Evolving Artificial Neural Networks Proceedings Of The Ieee

Evolving Artiﬁcial Neural Networks
XIN YAO, SENIOR MEMBER, IEEE
Invited Paper
Learning and evolution are two fundamental forms of adapta-
between ANN’s and EA’s for combinatorial optimization
tion. There has been a great interest in combining learning and
will be mentioned but not discussed in detail.
evolution with artiﬁcial neural networks (ANN’s) in recent years.
This paper: 1) reviews different combinations between ANN’s and
evolutionary algorithms (EA’s), including using EA’s to evolve
A. Artiﬁcial Neural Networks
ANN connection weights, architectures, learning rules, and input
features; 2) discusses different search operators which have been
1) Architectures: An ANN consists of a set of processing
used in various EA’s; and 3) points out possible future research
elements, also known as neurons or nodes, which are
directions. It is shown, through a considerably large literature
interconnected. It can be described as a directed graph in
review, that combinations between ANN’s and EA’s can lead to
which each node
performs a transfer function
of the
signiﬁcantly better intelligent systems than relying on ANN’s or
form
EA’s alone.
Keywords—Evolutionary computation, intelligent systems, neu-
ral networks.
(1)
I.
INTRODUCTION
where
is the output of the node
is the th input to
Evolutionary artiﬁcial neural networks (EANN’s) refer
the node, and
is the connection weight between nodes
to a special class of artiﬁcial neural networks (ANN’s) in
and .
is the threshold (or bias) of the node. Usually,
which evolution is another fundamental form of adaptation
is nonlinear, such as a heaviside, sigmoid, or Gaussian
in addition to learning [1]–[5]. Evolutionary algorithms
function.
(EA’s) are used to perform various tasks, such as con-
ANN’s can be divided into feedforward and recurrent
nection weight training, architecture design, learning rule
classes according to their connectivity. An ANN is feed-
adaptation, input feature selection, connection weight ini-
forward if there exists a method which numbers all the
tialization, rule extraction from ANN’s, etc. One distinct
nodes in the network such that there is no connection from
feature of EANN’s is their adaptability to a dynamic
a node with a large number to a node with a smaller number.
environment. In other words, EANN’s can adapt to an en-
All the connections are from nodes with small numbers to
vironment as well as changes in the environment. The two
nodes with larger numbers. An ANN is recurrent if such a
forms of adaptation, i.e., evolution and learning in EANN’s,
numbering method does not exist.
make their adaptation to a dynamic environment much more
In (1), each term in the summation only involves one
effective and efﬁcient. In a broader sense, EANN’s can be
input
. High-order ANN’s are those that contain high-
regarded as a general framework for adaptive systems, i.e.,
order nodes, i.e., nodes in which more than one input
systems that can change their architectures and learning
are involved in some of the terms of the summation. For
rules appropriately without human intervention.
example, a second-order node can be described as
This paper is most concerned with exploring possible
beneﬁts arising from combinations between ANN’s and
EA’s. Emphasis is placed on the design of intelligent
systems based on ANN’s and EA’s. Other combinations
Manuscript received July 10, 1998; revised February 18, 1999. This
work was supported in part by the Australian Research Council through
where all the symbols have similar deﬁnitions to those in
its small grant scheme.
(1).
The author is with the School of Computer Science, University
The architecture of an ANN is determined by its topo-
of Birmingham, Edgbaston, Birmingham B15 2TT U.K. (e-mail:
xin@cs.bham.ac.uk).
logical structure, i.e., the overall connectivity and transfer
Publisher Item Identiﬁer S 0018-9219(99)06906-6.
function of each node in the network.
0018–9219/99$10.00 © 1999 IEEE
PROCEEDINGS OF THE IEEE, VOL. 87, NO. 9, SEPTEMBER 1999
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Fig. 1.
A general framework of EA’s.
2) Learning in ANN’s: Learning in ANN’s is typically
gradient-based search algorithms. They do not depend
accomplished using examples. This is also called “training”
on gradient information and thus are quite suitable for
in ANN’s because the learning is achieved by adjusting the
problems where such information is unavailable or very
connection weights1 in ANN’s iteratively so that trained
costly to obtain or estimate. They can even deal with
(or learned) ANN’s can perform certain tasks. Learning in
problems where no explicit and/or exact objective function
ANN’s can roughly be divided into supervised, unsuper-
is available. These features make them much more robust
vised, and reinforcement learning. Supervised learning is
than many other search algorithms. Fogel [15] and B¨ack
based on direct comparison between the actual output of
et al. [16] give a good introduction to various evolutionary
an ANN and the desired correct output, also known as the
algorithms for optimization.
target output. It is often formulated as the minimization
of an error function such as the total mean square error
C. Evolution in EANN’s
between the actual output and the desired output summed
Evolution has been introduced into ANN’s at roughly
over all available data. A gradient descent-based optimiza-
three different levels: connection weights; architectures;
tion algorithm such as backpropagation (BP) [6] can then
and learning rules. The evolution of connection weights
be used to adjust connection weights in the ANN iteratively
introduces an adaptive and global approach to training,
in order to minimize the error. Reinforcement learning is a
especially in the reinforcement learning and recurrent net-
special case of supervised learning where the exact desired
work learning paradigm where gradient-based training al-
output is unknown. It is based only on the information of
gorithms often experience great difﬁculties. The evolution
whether or not the actual output is correct. Unsupervised
of architectures enables ANN’s to adapt their topologies
learning is solely based on the correlations among input
to different tasks without human intervention and thus
data. No information on “correct output” is available for
provides an approach to automatic ANN design as both
learning.
ANN connection weights and structures can be evolved.
The essence of a learning algorithm is the learning rule,
The evolution of learning rules can be regarded as a process
i.e., a weight-updating rule which determines how connec-
of “learning to learn” in ANN’s where the adaptation of
tion weights are changed. Examples of popular learning
learning rules is achieved through evolution. It can also be
rules include the delta rule, the Hebbian rule, the anti-
regarded as an adaptive process of automatic discovery of
Hebbian rule, and the competitive learning rule [7].
novel learning rules.
More detailed discussion of ANN’s can be found in [7].
D. Organization of the Article
B. EA’s
The remainder of this paper is organized as follows.
EA’s refer to a class of population-based stochastic
Section II discusses the evolution of connection weights.
search algorithms that are developed from ideas and princi-
The aim is to ﬁnd a near-optimal set of connection weights
ples of natural evolution. They include evolution strategies
globally for an ANN with a ﬁxed architecture using EA’s.
