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On the Locality of Neural Meta-Representations

Luís F. Simões

VU University Amsterdam

luis.simoes@vu.nl

A. E. Eiben

VU University Amsterdam

a.e.eiben@vu.nl

ABSTRACT

We consider the usage of artiﬁcial neural networks for repre-

senting genotype-phenotype maps, from and into continuous

decision variable domains. Through such an approach, ge-

netic representations become explicitly controllable entities,

amenable to adaptation. With a view towards understand-

ing the kinds of space transformations neural networks are

able to express, we investigate here the typical represen-

tation locality given by arbitrary neuro-encoded genotype-

phenotype maps. We consistently ﬁnd high locality space

transformations being carried out, across all tested feedfor-

ward neural network architectures, in 5, 10 and 30 dimen-

sional spaces.

Categories and Subject Descriptors

I.2.8 [Artiﬁcial Intelligence]: Problem Solving, Control

Methods, and Search—Heuristic methods

Keywords

Representations; Genotype-Phenotype map; Neural networks

1. INTRODUCTION

Adjusting the parameters and/or components of evolu-

tionary algorithms (EA) during a run oﬀers several advan-

tages [1]. Over the last two decades, we have witnessed

a plethora of methods being introduced for automatically

adapting all sorts of EA parameters and components, with

one notable exception: the genotype-phenotype (G-P) map-

ping, a.k.a. the representation. Few papers ever addressed

its adaptation.

We propose neural networks (NN) as a generic frame-

work for representing representations (i.e. NNs as a meta-

representation). Following such an approach, changes to

the architecture and/or weights of a NN providing a prob-

lem’s genotype-phenotype mapping will eﬀectively change

the representation space explored by evolution. Feasible ap-

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GECCO’14, July 12–16, 2014, Vancouver, BC, Canada.

ACM 978-1-4503-2881-4/14/07. .

proaches for implementing such representation learning in-

clude co-evolution, self-adaptation, and meta-evolution.

In this paper, we formally introduce the approach of us-

ing NNs as genotype-phenotype maps, and then follow with

an investigation into the locality of representations deﬁned

though these means.

2. NEURAL NETWORKS AS GENOTYPE-

PHENOTYPE MAPS

Neural networks form the basis of our approach. Being

proven global function approximators (given a suﬃciently

large number of hidden layer neurons), feedforward neural

networks are in principle capable of expressing any possible

G-P map. Also, the community has ample experience with

neuroevolution methods, which can be deployed for the ef-

fective learning of representations.

In general, evolutionary search is guided by a ﬁtness func-

tion fthat can be decomposed into a genotype-phenotype

mapping fg: Φg→Φpand a phenotype-ﬁtness mapping

fp: Φp→IR, cf. [2, Sec. 2.1.2]. When considering an EA

that searches in a continuous genotypic space, for solutions

that decode into continuous phenotypes, we then have that

Φg⊂IRm, and Φp⊂IRn, where mand nstand, respec-

tively, for the dimensionalities of the considered genotypic

and phenotypic spaces. A genotype-phenotype map is then

a transformation fg: IRm→IRn.

Let Nbe a fully connected, feedforward neural network

with llayers and dkneurons on its k-th layer (k= 1..l). If

Lkis the vector representing the states of the dkneurons in

its k-th layer, then the network’s output can be determined

through

Lk

i=σ(bk

i+

dk−1

X

j=1

wk

ij Lk−1

j), i = 1..dk,

where bkrepresents the biases for neurons in the k-th

layer, and wk

ithe weights given to signals neuron Lk

igets

from neurons in the preceding layer. A sigmoidal activation

function σ(y)=1/(1 + e−y) is used throughout this pa-

per. The network’s output, Ll, is then uniquely determined

through b,w, and L1, the input vector fed to its input layer.

Without loss of generality, in the sequel we assume that

the given phenotype space is an ndimensional hypercube.

(If needed, the interval [0,1] can be mapped with a trivial

linear transformation to the actual user speciﬁed lower and

upper bounds for each variable under optimization.) Using a

neural network as a G-P map, we then obtain a setup where

the number of output neurons dl=nand the mapping itself

is fg: [0,1]d1→[0,1]dl. To specify a given G-P mapping

199

network we will use the notation N(mp, ma), where mpand

maare the map parameters1and map arguments, deﬁned

as follows. The vector mp∈[−1,1]dcontains the deﬁnition

of all weights and biases in the network, while the vector ma

designates the input vector fed into the network. With this

notation, we obtain a formal framework where genotypes

are map arguments to the neural net and the representa-

tion is fg=N(mp, .). Given a genotype xg∈[0,1]d1, the

corresponding phenotype is fg(xg) = N(mp, xg)∈[0,1]dl.

