 1292 1841 2370 15/15

CMSA 1.4(0.9) 3.3(4) 11(31) 14(6) 19(22) 25(28) 163(176) 6/15

CC-ES 1.2(0.3) 18(1) 68(39) 193(254) 1088(1315) ∞ ∞ 2e5 0/15

CI-ES 37(3) 60(42) 183(77) 219(358) 348(1037) 452(522) 1206(823) 1/15

Tou-ES 5.3(0.4) 89(113) 135(168) 138(64) 151(67) 369(200) ∞ 2e5 0/15

FS-ES 1.9(0.6) 3.3(2) 14(6) 19(21) 25(10) 37(45) 75(37) 11/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f7 172 1611 4195 5099 5141 5141 5389 15/15

CMSA 4.0(6) 26(16) 85(66) ∞ ∞ ∞ ∞ 2e5 0/15

CC-ES 2.1(3) 42(29) 338(306) ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES 3.4(2) 34(34) 204(155) ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES3.2(5) 37(69) 323(155) ∞ ∞ ∞ ∞ 2e5 0/15

FS-ES 4.5(6) 78(33) 206(167) ∞ ∞ ∞ ∞ 2e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f8 326 921 1114 1217 1267 1315 1343 15/15

CMSA 3.3(2) 17(10) 18(9) 18(6) 18(6) 19(10) 19(9) 15/15

CC-ES 17(38) 378(428) 784(1249) 2310(1972) ∞ ∞ ∞ 2e5 0/15

CI-ES 75(163) 197(130) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES11(17) 68(18) 72(30) 76(30) 77(39) 78(33) 79(33) 15/15

FS-ES 4.2(8) 17(13) 18(8) 17(5) 17(13) 18(3) 18(8) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f9 200 648 857 993 1065 1138 1185 15/15

CMSA 2.3(2) 25(18) 24(3) 22(11) 22(11) 21(10) 21(4) 15/15

CC-ES 16(33) 2122(2603) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES 2.2(1) 323(319) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES 4.8(10) 91(31) 86(44) 86(46) 84(36) 84(27) 83(23) 15/15

FS-ES 3.5(2) 19(13) 19(5) 18(9) 18(13) 18(4) 18(12) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f10 1835 2172 2455 2728 2802 4543 4739 15/15

CMSA 6.5(3) 7.6(1) 7.7(3) 7.5(4) 7.6(3) 4.9(1) 4.9(2) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES111(72) 301(192) 608(815) ∞ ∞ ∞ ∞ 2e5 0/15

FS-ES 5.7(2) 7.9(1) 7.7(2) 7.5(1) 7.9(1) 5.2(1) 5.3(1) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f11 266 1041 2602 2954 3338 4092 4843 15/15

CMSA 14(3) 6.2(2) 3.2(1) 3.4(2) 3.5(1) 3.3(0.9) 3.0(2) 15/15

CC-ES 1.1e4(1e4) ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES152(258) 169(152) 150(143) 312(614) 277(312) 343(354) 596(1270) 1/15

FS-ES 16(5) 6.3(2) 3.2(0.9) 3.3(1) 3.3(1) 3.3(0.6) 3.0(0.6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f12 515 896 1240 1390 1569 3660 5154 15/15

CMSA 4.2(0.2) 10(6) 13(13) 15(13) 16(9) 8.8(5) 7.8(2) 15/15

CC-ES 11(18) 37(15) 68(142) 161(241) 572(444) ∞ ∞ 2e5 0/15

CI-ES 29(97) 615(669) 2258(4031) 2020(1655) ∞ ∞ ∞ 2e5 0/15

Tou-ES76(228) 177(170) 722(1674) 1016(1115) 910(970) 799(464) 569(378) 1/15

FS-ES 2.3(2) 10(12) 13(8) 15(8) 16(10) 10(5) 10(6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f13 387 596 797 1014 4587 6208 7779 15/15

CMSA 15(11) 19(34) 31(33) 58(115) 28(23) 89(114) 185(129) 2/15

CC-ES 19(36) 59(71) 186(311) 610(1661) 637(1243) ∞ ∞ 2e5 0/15

CI-ES 32(6) 57(118) 87(46) 229(245) 145(147) ∞ ∞ 2e5 0/15

Tou-ES22(45) 50(37) 117(68) 500(786) 307(262) ∞ ∞ 2e5 0/15

FS-ES 5.1(0.5) 23(32) 35(49) 66(52) 31(55) 98(128) 183(238) 2/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f14 37 98 133 205 392 687 4305 15/15