(ES) [8], [9], evolutionary programming (EP) [10], [11],
Various methods of encoding connection weights and dif-
[12], and genetic algorithms (GA’s) [13], [14]. One im-
ferent search operators used in EA’s will be discussed.
portant feature of all these algorithms is their population-
Comparisons between the evolutionary approach and con-
based search strategy. Individuals in a population compete
ventional training algorithms, such as BP, will be made.
and exchange information with each other in order to
In general, no single algorithm is an overall winner for all
perform certain tasks. A general framework of EA’s can
kinds of networks. The best training algorithm is problem
be described by Fig. 1.
dependent.
EA’s are particularly useful for dealing with large com-
Section III is devoted to the evolution of architectures,
plex problems which generate many local optima. They are
i.e., ﬁnding a near-optimal ANN architecture for the tasks
less likely to be trapped in local minima than traditional
at hand. It is known that the architecture of an ANN
1 Thresholds (biases) can be viewed as connection weights with ﬁxed
determines the information processing capability of the
input 01.
ANN. Architecture design has become one of the most
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Fig. 2.
A typical cycle of the evolution of connection weights.
important tasks in ANN research and application. Two most
a global minimum if the error function is multimodal
important issues in the evolution of architectures, i.e., the
and/or nondifferentiable. A detailed review of BP and other
representation and search operators used in EA’s, will be
learning algorithms can be found in [7], [17], and [25].
addressed in this section. It is shown that evolutionary
One way to overcome gradient-descent-based training
algorithms relying on crossover operators do not perform
algorithms’ shortcomings is to adopt EANN’s, i.e., to for-
very well in searching for a near-optimal ANN architecture.
mulate the training process as the evolution of connection
Reasons and empirical results will be given in this section
weights in the environment determined by the architecture
to explain why this is the case.
and the learning task. EA’s can then be used effectively
If imagining ANN’s connection weights and architectures
in the evolution to ﬁnd a near-optimal set of connection
as their “hardware,” it is easier to understand the importance
weights globally without computing gradient information.
of the evolution of ANN’s “software”—learning rules.
The ﬁtness of an ANN can be deﬁned according to different
Section IV addresses the evolution of learning rules in
needs. Two important factors which often appear in the
ANN’s and examines the relationship between learning
ﬁtness (or error) function are the error between target and
and evolution, e.g., how learning guides evolution and
actual outputs and the complexity of the ANN. Unlike
how learning itself evolves. It is demonstrated that an
the case in gradient-descent-based training algorithms, the
ANN’s learning ability can be improved through evolution.
ﬁtness (or error) function does not have to be differentiable
Although research on this topic is still in its early stages,
or even continuous since EA’s do not depend on gradient
further studies will no doubt beneﬁt research in ANN’s and
information. Because EA’s can treat large, complex, non-
machine learning as a whole.
differentiable, and multimodal spaces, which are the typical
Section V summarizes some other forms of combinations
case in the real world, considerable research and application
between ANN’s and EA’s. They do not intend to be
has been conducted on the evolution of connection weights
exhaustive, simply indicative. They demonstrate the breadth
[24], [26]–[112].
of possible combinations between ANN’s and EA’s.
The evolutionary approach to weight training in ANN’s
Section VI ﬁrst describes a general framework of
consists of two major phases. The ﬁrst phase is to decide
EANN’s in terms of adaptive systems where interactions
the representation of connection weights, i.e., whether in
among three levels of evolution are considered. The
the form of binary strings or not. The second one is
framework provides a common basis for comparing
the evolutionary process simulated by an EA, in which
different EANN models. The section then gives a brief
search operators such as crossover and mutation have to
summary of the paper and concludes with a few remarks.
be decided in conjunction with the representation scheme.
Different representations and search operators can lead to
II.
THE EVOLUTION OF CONNECTION WEIGHTS
quite different training performance. A typical cycle of the
evolution of connection weights is shown in Fig. 2. The
Weight training in ANN’s is usually formulated as min-
evolution stops when the ﬁtness is greater than a predeﬁned
imization of an error function, such as the mean square
value (i.e., the training error is smaller than a certain value)
error between target and actual outputs averaged over all
or the population has converged.
examples, by iteratively adjusting connection weights. Most
training algorithms, such as BP and conjugate gradient
algorithms [7], [17]–[19], are based on gradient descent.
A. Binary Representation
There have been some successful applications of BP in
The canonical genetic algorithm (GA) [13], [14] has
various areas [20]–[22], but BP has drawbacks due to its use
always used binary strings to encode alternative solutions,
of gradient descent [23], [24]. It often gets trapped in a local
often termed chromosomes. Some of the early work in
minimum of the error function and is incapable of ﬁnding
evolving ANN connection weights followed this approach
YAO: EVOLVING ARTIFICIAL NEURAL NETWORKS
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(a)
(b)
(a)
(b)
Fig. 3.
(a) An ANN with connection weights shown. (b) A
Fig. 4.
(a) An ANN which is equivalent to that given in Fig. 3(a).
binary representation of the weights, assuming that each weight
(b) Its binary representation under the same representation scheme.
is represented by four bits.
many-to-one mapping from the representation (genotype)
[24], [26], [28], [37], [38], [41], [52], [53]. In such a repre-
to the actual ANN (phenotype) since two ANN’s that order
sentation scheme, each connection weight is represented by
their hidden nodes differently in their chromosomes will
a number of bits with certain length. An ANN is encoded by
still be equivalent functionally. For example, ANN’s shown
concatenation of all the connection weights of the network
by Figs. 3(a) and 4(a) are equivalent functionally, but they
in the chromosome.
have different chromosomes as shown by Figs. 3(b) and
A heuristic concerning the order of the concatenation
4(b). In general, any permutation of the hidden nodes
is to put connection weights to the same hidden/output
will produce functionally equivalent ANN’s with differ-
node together. Hidden nodes in ANN’s are in essence
ent chromosome representations. The permutation problem
feature extractors and detectors. Separating inputs to the
makes crossover operator very inefﬁcient and ineffective in
same hidden node far apart in the binary representation
producing good offspring.
would increase the difﬁculty of constructing useful feature
detectors because they might be destroyed by crossover
B. Real-Number Representation
operators. It is generally very difﬁcult to apply crossover
There have been some debates on the cardinality of
operators in evolving connection weights since they tend
the genotype alphabet. Some have argued that the mini-
to destroy feature detectors found during the evolutionary
mal cardinality, i.e., the binary representation, might not
process.
be the best [48], [114]. Formal analysis of nonstandard
Fig. 3 gives an example of the binary representation of
representations and operators based on the concept of
an ANN whose architecture is predeﬁned. Each connection
equivalent classes [115], [116] has given representations
weight in the ANN is represented by 4 bits, the whole ANN
other than ary strings a more solid theoretical foundation.
is represented by 24 bits where weight 0000 indicates no
Real numbers have been proposed to represent connection
connection between two nodes.