As shorthand for a considered NN architecture, we will

use notation such as 30-5-10, to indicate a fully connected

feeedforward neural network with l= 3 layers, having d1=

30 neurons in its input layer, d2= 5 neurons in the hid-

den layer, and d3= 10 neurons in the output layer (Φg=

[0,1]30,Φp⊂IR10).

3. LOCALITY

In the sizing of the neural network employed to represent

a G-P map, we are given a trade-oﬀ between its capacity

to deﬁne arbitrary space transformations, and our capacity

to eﬀectively train it. To better understand which kinds

of mappings might be deﬁned in practice through this ap-

proach, we focus on one often-studied representation prop-

erty, locality, and address the following research question:

•what is the locality of representations given by small

to medium sized neural networks?

A representation’s locality [2, Sec. 3.3] describes how

well neighboring genotypes correspond to neighboring phe-

notypes. In a representation with perfect (high) locality,

all neighboring genotypes correspond to neighboring pheno-

types. Theoretical and experimental evidence [2] support

the view that high locality representations are important

for eﬃcient evolutionary search, as they do not modify the

complexity of the problems they are used for.

Map characterization.

The G-P map design space is explored by randomly sam-

pling (with uniform probability) NN weights and biases, in

the range [−1,1], thus providing the deﬁnition of map pa-

rameters, mp. We follow by generating a large number of

map arguments, ma(10000, to be precise), in the range

[0,1], according to a quasi-random distribution. Sobol se-

quences are used to sample the genotype space (ma∈Φg),

so as to obtain a more evenly spread coverage. The mascat-

tered in the genotype space are subsequently mapped into

the phenotype space. The Euclidean metric is used to mea-

sure distances between points within the genotype space, as

well as within the phenotype space.

We characterize the locality of representations deﬁnable

by a given neural network architecture, by randomly sam-

pling the space of possible network conﬁgurations (mp) in

that architecture. A sample of 1000 points is taken, out

of the 10000 magiven by the Sobol sequence (mentioned

above), and a mutated version generated. A mutation is al-

ways a random point along the surface of the hypersphere

centered on the original genotype, and having a radius equal

to 1% the maximum possible distance in the Φghypercube.

The mutated mais mapped into the phenotype space, and

1

“Map parameters” named by analogy with the strategy pa-

rameters (e.g., standard deviations of a Gaussian mutation)

traditionally used in Evolution Strategies.

Figure 1: Locality of representations expressible by

diﬀerent sized neural networks. Shown: empiri-

cal cumulative distribution functions of distances in

phenotype space between pairs of neighboring geno-

types.

its distance there to the original point’s phenotype mea-

sured. Given we wish to consider phenotype spaces having

distinct numbers of dimensions, and importantly, given the

fact that each diﬀerent G-P map encodes a diﬀerent subset

of the phenotype space, it becomes important to normalize

phenotypic distances, in a way that makes then comparable.

To that end, we identify the hyperrectangle that encloses

all the phenotypes identiﬁed in the initial scatter of 10000

points, and use the maximum possible distance value there

to normalize phenotype distances.

Results.

Figure 1 characterizes the locality of representations deﬁn-

able by diﬀerent NN architectures. Each of the shown distri-

butions was obtained by analyzing 1000 randomly generated

G-P maps having that architecture, and thus represents a

total of 106measured phenotype distances. We consistently

observe high locality representations resulting from all stud-

ied NN architectures: a mutation step of 1% the maximum

possible distance in genotype space is in all cases expected

to take us across a distance in phenotype space of at most

∼1% the maximum possible distance among phenotypes

representable by the considered G-P map.

4. CONCLUSION

We investigated the usage of neural networks as a meta-

representation, suited to the encoding of genotype-phenotype

maps for arbitrary pairings of ﬁtness landscapes and meta-

heuristics that are to search on them.

Small to moderately sized feedforward neural networks

were found to deﬁne, on average, high locality representa-

tions (where structure of the phenotypic ﬁtness landscape is

locally preserved in the genotype space).

5. ACKNOWLEDGMENTS

Lu´ıs F. Sim˜oes was supported by FCT (Minist´erio da

Ciˆencia e Tecnologia) fellowship SFRH/BD/84381/2012.

6. REFERENCES

[1] A. E. Eiben, R. Hinterding, and Z. Michalewicz.

Parameter Control in Evolutionary Algorithms. IEEE

Transactions on Evolutionary Computation,

3(2):124–141, 1999.

[2] F. Rothlauf. Representations for Genetic and

Evolutionary Algorithms. Springer, Berlin, Heidelberg,

2nd edition, Nov. 2006.

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