CMSA 1.1(1) 2.2(0.6) 2.8(0.5) 3.5(1) 4.3(0.5) 8.7(3) 4.8(3) 15/15

CC-ES 0.98(1) 2.4(1) 2.9(1) 3.6(2) 72(70) ∞ ∞ 2e5 0/15

CI-ES 1.2(0.9) 2.4(1) 2.9(1) 3.6(3) 24(6) ∞ ∞ 2e5 0/15

Tou-ES1.4(2) 2.7(0.5) 3.5(2) 5.4(2) 24(20) 241(169) 339(314) 2/15

FS-ES 1.4(1) 2.6(1) 3.0(1) 3.7(1) 4.5(1.0) 10(4) 4.5(2) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f16 425 7029 15779 45669 51151 65798 71570 15/15

CMSA 1.0(0.7) 1.8(2) 13(14) 31(12) ∞ ∞ ∞ 2e5 0/15

CC-ES 1.1(2) 2.3(1) 43(35) ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES 0.76(1.0) 19(14) 180(133) ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES1.2(1) 8.7(18) 179(139) ∞ ∞ ∞ ∞ 2e5 0/15

FS-ES 1.7(0.7) 0.55(0.7) 23(27) ∞ ∞ ∞ ∞ 2e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f17 26 429 2203 6329 9851 20190 26503 15/15

CMSA 0.71(0.4) 18(26) 23(44) 30(17) 140(86) ∞ ∞ 2e5 0/15

CC-ES 0.72(0.5) 20(12) 20(26) 39(114) 304(436) ∞ ∞ 2e5 0/15

CI-ES 1.2(0.5) 171(117) 182(227) ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES1.6(1) 52(43) 75(115) 141(174) ∞ ∞ ∞ 2e5 0/15

FS-ES 1.00(3) 26(29) 27(48) 77(107) 424(559) ∞ ∞ 2e5 0/22

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f18 238 836 7012 15928 27536 37234 42708 15/15

CMSA 68(47) 129(298) 124(54) ∞ ∞ ∞ ∞ 2e5 0/15

CC-ES 8.8(28) 38(71) 55(60) ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES 3.3(17) 136(124) 124(235) ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES 8.5(1.0) 187(237) 211(135) ∞ ∞ ∞ ∞ 2e5 0/15

FS-ES 2.6(0.8) 98(79) 94(124) ∞ ∞ ∞ ∞ 2e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f20 32 15426 5.5e5 5.7e5 5.7e5 5.8e5 5.9e5 15/15

CMSA 1.9(1) 25(37) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CC-ES 1.9(1) 33(17) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

CI-ES 2.1(1) 38(55) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

Tou-ES1.8(1) 56(101) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

FS-ES 1.5(0.7) 24(32) ∞ ∞ ∞ ∞ ∞ 2e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f21 130 2236 4392 4487 4618 5074 11329 8/15

CMSA 9.5(11) 23(24) 20(15) 20(20) 19(28) 17(40) 7.8(9) 12/15

CC-ES 33(54) 18(10) 13(18) 17(13) 16(10) 36(19) 35(49) 6/15

CI-ES 47(133) 109(268) 58(56) 57(92) 55(102) 50(50) 23(46) 8/15

Tou-ES50(232) 39(98) 32(76) 31(23) 30(44) 28(74) 14(6) 9/15

FS-ES 13(50) 16(27) 10(13) 9.4(13) 9.2(12) 8.4(6) 3.8(6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f22 98 2839 6353 6620 6798 8296 10351 6/15

CMSA 25(45) 6.3(9) 13(23) 12(16) 12(19) 10(14) 8.1(9) 13/15

CC-ES 48(63) 13(12) 23(21) 49(21) 133(238) ∞ ∞ 2e5 0/15

CI-ES 149(369) 29(72) 24(43) 25(26) 25(22) 30(48) ∞ 2e5 0/15

Tou-ES 61(132) 10(12) 17(14) 17(17) 17(22) 16(24) 14(19) 12/15

FS-ES 22(68) 14(22) 11(13) 11(20) 11(18) 9.0(5) 7.3(11) 13/15

Schwefel (id 20). In these cases, all strategies are un-

able to progress further than the ﬁrst intermediate pre-

cision of 10

. Additionally, for the multi-modal func-

tions, 19 (Composite Griewank-Rosenbrock Func-

tion) and 23 (Katsuura), no strategy could reach 10

−1

The functions in question are removed from the ta-

bles.