weights directly, i.e., one real number per connection
The advantages of the binary representation lie in its
weight [27], [29], [30], [48], [63]–[65], [74], [95], [96],
simplicity and generality. It is straightforward to apply
[102], [110], [111], [117], [118]. For example, a real-
classical crossover (such as one-point or uniform crossover)
number representation of the ANN given by Fig. 3(a) could
and mutation to binary strings. There is little need to design
be (4.0,10.0,2.0,0.0,7.0,3.0).
complex and tailored search operators. The binary repre-
As connection weights are represented by real numbers,
sentation also facilitates digital hardware implementation
each individual in an evolving population will be a real
of ANN’s since weights have to be represented in terms of
vector. Traditional binary crossover and mutation can no
bits in hardware with limited precision.
longer be used directly. Special search operators have
There are several encoding methods, such as uniform,
to be designed. Montana and Davis [27] deﬁned a large
Gray, exponential, etc., that can be used in the binary
number of tailored genetic operators which incorporated
representation. They encode real values using different
many heuristics about training ANN’s. The idea was to
ranges and precisions given the same number of bits.
retain useful feature detectors formed around hidden nodes
However, a tradeoff between representation precision and
during evolution. Their results showed that the evolutionary
the length of chromosome often has to be made. If too
training approach was much faster than BP for the problems
few bits are used to represent each connection weight,
they considered. Bartlett and Downs [30] also demonstrated
training might fail because some combinations of real-
that the evolutionary approach was faster and had better
valued connection weights cannot be approximated with
scalability than BP.
sufﬁcient accuracy by discrete values. On the other hand,
A natural way to evolve real vectors would be to use
if too many bits are used, chromosomes representing large
EP or ES since they are particularly well-suited for treating
ANN’s will become extremely long and the evolution in
continuous optimization. Unlike GA’s, the primary search
turn will become very inefﬁcient.
operator in EP and ES is mutation. One of the major
One of the problems faced by evolutionary training of
advantages of using mutation-based EA’s is that they can
ANN’s is the permutation problem [32], [113], also known
reduce the negative impact of the permutation problem.
as the competing convention problem. It is caused by the
Hence the evolutionary process can be more efﬁcient. There
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have been a number of successful examples of applying EP
is unavailable or very costly to obtain or estimate. For
or ES to the evolution of ANN connection weights [29],
example, the evolutionary approach has been used to train
[63]–[65], [67], [68], [95], [96], [102], [106], [111], [117],
recurrent ANN’s [41], [60], [65], [100], [102], [103], [106],
[119], [120]. In these examples, the primary search operator
[117], [126]–[128], higher order ANN’s [52], [53], and
has been Gaussian mutation. Other mutation operators, such
fuzzy ANN’s [76], [77], [129], [130]. Moreover, the same
as Cauchy mutation [121], [122], can also be used. EP
EA can be used to train many different networks regardless
and ES also allow self adaptation of strategy parameters.
of whether they are feedforward, recurrent, or higher order
Evolving connection weights by EP can be implemented
ANN’s. The general applicability of the evolutionary ap-
as follows.
proach saves a lot of human efforts in developing different
training algorithms for different types of ANN’s.
1) Generate an initial population of
individuals at
The evolutionary approach also makes it easier to gener-
random and set
. Each individual is a pair
ate ANN’s with some special characteristics. For example,
of real-valued vectors,
,
the ANN’s complexity can be decreased and its generaliza-
where
’s are connection weight vectors and
’s are
tion increased by including a complexity (regularization)
variance vectors for Gaussian mutations (also known
term in the ﬁtness function. Unlike the case in gradient-
as strategy parameters in self-adaptive EA’s). Each
based training, this term does not need to be differentiable
individual corresponds to an ANN.
or even continuous. Weight sharing and weight decay can
2) Each individual
, creates a
also be incorporated into the ﬁtness function easily.
single offspring
by: for
Evolutionary training can be slow for some problems in
comparison with fast variants of BP [131] and conjugate
(2)
gradient algorithms [19], [132]. However, EA’s are gen-
(3)
erally much less sensitive to initial conditions of training.
They always search for a globally optimal solution, while
where
, and
denote the
a gradient descent algorithm can only ﬁnd a local optimum
th component of the vectors
and
,
in a neighborhood of the initial solution.
respectively.
denotes a normally distributed
For some problems, evolutionary training can be signif-
one-dimensional random number with mean zero and
icantly faster and more reliable than BP [30], [34], [40],
variance one.
indicates that the random
[63], [83], [89]. Prados [34] described a GA-based training
number is generated anew for each value of . The
algorithm which is “signiﬁcantly faster than methods that
parameters
and
are commonly set to
use the generalized delta rule (GDR).” For the three tests
and
[15], [123].
in (3) may be
reported in his paper [34], the GA-based training algorithm
replaced by Cauchy mutation [121], [122], [124] for
“took a total of about 3 hours and 20 minutes, and the GDR
faster evolution.
took a total of about 23 hours and 40 minutes.” Bartlett and
3) Determine the ﬁtness of every individual, including
Downs [30] also gave a modiﬁed GA which was “an order
all parents and offspring, based on the training error.
of magnitude” faster than BP for the 7-bit parity problem.
Different error functions may be used here.
The modiﬁed GA seemed to have better scalability than
4) Conduct pairwise comparison over the union of par-
BP since it was “around twice” as slow as BP for the
ents
and offspring
.
XOR problem but faster than BP for the larger 7-bit parity
For each individual,
opponents are chosen uni-
problem.
formly at random from all the parents and offspring.
Interestingly, quite different results were reported by Ki-
For each comparison, if the individual’s ﬁtness is
tano [133]. He found that the GA–BP method, a technique
no smaller than the opponent’s, it receives a “win.”
that runs a GA ﬁrst and then BP, “is, at best, equally
Select
individuals out of
and
efﬁcient to faster variants of back propagation in very small
, that have most wins to form the
scale networks, but far less efﬁcient in larger networks.”
next generation. (This tournament selection scheme
The test problems he used included the XOR problem,
may be replaced by other selection schemes, such as
various size encoder/decoder problems, and the two-spiral
[125].)
problem. However, there have been many other papers
5) Stop if the halting criterion is satisﬁed; otherwise,
which report excellent results using hybrid evolutionary and
and go to Step 2).
gradient descent algorithms [32], [67], [70], [71], [74], [80],
[81], [86], [103], [105], [110]–[112].
C. Comparison Between Evolutionary Training
The discrepancy between two seemingly contradictory
and Gradient-Based Training
results can be attributed at least partly to the different
As indicated at the beginning of Section II, the evolu-
EA’s and BP compared. That is, whether the comparison
tionary training approach is attractive because it can handle
is between a classical binary GA and a fast BP algorithm,
the global search problem better in a vast, complex, mul-
or between a fast EA and a classical BP algorithm. The
timodal, and nondifferentiable surface. It does not depend
discrepancy also shows that there is no clear winner in
on gradient information of the error (or ﬁtness) function
terms of the best training algorithm. The best one is always
and thus is particularly appealing when this information
problem dependent. This is certainly true according to the
YAO: EVOLVING ARTIFICIAL NEURAL NETWORKS
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no-free-lunch theorem [134]. In general, hybrid algorithms
tend to perform better than others for a large number of
problems.
D. Hybrid Training
Most EA’s are rather inefﬁcient in ﬁne-tuned local search
although they are good at global search. This is especially
true for GA’s. The efﬁciency of evolutionary training can
be improved signiﬁcantly by incorporating a local search
procedure into the evolution, i.e., combining EA’s global
search ability with local search’s ability to ﬁne tune. EA’s
can be used to locate a good region in the space and then
a local search procedure is used to ﬁnd a near-optimal
Fig. 5.
An illustration of using an EA to ﬁnd good initial weights
solution in this region. The local search algorithm could
such that a local search algorithm can ﬁnd the globally optimal
be BP [32], [133] or other random search algorithms [30],
weights easily. wI2 in this ﬁgure is an optimal initial weight
[135]. Hybrid training has been used successfully in many
because it can lead to the global optimum wB using a local search
algorithm.
application areas [32], [67], [70], [71], [74], [80], [81], [86],
[103], [105], [110]–[112].
Lee [81] and many others [32], [136]–[138] used GA’s to
a large number of connections and nonlinear nodes may
search for a near-optimal set of initial connection weights
overﬁt noise in the training data and fail to have good
and then used BP to perform local search from these
generalization ability.
initial weights. Their results showed that the hybrid GA/BP
Up to now, architecture design is still very much a human
approach was more efﬁcient than either the GA or BP al-
expert’s job. It depends heavily on the expert experience
gorithm used alone. If we consider that BP often has to run
and a tedious trial-and-error process. There is no systematic
several times in practice in order to ﬁnd good connection
way to design a near-optimal architecture for a given task
weights due to its sensitivity to initial conditions, the hybrid
automatically. Research on constructive and destructive al-
training algorithm will be quite competitive. Similar work
gorithms represents an effort toward the automatic design of
on the evolution of initial weights has also been done on
architectures [141]–[148]. Roughly speaking, a constructive
competitive learning neural networks [139] and Kohonen
algorithm starts with a minimal network (network with
networks [140].
minimal number of hidden layers, nodes, and connections)
It is interesting to consider ﬁnding good initial weights
and adds new layers, nodes, and connections when nec-
as locating a good region in the weight space. Deﬁning that
essary during training while a destructive algorithm does
basin of attraction of a local minimum as being composed
the opposite, i.e., starts with the maximal network and
of all the points, sets of weights in this case, which can
deletes unnecessary layers, nodes, and connections during
converge to the local minimum through a local search
training. However, as indicated by Angeline et al. [149],
algorithm, then a global minimum can easily be found by
“Such structural hill climbing methods are susceptible to
the local search algorithm if an EA can locate a point, i.e., a
becoming trapped at structural local optima.” In addition,
set of initial weights, in the basin of attraction of the global
they “only investigate restricted topological subsets rather
minimum. Fig. 5 illustrates a simple case where there is
than the complete class of network architectures” [149].
only one connection weight in the ANN. If an EA can ﬁnd
Design of the optimal architecture for an ANN can be
an initial weight such as
, it would be easy for a local
formulated as a search problem in the architecture space
search algorithm to arrive at the globally optimal weight
where each point represents an architecture. Given some
even though
itself is not as good as
.
performance (optimality) criteria, e.g., lowest training error,
lowest network complexity, etc., about architectures, the
III.
THE EVOLUTION OF ARCHITECTURES
performance level of all architectures forms a discrete
Section II assumed that the architecture of an ANN is
surface in the space. The optimal architecture design is
predeﬁned and ﬁxed during the evolution of connection
equivalent to ﬁnding the highest point on this surface. There
weights. This section discusses the design of ANN architec-
are several characteristics of such a surface as indicated by
tures. The architecture of an ANN includes its topological
Miller et al. [150] which make EA’s a better candidate for
structure, i.e., connectivity, and the transfer function of each
searching the surface than those constructive and destruc-
node in the ANN. As indicated in the beginning of this
tive algorithms mentioned above. These characteristics are
paper, architecture design is crucial in the successful ap-
[150]:
plication of ANN’s because the architecture has signiﬁcant
1) the surface is inﬁnitely large since the number of
impact on a network’s information processing capabilities.
possible nodes and connections is unbounded;
Given a learning task, an ANN with only a few connections
2) the surface is nondifferentiable since changes in the
and linear nodes may not be able to perform the task
number of nodes or connections are discrete and can
at all due to its limited capability, while an ANN with
have a discontinuous effect on EANN’s performance;
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Fig. 6.
A typical cycle of the evolution of architectures.
3) the surface is complex and noisy since the mapping
term topological structure (topology) in these two sec-
from an architecture to its performance is indirect,
tions. Section III-C discusses the evolution of node transfer
strongly epistatic, and dependent on the evaluation
functions brieﬂy. Then we explain why the simultaneous
method used;
evolution of ANN connection weights and architectures
4) the surface is deceptive since similar architectures
is beneﬁcial and what search operators should be used in
may have quite different performance;
evolving architectures in Section III-D.
5) the surface is multimodal since different architectures
A. The Direct Encoding Scheme
may have similar performance.
Two different approaches have been taken in the direct
Similar to the evolution of connection weights, two major
encoding scheme. The ﬁrst separates the evolution of archi-
phases involved in the evolution of architectures are the
tectures from that of connection weights [24], [150], [153],
genotype representation scheme of architectures and the EA
[154], [165], [167], [169], [170]. The second approach
used to evolve ANN architectures. One of the key issues
evolves architectures and connection weights simultane-
in encoding ANN architectures is to decide how much
ously [149], [179], [180], [182], [185]–[200]. This section
information about an architecture should be encoded in the
will focus on the ﬁrst approach. The second approach will
chromosome. At one extreme, all the details, i.e., every
be discussed in Section III-D.
connection and node of an architecture, can be speciﬁed
In the ﬁrst approach, each connection of an architecture
by the chromosome. This kind of representation scheme
is directly speciﬁed by its binary representation [24], [150],
is called direct encoding. At the other extreme, only the
[153], [154], [165], [167], [169], [170], [202]. For example,
most important parameters of an architecture, such as the
an
matrix
can represent an ANN
number of hidden layers and hidden nodes in each layer,
architecture with
nodes, where
indicates presence or
are encoded. Other details about the architecture are left to
absence of the connection from node to node . We can use
the training process to decide. This kind of representation
to indicate a connection and
to indicate no
scheme is called indirect encoding. After a representation
connection. In fact,
can represent real-valued connection
scheme has been chosen, the evolution of architectures can
weights from node
to node
so that the architecture and
progress according to the cycle shown in Fig. 6. The cycle
connection weights can be evolved simultaneously [37],
stops when a satisfactory ANN is found.