To analyze the remaining functions, the four

groups of the test suite are taken into account. The

ﬁrst class comprises the separable functions with id

1 to id 5. The three remaining functions, the sphere

(f1), the separable ellipsoidal function (f2), and the

linear slope (f5) differ in the degree of difﬁculty for

the strategies. All strategies do not show any prob-

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

112

Table 2: Expected running time (ERT in number of function evaluations) divided by the respective best ERT measured during

BBOB-2009 in dimension 20. The ERT and in braces, as dispersion measure, the half difference between 90 and 10%-tile

of bootstrapped run lengths appear for each algorithm and target, the corresponding best ERT in the ﬁrst row. The different

target ∆f -values are shown in the top row. #succ is the number of trials that reached the (ﬁnal) target f

opt

+10

−8

. The median

number of conducted function evaluations is additionally given in italics, if the target in the last column was never reached.

20-D

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f1 43 43 43 43 43 43 43 15/15

CMSA 4.9(2) 10(0.9) 15(2) 19(2) 25(2) 34(2) 45(2) 15/15

CC-ES 4.7(1) 8.9(2) 14(1) 19(2) 23(2) 33(4) 42(3) 15/15

CI-ES 5.0(1) 10(1) 15(2) 19(2) 24(2) 34(2) 44(2) 15/15

Tou-ES5.1(0.9) 10(1) 14(2) 19(2) 25(3) 35(5) 46(4) 15/15

FS-ES 4.8(2) 10(1) 14(3) 19(2) 24(2) 34(3) 44(4) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f2 385 386 387 388 390 391 393 15/15

CMSA 173(26) 240(36) 265(23) 273(41) 277(37) 285(28) 293(37) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 162(21) 224(70) 263(40) 278(28) 285(36) 296(19) 308(52) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f5 41 41 41 41 41 41 41 15/15

CMSA 12(4) 15(7) 15(4) 15(7) 15(7) 15(5) 15(6) 15/15

CC-ES 13(7) 17(8) 18(7) 18(10) 18(7) 18(9) 18(10) 15/15

CI-ES 14(8) 18(7) 19(6) 19(7) 19(15) 19(22) 19(8) 15/15

Tou-ES11(3) 14(3) 14(5) 14(4) 14(4) 14(6) 14(4) 15/15

FS-ES 12(5) 14(4) 15(4) 15(4) 15(4) 15(7) 15(4) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f6 1296 2343 3413 4255 5220 6728 8409 15/15

CMSA 1.5(0.4) 2.5(2) 4.6(2) 12(23) 34(15) 80(90) 331(393) 2/15

CC-ES 80(80) 471(640) 1643(1465) ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 29(232) 88(128) 108(177) 154(236) 144(230) 253(253) 206(187) 3/15

Tou-ES56(27) 275(515) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 2.0(0.7) 3.9(4) 6.1(6) 19(9) 64(58) 265(357) ∞ 4e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f8 2039 3871 4040 4148 4219 4371 4484 15/15

CMSA 11(9) 30(3) 31(54) 31(8) 31(99) 31(6) 31(47) 13/15

CC-ES 2752(4660) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 106(132) 727(1259) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES 96(46) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 13(6) 31(6) 33(52) 33(31) 33(53) 32(25) 32(44) 13/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f9 1716 3102 3277 3379 3455 3594 3727 15/15

CMSA 17(6) 40(63) 41(63) 41(91) 41(88) 40(31) 40(29) 13/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 158(161) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES 88(32) 633(291) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 17(7) 31(2) 33(4) 33(5) 33(32) 33(30) 32(2) 14/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f10 7413 8661 10735 13641 14920 17073 17476 15/15

CMSA 10(3) 11(2) 9.2(2) 7.8(2) 7.3(1) 6.7(0.8) 6.8(1.0) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES796(445) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 8.3(3) 10(1) 8.4(2) 7.0(1) 6.8(0.7) 6.2(0.5) 6.3(0.6) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f11 1002 2228 6278 8586 9762 12285 14831 15/15