[42], [45], [165], [166], [169]–[171].
Considerable research on evolving ANN architectures
Each matrix
has a direct one-to-one mapping to
has been carried out in recent years [33], [42], [45],
the corresponding ANN architecture. The binary string
[118], [127], [128], [130], [138], [149]–[225]. Most of
representing an architecture is the concatenation of rows
the research has concentrated on the evolution of ANN
(or columns) of the matrix. Constraints on architectures
topological structures. Relatively little has been done on
being explored can easily be incorporated into such a rep-
the evolution of node transfer functions, let alone the
resentation scheme by imposing constraints on the matrix,
simultaneous evolution of both topological structures and
e.g., a feedforward ANN will have nonzero entries only
node transfer functions. In this paper, we will analyze
in the upper-right triangle of the matrix. Figs. 7 and 8
the genotypical representation scheme of topological struc-
give two examples of the direct encoding scheme of ANN
tures in Sections III-A and III-B. For convenience, the
architectures. It is obvious that such an encoding scheme
term architecture will be used interchangeably with the
can handle both feedforward and recurrent ANN’s.
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(a)
(b)
(c)
Fig. 7.
An example of the direct encoding of a feedforward ANN. (a), (b), and (c) show the
architecture, its connectivity matrix, and its binary string representation, respectively. Because only
feedforward architectures are under consideration, the binary string representation only needs to
consider the upper-right triangle of the matrix.
(a)
(b)
(c)
Fig. 8.
An example of the direct encoding of a recurrent ANN. (a), (b), and (c) show the
architecture, its connectivity matrix, and its binary string representation, respectively.
Fig. 7(a) shows a feedforward ANN with two inputs and
is possible if we want to explore the whole connectivity
one output. Its connectivity matrix is given by Fig. 7(b),
space. The EA used to evolve recurrent ANN’s can be the
where entry
indicates the presence or absence of a
same as that used to evolve feedforward ones.
connection from node
to node . For example, the ﬁrst
The direct encoding scheme as described above is quite
row indicates connections from node 1 to all other nodes.
straightforward to implement. It is very suitable for the
The ﬁrst two columns are 0’s because there is no connection
precise and ﬁne-tuned search of a compact ANN architec-
from node 1 to itself and no connection to node 2. However,
ture, since a single connection can be added or removed
node 1 is connected to nodes 3 and 4. Hence columns 3
from the ANN easily. It may facilitate rapid generation and
and 4 have 1’s. Converting this connectivity matrix to a
optimization of tightly pruned interesting designs that no
chromosome is straightforward. We can concatenate all the
one has hit upon so far [150].
rows (or columns) and obtain
Another ﬂexibility provided by the evolution of architec-
tures stems from the ﬁtness deﬁnition. There is virtually
00 110 00 101 00 001 00 001 00 000.
no limitation such as being differentiable or continuous
Since the ANN is feedforward, we only need to represent
on how the ﬁtness function should be deﬁned at Step 3
the upper triangle of the connectivity matrix in order to
in Fig. 6. The training result pertaining to an architecture
reduce the chromosome length. The reduced chromosome is
such as the error and the training time is often used in
given by Fig. 7(c). An EA can then be employed to evolve
the ﬁtness function. The complexity measurement such as
a population of such chromosomes. In order to evaluate
the number of nodes and connections is also used in the
the ﬁtness of each chromosome, we need to convert a
ﬁtness function. As a matter of fact, many criteria based
chromosome back to an ANN, initialize it with random
on the information theory or statistics [226]–[228] can
weights, and train it. The training error will be used to
readily be introduced into the ﬁtness function without much
measure the ﬁtness. It is worth noting that the ANN in
difﬁculty. Improvement on ANN’s generalization ability
Fig. 7 has a shortcut connection from the input to output.
can be expected if these criteria are adopted. Schaffer
Such shortcuts pose no problems to the representation and
et al. [153] have presented an experiment which showed
evolution. An EA is capable of exploring all possible
that an ANN designed by the evolutionary approach had
connectivities.
better generalization ability than one trained by BP using
Fig. 8 shows a recurrent ANN. Its representation is
a human-designed architecture.
basically the same as that for feedforward ANN’s. The
One potential problem of the direct encoding scheme
only difference is that no reduction in chromosome length
is scalability. A large ANN would require a very large
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matrix and thus increase the computation time of the
chromosomes to specify independently the whole nervous
evolution. One way to cut down the size of matrices is
system according to the discoveries of neuroscience.
to use domain knowledge to reduce the search space. For
1) Parametric Representation: ANN architectures may
example, if complete connection is to be used between two
be speciﬁed by a set of parameters such as the number
neighboring layers in a feedforward ANN, its architecture
of hidden layers, the number of hidden nodes in each
can be encoded by just the number of hidden layers and
layer, the number of connections between two layers, etc.
nodes in each hidden layer. The length of chromosome can
These parameters can be encoded in various forms in a
be reduced greatly in this case [153], [154]. However, doing
chromosome. Harp et al. [152], [156] used a “blueprint” to
so requires sufﬁcient domain knowledge and expertise,
represent an architecture which consists of one or more
which are difﬁcult to obtain in practice. We also run the
segments representing an area (layer) and its efferent
risk of missing some very good solutions when we restrict
connectivity (projections). The ﬁrst and last area are
the search space manually.
constrained to be the input and output area, respectively.
The permutation problem as illustrated by Figs. 3 and
Each segment includes two parts of the information: 1)
4 in Section II-A still exists and causes unwanted side
that about the area itself, such as the number of nodes in
effects in the evolution of architectures. Because two func-
the area and the spatial organization of the area, and 2)
tionally equivalent ANN’s which order their hidden nodes
that about the efferent connectivity. It should be noted that
differently have two different genotypical representations,
only the connectivity pattern instead of each connection
the probability of producing a highly ﬁt offspring by
is speciﬁed here. The detailed node-to-node connection is
recombining them is often very low. Some researchers
speciﬁed by implicit developmental rules, e.g., the network
thus avoided crossover and adopted only mutations in
instantiation software used by Harp et al. [152], [156].
the evolution of architectures [45], [128], [149], [179],
Similar parametric representation methods with different
[185]–[197], [217], [223], although it has been shown that
sets of parameters have also been proposed by others [155],
crossover may be useful and important in increasing the
[159]. An interesting aspect of Harp et al.’s work is their
efﬁciency of evolution for some problems [48], [113],
combination of learning parameters with architectures in
[212], [229]. Hancock [113] suggested that the permutation
the genotypical representation. The learning parameters can
problem might “not be as severe as had been supposed”
evolve along with architecture parameters. The interaction
with the population size and the selection mechanism he
between the two can be explored through evolution.
used because “The increased number of ways of solving
Although the parametric representation method can re-
the problem outweigh the difﬁculties of bringing building
duce the length of binary chromosome specifying ANN’s
blocks together.” Thierens [101] proposed a genetic encod-
architectures, EA’s can only search a limited subset of
ing scheme of ANN’s which can avoid the permutation
problem, however, only very limited experimental results
the whole feasible architecture space. For example, if we
were presented. It is worth indicating that most studies on
encode only the number of hidden nodes in the hidden layer,
the permutation problem concentrate on the GA used, e.g.,
we basically assume strictly layered feedforward ANN’s
genetic operators, population sizes, selection mechanisms,
with a single hidden layer. We will have to assume two
etc. While it is necessary to investigate the algorithm, it is
neighboring layers are fully connected as well. In general,
equally important to study the genotypical representation
the parametric representation method will be most suitable
scheme, since the performance surface deﬁned in the be-
when we know what kind of architectures we are trying
ginning of Section III is determined by the representation.
to ﬁnd.
More research is needed to further understand the impact of
2) Developmental Rule Representation: A quite different
the permutation problem on the evolution of architectures.
indirect encoding method from the above is to encode de-
velopmental rules, which are used to construct architectures,
B. The Indirect Encoding Scheme
in chromosomes [151], [168], [184], [205], [230], [232].
In order to reduce the length of the genotypical represen-
The shift from the direct optimization of architectures to
tation of architectures, the indirect encoding scheme has
the optimization of developmental rules has brought some
been used by many researchers [151], [152], [155], [156],
beneﬁts, such as more compact genotypical representation,
[159], [160], [168], [184], [205], [208], [211], [230]–[232]
to the evolution of architectures. The destructive effect of
where only some characteristics of an architecture are
crossover will also be lessened since the developmental rule
encoded in the chromosome. The details about each con-
representation is capable of preserving promising building
nection in an ANN is either predeﬁned according to prior
blocks found so far [151]. But this method also has some
knowledge or speciﬁed by a set of deterministic devel-
problems [233].
opmental rules. The indirect encoding scheme can pro-
A developmental rule is usually described by a recursive
duce more compact genotypical representation of ANN
equation [230] or a generation rule similar to a production
architectures, but it may not be very good at ﬁnding
rule in a production system with a left-hand side (LHS) and
a compact ANN with good generalization ability. Some
a right-hand side (RHS) [151]. The connectivity pattern of
[151], [230], [231] have argued that the indirect encoding
the architecture in the form of a matrix is constructed from a
scheme is biologically more plausible than the direct one,
basis, i.e., a single-element matrix, by repetitively applying
because it is impossible for genetic information encoded in
suitable developmental rules to nonterminal elements in
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Fig. 9.
Examples of some developmental rules used to construct a connectivity matrix. S is the
initial element (or state).
the current matrix until the matrix contains only terminal2
elements which indicate the presence or absence of a
connection, that is, until a connectivity pattern is fully
speciﬁed.
Some examples of developmental rule are given in Fig. 9.
(a)
(b)
(c)
Each developmental rule consists of a LHS which is a
nonterminal element and a RHS which is a 2
2 matrix
with either terminal or nonterminal elements. A typical step
of constructing a connection matrix is to ﬁnd rules whose
LHS’s appear in the current matrix and replace all the
appearance with respective RHS’s. For example, given a
set of rules as described by Fig. 9, where
is the starting
symbol (state), one step application of these rules will
produce the matrix
(d)
(e)
by replacing
with
. If we apply these rules again,
we can generate the following matrix:
Fig. 10.
Development of an EANN architecture using the rules
given in Fig. 9: (a) the initial state; (b) step 1; (c) step 2; (d) step 3
when all the entries in the matrix are terminal elements, i.e., either
1 or 0; and (e) the architecture. The nodes in the architecture are
numbered from one to eight. Isolated nodes are not shown.
Note that nodes 2, 4, and 6 do not appear in the ANN
by replacing
with
with
with
,
because they are not connected to any other nodes.
and
with
. Another step of applying these rules
The example described by Figs. 9 and 10 illustrates how
would lead us to the matrix
an ANN architecture can be deﬁned given a set of rules.
The question now is how to get such a set of rules to
construct an ANN. One answer is to evolve them. We can
encode the whole rule set as an individual [151] (the so-
called Pitt approach) or encode each rule as an individual
[184] (the so-called Michigan approach). Each rule may
be represented by four allele positions corresponding to
four elements in the RHS of the rule. The LHS can
be represented implicitly by the rule’s position in the
chromosome. Each position in a chromosome can take
by replacing
with
with
with
, and
one of many different values, depending on how many
with
. Since the above matrix consists of ones and
nonterminal elements (symbols) we use in the rule set.
zeros only (i.e., terminals only), there will be no further
For example, the nonterminals may range from “ ” to
application of developmental rules. The above matrix will
“ ” and “ ” to “ .” The 16 rules with “ ” to “ ” on the
be an ANN’s connection matrix.
LHS and 2
2 matrices with only ones and zeros on the
Fig. 10 summarizes the previous three rule-rewriting
RHS are predeﬁned and do not participate in evolution in
steps and the ﬁnal ANN generated according to Fig. 10(d).
order to guarantee that different connectivity patterns can
2 In this paper, a terminal element is either 1 (existence of a connection)
be reached. Since there are 26 different rules, whose LHS
or 0 (nonexistence of a connection) and a nonterminal element is a symbol
is “ ,” “ ,”
,“ ,” respectively, a chromosome encoding
other than 1 and 0. These deﬁnitions are slightly different from those used
by others [151].
all of them would need 26
4
104 alleles, four per rule.