CMSA 12(2) 7.5(0.9) 3.1(0.1) 2.6(0.5) 2.6(0.4) 2.5(0.6) 2.5(0.4) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES2831(3585) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 12(2) 6.7(0.8) 2.7(0.6) 2.4(0.6) 2.5(0.4) 2.4(0.6) 2.3(0.4) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f12 1042 1938 2740 3156 4140 12407 13827 15/15

CMSA 2.5(0.1) 10(9) 13(8) 15(6) 14(5) 5.9(2) 6.2(2) 15/15

CC-ES 97(480) 569(568) 2045(2299) 1775(887) ∞ ∞ ∞ 4e5 0/15

CI-ES 60(96) 182(206) 585(365) 1775(3993) ∞ ∞ ∞ 4e5 0/15

Tou-ES97(192) 137(139) 315(357) 1775(1901) ∞ ∞ ∞ 4e5 0/15

FS-ES 4.9(0.2) 12(11) 14(10) 16(12) 15(7) 6.3(3) 6.5(3) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f13 652 2021 2751 3507 18749 24455 30201 15/15

CMSA 156(154) 545(594) 2037(1600) ∞ ∞ ∞ ∞ 4e5 0/15

CC-ES 97(307) 397(495) 2037(2181) ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 97(155) 397(396) 584(763) ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES309(460) 545(742) 947(1309) ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 97(460) 298(396) 2037(2509) ∞ ∞ ∞ ∞ 4e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f14 75 239 304 451 932 1648 15661 15/15

CMSA 1.8(0.9) 1.9(0.7) 2.5(0.5) 3.2(0.6) 5.2(0.8) 11(3) 4.2(1) 15/15

CC-ES 1.8(0.4) 1.9(0.6) 2.4(0.6) 3.1(0.6) ∞ ∞ ∞ 4e5 0/15

CI-ES 1.8(0.6) 1.8(0.4) 2.4(0.5) 3.4(1.0) 23(4) ∞ ∞ 4e5 0/15

Tou-ES1.5(0.4) 1.7(0.5) 2.3(0.7) 3.5(0.7) 27(21) ∞ ∞ 4e5 0/15

FS-ES 2.0(1) 1.9(0.5) 2.4(0.5) 3.2(0.8) 4.8(0.3) 12(2) 3.9(0.5) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f16 1384 27265 77015 1.4e5 1.9e5 2.0e5 2.2e5 15/15

CMSA 22(145) 95(106) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CC-ES 45(145) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 45(217) 95(114) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES21(0.9) 205(308) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 45(218) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f17 63 1030 4005 12242 30677 56288 80472 15/15

CMSA 0.93(0.9) 196(97) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CC-ES 1.3(0.5) 341(194) 650(949) ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 1.1(0.8) 445(486) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES1.2(0.6) 778(486) 1399(1473) ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 1.7(3) 260(583) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f18 621 3972 19561 28555 67569 1.3e5 1.5e5 15/15

CMSA 48(0.9) ∞ ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CC-ES 1.3(1) 277(453) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 47(0.9) 1410(2946) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES 1.3(1) 655(579) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

FS-ES 101(484) 655(529) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f21 561 6541 14103 14318 14643 15567 17589 15/15

CMSA 179(178) 245(138) 113(113) 112(70) 109(109) 103(128) 91(85) 3/15

CC-ES 357(713) 168(122) 113(50) 112(105) 109(184) 103(218) 91(91) 3/15

CI-ES 357(1247) 398(413) 184(128) 182(321) 178(96) 167(212) 148(159) 2/15

Tou-ES179(356) 122(306) 78(135) 77(77) 75(137) 71(71) 63(159) 4/15

FS-ES 179(178) 122(61) 78(113) 77(77) 75(41) 71(96) 63(57) 4/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f22 467 5580 23491 24163 24948 26847 1.3e5 12/15

CMSA 572(1500) 287(215) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CC-ES 312(1071) 108(125) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

CI-ES 980(1285) 466(627) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

Tou-ES572(857) 287(197) 239(315) 232(128) 225(341) 211(183) 43(36) 1/15

FS-ES 751(1286) 466(502) ∞ ∞ ∞ ∞ ∞ 4e5 0/15

lems on the sphere or on the slope. Here, several

shrinkage variants surpass the original version. The

more difﬁcult ellipsoidal function, however, cannot

be solved by the CC-ES, the CI-ES, and for N =20

and N =40 by the Tou-ES. Interestingly, the FS-ES

achieves successful runs for all search space dimen-

sionalities. Its performance is similar to the origi-

nal CMSA-ES. However, the latter shows the better

performance in many cases. Two effects may play

a role. The ellipsoidal function, deﬁned by f (x)=

∑

i=1

6(i−1)/(N−1)

, is not solved well by ESs with

covariance matrices treating all directions with the

same weight. The matrix used in the basis transforma-

tion may not be sufﬁcient to provide the variability re-

quired if combined with structures that are restrictive.

The structures supplied by the CC-ES and the CI-ES

may be too regular for the ES to be able to adapt with

the necessary velocity. The failure of the Tou-ES to

achieve the ﬁnal precision target may be due to the

shrinkage intensity since this is the only point where

it differs from the FS-ES. In future research, exper-

iments will be performed that use lower shrinkage

intensities. Also the interaction with the parameter

will be taken into account and investigated more

Evolution Strategies and Covariance Matrix Adaptation - Investigating New Shrinkage Techniques

113

Table 3: Expected running time (ERT in number of function evaluations) divided by the respective best ERT measured during

BBOB-2009 in dimension 40. The ERT and in braces, as dispersion measure, the half difference between 90 and 10%-tile

of bootstrapped run lengths appear for each algorithm and target, the corresponding best ERT in the ﬁrst row. The different

target ∆f -values are shown in the top row. #succ is the number of trials that reached the (ﬁnal) target f

opt

+10

−8

. The median

number of conducted function evaluations is additionally given in italics, if the target in the last column was never reached.

40-D

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f1 83 83 83 83 83 83 83 30/30

CMSA 5.8(1) 10(0.9) 14(2) 18(1) 23(2) 32(3) 40(2) 15/15

CC-ES 5.1(0.9) 9.3(1) 14(2) 18(1) 23(1) 31(3) 40(3) 15/15

CI-ES 5.5(1) 10(1) 14(2) 18(2) 23(2) 31(2) 40(1) 15/15

Tou-ES6.3(2) 11(2) 15(0.8) 19(2) 24(2) 33(1) 42(3) 15/15

FS-ES 5.4(0.7) 10(2) 14(2) 18(2) 22(2) 31(2) 40(2) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f2 796 797 799 799 800 802 804 15/15

CMSA 367(65) 541(94) 665(90) 736(79) 786(92) 817(59) 837(49) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 391(124) 559(130) 651(93) 718(168) 787(81) 823(46) 837(80) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f5 98 116 120 121 121 121 121 15/15

CMSA 11(5) 12(4) 12(2) 12(8) 12(4) 12(4) 12(6) 15/15

CC-ES 11(7) 11(9) 11(11) 11(6) 11(8) 11(5) 11(7) 15/15

CI-ES 10(4) 11(6) 11(6) 11(6) 11(6) 11(5) 11(5) 15/15

Tou-ES 9.3(3) 10(2) 10(2) 10(2) 10(3) 10(3) 10(3) 15/15

FS-ES 8.4(2) 8.3(2) 8.2(2) 8.2(2) 8.2(2) 8.2(2) 8.2(1) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f6 3507 5523 7168 9470 11538 15007 19222 15/15

CMSA 3.9(2) 10(5) 22(18) 87(86) 312(258) ∞ ∞ 8e5 0/15

CC-ES 206(229) 948(904) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 285(171) 946(905) 1568(2093) 1191(1478) 1033(832) ∞ ∞ 8e5 0/15

Tou-ES645(684) ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 4.0(2) 10(12) 28(33) 51(37) 142(113) ∞ ∞ 8e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f8 7080 10655 11012 11265 11430 11701 11969 15/15

CMSA 30(12) 35(11) 36(28) 37(10) 37(90) 37(9) 36(26) 14/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 810(831) 1117(657) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES1665(3051) ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 33(17) 37(17) 38(14) 38(9) 38(10) 38(14) 38(8) 14/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f9 6122 12982 13300 13496 13651 13909 14142 15/15