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The LHS of a rule is implicitly determined by its position
current architecture cannot reduce the training error below
in the chromosome. For example, the rule set in Fig. 9 can
certain threshold. Each hidden layer is constructed auto-
be represented by the following chromosome:
matically through an evolutionary process which employs
ABCDaaaaiiiaiaacaeae
the GA with ﬁtness sharing. Fitness sharing encourages the
where the ﬁrst four elements indicate the RHS of rule “ ”,
formation of different feature detectors (hidden nodes) in
the second four indicate the RHS of rule “ ,” etc.
the population. The number of hidden nodes in each hidden
Some good results from the developmental rule repre-
layer can vary.
sentation method have been reported [151] using various
One limitation of this approach [236] is that it could
size encoder/decoder problems. However, the method has
only deal with strictly layered feedforward ANN’s. Another
some limitations. It often needs to predeﬁne the number of
limitation is that there are usually several hidden nodes in
rewriting steps. It does not allow recursive rules. It is not
the same species which have very similar functionality,
very good at evolving detailed connectivity patterns among
i.e., which are basically the same feature detector in a
individual nodes. A compact genotypical representation
population. Such redundancy needs to be removed by an
does not imply a compact phenotypical representation, i.e.,
additional clean-up algorithm.
a compact ANN architecture. Recent work by Siddiqi and
Smith and Cribbs [181], [237] also used an individual
Lucas [233] shows that the direct encoding scheme can be
to represent a hidden node rather than the whole ANN.
at least as good as the developmental rule method. They
Their approach can only deal with strictly three-layered
have reimplemented Kitano’s system and discovered that
feedforward ANN’s.
the performance difference between the direct and indirect
encoding schemes was not caused by the encoding scheme
C. The Evolution of Node Transfer Functions
itself, but by how sparsely connected the initial ANN
The discussion on the evolution of architectures so far
architectures were in the initial population [151]. The direct
only deals with the topological structure of an architecture.
encoding scheme achieved the same performance as that
The transfer function of each node in the architecture has
achieved by the developmental rule representation when
been assumed to be ﬁxed and predeﬁned by human experts,
the initial conditions were the same [233].
yet the transfer function has been shown to be an important
The developmental rule representation method normally
part of an ANN architecture and have signiﬁcant impact on
separates the evolution of architectures from that of con-
ANN’s performance [238]–[240]. The transfer function is
nection weights. This creates some problems for evolution.
often assumed to be the same for all the nodes in an ANN,
Section III-D will discuss these in more detail.
at least for all the nodes in the same layer.
Mjolsness et al. [230] described a similar rule encoding
Stork et al. [241] were, to our best knowledge, the ﬁrst to
method where rules are represented by recursive equa-
apply EA’s to the evolution of both topological structures
tions which specify the growth of connection matrices.
and node transfer functions even though only simple ANN’s
Coefﬁcients of these recursive equations, represented by
with seven nodes were considered. The transfer function
decomposition matrices, are encoded in genotypes and op-
was speciﬁed in the structural genes in their genotypic
timized by simulated annealing instead of EA’s. Connection
representation. It was much more complex than the usual
weights are optimized along with connectivity by simulated
sigmoid function because they tried to model a biological
annealing since each entry of a connection matrix can have
neuron in the tailﬂip circuitry of crayﬁsh.
a real-valued weight. One advantage of using simulated
White and Ligomenides [171] adopted a simpler ap-
annealing instead of GA’s in the evolution is the avoidance
proach to the evolution of both topological structures and
of the destructive effect of crossover. Wilson [234] also
node transfer functions. For each individual (i.e., ANN)
used simulated annealing in ANN architecture design.
in the initial population, 80% nodes in the ANN used
3) Fractal Representation: Merrill and Port [231] pro-
the sigmoid transfer function and 20% nodes used the
posed another method for encoding architectures which
Gaussian transfer function. The evolution was used to
is based on the use of fractal subsets of the plane. They
decide the optimal mixture between these two transfer
argued that the fractal representation of architectures was
functions automatically. The sigmoid and Gaussian transfer
biologically more plausible than the developmental rule
function themselves were not evolvable. No parameters of
representation. They used three real-valued parameters, i.e.,
the two functions were evolved.
an edge code, an input coefﬁcient, and an output coefﬁcient
Liu and Yao [191] used EP to evolve ANN’s with
to specify each node in an architecture. In a sense, this
both sigmoidal and Gaussian nodes. Rather than ﬁxing
encoding method is closer to the direct encoding scheme
the total number of nodes and evolve mixture of different
rather to the indirect one. Fast simulated annealing [235]
nodes, their algorithm allowed growth and shrinking of the
was used in the evolution.
whole ANN by adding or deleting a node (either sigmoidal
4) Other Representations: A very different approach to
or Gaussian). The type of node added or deleted was
the evolution of architectures has been proposed by Ander-
determined at random. Good performance was reported for
sen and Tsoi [236]. Their approach is unique in that each
some benchmark problems [191]. Hwang et al. [225] went
individual in a population represents a hidden node rather
one step further. They evolved ANN topology, node transfer
than the whole architecture. An architecture is built layer
function, as well as connection weights for projection neural
by layer, i.e., hidden layers are added one by one if the
networks.
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Sebald and Chellapilla [242] used the evolution of node
[182], [185]–[200], [230], [232]. In this case, each individ-
transfer function as an example to show the importance of
ual in a population is a fully speciﬁed ANN with complete
evolving representations. Representation and search are the
weight information. Since there is a one-to-one mapping
two key issues in problem solving. Co-evolving solutions
between a genotype and its phenotype, ﬁtness evaluation
and their representations may be an effective way to tackle
is accurate.
some difﬁcult problems where little human expertise is
One issue in evolving ANN’s is the choice of search
available.
operators used in EA’s. Both crossover-based and mutation-
based EA’s have been used. However, use of crossover
appears to contradict the basic ideas behind ANN’s, because
D. Simultaneous Evolution of Architectures
crossover works best when there exist “building blocks” but
and Connection Weights
it is unclear what a building block might be in an ANN since
The evolutionary approaches discussed so far in design-
ANN’s emphasize distributed (knowledge) representation
ing ANN architectures evolve architectures only, without
[244]. The knowledge in an ANN is distributed among all
any connection weights. Connection weights have to be
the weights in the ANN. Recombining one part of an ANN
learned after a near-optimal architecture is found. This is es-
with another part of another ANN is likely to destroy both
pecially true if one uses the indirect encoding scheme, such
ANN’s.
as the developmental rule method. One major problem with
However, if ANN’s do not use a distributed represen-
the evolution of architectures without connection weights
tation but rather a localized one, such as radial basis
is noisy ﬁtness evaluation [194]. In other words, ﬁtness
function (RBF) networks or nearest-neighbor multilayer
evaluation as described in step 3 of Fig. 6 is very inaccurate
perceptrons, crossover might be a very useful operator.
and noisy because a phenotype’s (i.e., an ANN with a
There has been some work in this area where good results
full set of weights) ﬁtness was used to approximate its
were reported [119], [120], [245]–[253]. In general, ANN’s
genotype’s (i.e., an ANN without any weight information)
using distributed representation are more compact and have
ﬁtness. There are two major sources of noise [194].
better generalization capability for most practical problems.