CMSA 43(9) 38(36) 40(32) 40(32) 40(7) 40(18) 40(30) 13/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 43(9) 38(48) 39(32) 39(6) 39(32) 39(32) 39(17) 13/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f10 25890 30368 36796 51579 56007 65128 70824 15/15

CMSA 11(2) 14(4) 14(1) 11(1) 11(0.6) 10(0.6) 9.4(0.6) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 12(3) 14(1) 14(1) 11(1) 11(0.9) 10(0.5) 9.3(0.5) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f11 2368 4855 11681 25315 29749 38949 48211 15/15

CMSA 16(1) 9.2(0.6) 4.2(0.5) 2.2(0.3) 2.0(0.2) 1.8(0.4) 1.6(0.3) 15/15

CC-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES ∞ ∞ ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 16(2) 9.3(0.9) 4.3(0.6) 2.2(0.4) 2.0(0.2) 1.8(0.2) 1.6(0.3) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f12 4169 7452 9174 10751 13146 22758 25192 15/15

CMSA 5.0(3) 7.8(9) 10(7) 11(7) 11(4) 7.6(2) 7.7(2) 15/15

CC-ES 49(48) 296(537) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 97(240) 698(537) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES28(48) 94(54) 567(981) ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 4.6(8) 10(8) 12(8) 12(4) 12(6) 8.1(2) 8.2(2) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f13 2029 6916 8734 11861 71936 98467 1.2e5 15/15

CMSA 145(197) 102(260) 1283(710) ∞ ∞ ∞ ∞ 8e5 0/15

CC-ES 145(99) 174(174) 596(1214) ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 199(493) 174(145) 596(870) 945(1872) 156(158) ∞ ∞ 8e5 0/15

Tou-ES 62(197) 102(174) 596(366) 945(759) ∞ ∞ ∞ 8e5 0/15

FS-ES 62(99) 133(145) 596(733) 439(742) 156(259) ∞ ∞ 8e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f14 304 616 777 1105 2207 4825 57711 15/15

CMSA 1.2(0.4) 1.5(0.3) 1.9(0.2) 2.9(0.6) 6.2(0.8) 15(2) 4.6(0.5) 15/15

CC-ES 1.2(0.3) 1.6(0.6) 1.9(0.5) 2.9(0.7) ∞ ∞ ∞ 8e5 0/15

CI-ES 1.3(0.2) 1.6(0.2) 1.9(0.2) 3.1(0.9) 31(5) ∞ ∞ 8e5 0/15

Tou-ES1.3(0.4) 1.6(0.3) 2.0(0.3) 3.1(0.5) 32(5) ∞ ∞ 8e5 0/15

FS-ES 1.4(0.4) 1.7(0.7) 2.0(0.4) 2.9(0.3) 6.4(0.5) 16(3) 4.4(0.7) 15/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f17 399 4220 14158 34948 51958 1.3e5 2.7e5 14/15

CMSA 0.48(0.7) 1233(1517) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CC-ES 0.40(0.5)

↓3

759(758) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 0.38(0.4)

↓4

759(332) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES0.42(0.5)

↓2

759(1280) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 0.55(0.7) 759(711) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f21 1044 21144 1.0e5 1.0e5 1.0e5 1.0e5 1.0e5 26/30

CMSA 55(192) 104(85) 32(26) 32(26) 32(28) 31(22) 31(20) 3/15

CC-ES 192(767) 530(766) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 512(575) 151(255) 52(77) 52(34) 51(97) 51(29) 51(61) 2/15

Tou-ES192(767) 246(568) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

FS-ES 384(575) 530(416) 111(42) 111(103) 111(122) 110(135) 109(121) 1/15

∆ f

opt

1e1 1e0 1e-1 1e-2 1e-3 1e-5 1e-7 #succ

f22 3090 35442 6.5e5 6.5e5 6.5e5 6.5e5 6.5e5 8/30

CMSA 65(129) 34(51) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CC-ES 94(194) 45(135) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

CI-ES 40(65) 45(34) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

Tou-ES 65(130) 45(68) 17(18) 17(27) 17(24) 17(20) 18(19) 1/15

FS-ES 296(647) 147(226) ∞ ∞ ∞ ∞ ∞ 8e5 0/15

closely. Since the learning rate approaches inﬁnity for

the typical µN ratios and increasing dimensionalities,

the inﬂuence of the sample covariance lessens. Regu-

larizing the covariance matrix may therefore be more

important for smaller to medium search space dimen-

sionalities. Concerning the question whether shrink-

age improves the performance, no clear answer can be

provided for the group of separable functions since

the ellipsoidal function apparently requires a faster

adaptation than the current versions supply.

The second group of functions consists of the at-

tractive sector function (id 6), the step ellipsoidal

function (id 7), the original Rosenbrock function (id

8), and a rotated Rosenbrock function (id 9). These

functions have low to moderate conditioning. The

sector function is difﬁcult to solve for all strategies.

For N =40, no ES is able to reach the ﬁnal precision

target with the number of evaluations allowed. For

N =20, successful runs were recorded for the CMSA-

ES and the CI-ES but only for a few cases, i.e., two or

three. Therefore, the question arises whether initial-

ization effects may have played a role and a compar-

ison is not attempted. For N =10, the FS-ES reaches

the ﬁnal target precision of 10

−8

in eleven of 15 runs,

a result not mirrowed by the other strategies. Con-

cerning the Rosenbrock functions (f8 and f9), the

CMSA-ES and the FS-ES perform best, with the latter

resulting in lower ERT values on the rotated version

and the former on the original (with the exception of

N =10 where the shrinkage version surpasses the orig-

inal). Thus, the FS-ES may be preferable in the case

of the second group at least for smaller dimensional-

ICAART 2016 - 8th International Conference on Agents and Artiﬁcial Intelligence

114

ities. However, more experiments should be carried

out.

Functions with high conditioning constitute the

next group. The ellipsoidal function (f10), the dis-

cus (f11), the bent cigar (f12), the sharp ridge (f13),

and the different powers function (f14). Here, simi-

lar behaviors can be observed for the CMSA-ES and

the FS-ES with no clear advantage for either variant.

Both are able to reach the ﬁnal target precision on the

functions with the exception of the sharp ridge where

only two successful runs are recorded for N =10 in

both cases. Again, these may be due to initialization

effects and should be treated therefore with care.

In the case of the multi-modal functions (ids 15-

24), all ESs encounter problems. Only for the two

Gallagher’s problems (id 21 and id 22) successes are

recorded. Here, the CMSA-ES and the versions that

use the diagonal elements of the sample covariance as

shrinkage target show the best results. The FS-ES ap-

pears to be a good choice for N =10. For N =20 and

N =40, only a few runs of all ESs reach the ﬁnal tar-

get precision. Therefore, a comparison for the higher-

dimensional search spaces is difﬁcult and is not car-

ried out in this paper. We will conduct further exper-

iments with a higher setting for the maximal number

of ﬁtness evaluations in future work.

6 CONCLUSIONS

Evolution strategies rely on mutation as their main

search operator. This necessitates control and adap-

tation mechanisms. This paper considered the covari-

ance matrix adaptation together with self-adaptation.

Here, as for other approaches, the sample covariance

plays an important role. This estimate should be

treated with care, however, since it may not be reliable

in all cases. This holds especially if the sample size is

small with respect to the search space dimensionality

and therefore for most application cases of evolution

strategies. The paper provided an experimental anal-

ysis of shrinkage operators, an approach introduced

in statistics for correcting the sample covariance. The

covariance is shrunk towards a target and thus “cor-

rected”. The choice of the target and the combination

weight, the shrinkage intensity, are crucial. Since the

functions to be optimized may assume various struc-

tures, the approach must remain sufﬁciently adapt-

able. To achieve this, we considered a transformation

of the original search space. The experimental anal-

ysis took several shrinkage targets into account us-

ing the intensity settings of the original publications.

Pending further experiments that shall provide more

information regarding the shrinkage intensity which

may have interfered with the ﬁndings, shrinkage tar-

gets in the transformed space that use a diagonal ma-

trix consisting of the different entries of the trans-

formed sample covariance appear as the best choices.

Since the original covariance matrix adaptation per-

forms a further type of shrinkage which lessens the

inﬂuence of the sample covariance when the search

space dimensionality increases, effects may be more

pronounced for midsize dimensionalities. Future re-

search will focus on the shrinkage intensity and its

interaction with the covariance matrix adaptation.

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