1) The ﬁrst source is the random initialization of the
Yao and Liu [193], [194] developed an automatic system,
weights. Different random initial weights may pro-
EPNet, based on EP for simultaneous evolution of ANN
duce different training results. Hence, the same geno-
architectures and connection weights. EPNet does not use
type may have quite different ﬁtness due to different
any crossover operators for the reason given above. It relies
random initial weights used in training.
on a number of mutation operators to modify architectures
2) The second source is the training algorithm. Different
and weights. Behavioral (i.e., functional) evolution, rather
training algorithms may produce different training
genetic evolution, is emphasized in EPNet. A number of
results even from the same set of initial weights. This
techniques were adopted to maintain the behavioral link
is especially true for multimodal error functions. For
between a parent and its offspring [190]. Fig. 11 shows the
example, BP may reduce an ANN’s error to 0.05
main structure of EPNet.
through training, but an EA could reduce the error
EPNet uses rank-based selection [125] and ﬁve muta-
to 0.001 due to its global search capability.
tions: hybrid training; node deletion; connection deletion;
connection addition; and node addition [188], [194], [254].
Such noise may mislead evolution because the fact that
Hybrid training is the only mutation in EPNet which mod-
the ﬁtness of a phenotype generated from genotype
is
iﬁes ANN’s weights. It is based on a modiﬁed BP (MBP)
higher than that generated from genotype
does not mean
algorithm with an adaptive learning rate and simulated
that
truly has higher quality than
. In order to reduce
annealing. The other four mutations are used to grow and
such noise, an architecture usually has to be trained many
prune hidden nodes and connections.
times from different random initial weights. The average
The number of epochs used by MBP to train each ANN’s
result is then used to estimate the genotype’s mean ﬁtness.
in a population is deﬁned by two user-speciﬁed parameters.
This method increases the computation time for ﬁtness
There is no guarantee that an ANN will converge to even
evaluation dramatically. It is one of the major reasons why
a local optimum after those epochs. Hence this training
only small ANN’s were evolved in previous studies.
process is called partial training. It is used to bridge the
In essence, the noise identiﬁed in this paper is caused by
behavioral gap between a parent and its offspring.
the one-to-many mapping from genotypes to phenotypes.
The ﬁve mutations are attempted sequentially. If one
Angeline et al. [149] and Fogel [12], [243] have provided a
mutation leads to a better offspring, it is regarded as
more general discussion on the mapping between genotypes
successful. No further mutation will be applied. Otherwise
and phenotypes. It is clear that the evolution of architectures
the next mutation is attempted. The motivation behind
without any weight information has difﬁculties in evaluat-
ordering mutations is to encourage the evolution of compact
ing ﬁtness accurately. As a result, the evolution would be
ANN’s without sacriﬁcing generalization. A validation set
very inefﬁcient.
is used in EPNet to measure the ﬁtness of an individual.
One way to alleviate this problem is to evolve ANN
EPNet has been tested extensively on a number of bench-
architectures and connection weights simultaneously [37],
mark problems and achieved excellent results, including
[42], [45], [149], [165], [166], [169]–[172], [179], [180],
parity problems of size from four to eight, the two-spiral
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PROCEEDINGS OF THE IEEE, VOL. 87, NO. 9, SEPTEMBER 1999

Fig. 11.
The main structure of EPNet.
problem, the breast cancer problem, the diabetes problem,
all ANN’s. In fact, what is needed from an ANN is its
the heart disease problem, the thyroid problem, the Aus-
ability to adjust its learning rule adaptively according to its
tralian credit card problem, the Mackey–Glass time series
architecture and the task to be performed. In other words,
prediction problem, etc. Very compact ANN’s with good
an ANN should learn its learning rule dynamically rather
generalization ability have been evolved [185]–[195]. There
than have it designed and ﬁxed manually. Since evolution
are some other different EP-based systems for designing
is one of the most fundamental forms of adaptation, it is
ANN’s [128], [129], [217], [223], but none has been tested
not surprising that the evolution of learning rules has been
on as many different benchmark problems.
introduced into ANN’s in order to learn their learning rules.
The relationship between evolution and learning is
IV. THE EVOLUTION OF LEARNING RULES
extremely complex. Various models have been proposed
An ANN training algorithm may have different perfor-
[257]–[271], but most of them deal with the issue of
mance when applied to different architectures. The design
how learning can guide evolution [257]–[260] and the
of training algorithms, more fundamentally the learning
relationship between the evolution of architectures and
rules used to adjust connection weights, depends on the
that of connection weights [261]–[263]. Research into
type of architectures under investigation. Different variants
the evolution of learning rules is still in its early stages
of the Hebbian learning rule have been proposed to deal
[264]–[267], [269], [270]. This research is important not
with different architectures. However, designing an optimal
only in providing an automatic way of optimizing learning
learning rule becomes very difﬁcult when there is little
rules and in modeling the relationship between learning
prior knowledge about the ANN’s architecture, which is
and evolution, but also in modeling the creative process
often the case in practice. It is desirable to develop an
since newly evolved learning rules can deal with a complex
automatic and systematic way to adapt the learning rule to
and dynamic environment. This research will help us to
an architecture and the task to be performed. Designing a
understand better how creativity can emerge in artiﬁcial
learning rule manually often implies that some assumptions,
systems, like ANN’s, and how to model the creative process
which are not necessarily true in practice, have to be made.
in biological systems. A typical cycle of the evolution of
For example, the widely accepted Hebbian learning rule
learning rules can be described by Fig. 12. The iteration
has recently been shown to be outperformed by a new rule
stops when the population converges or a predeﬁned
proposed by Artola et al. [255] in many cases [256]. The
maximum number of iterations has been reached.
new rule can learn more patterns than the optimal Hebbian
Similar to the reason explained in Section III-D, the
rule and can learn exceptions as well as regularities. It is,
ﬁtness evaluation of each individual, i.e., the encoded
however, still difﬁcult to say that this rule is optimal for
learning rule, is very noisy because we use phenotype’s
YAO: EVOLVING ARTIFICIAL NEURAL NETWORKS
1435

Fig. 12.
A typical cycle of the evolution of learning rules.
ﬁtness (i.e., an ANN’s training result) to approximate
Unlike the evolution of connection weights and architec-
genotype’s ﬁtness (i.e., a learning rule’s ﬁtness). Such
tures which only deal with static objects in an ANN, i.e.,
approximation may be inaccurate. Some techniques have
weights and architectures, the evolution of learning rules
been used to alleviate this problem, e.g., using a weighted
has to work on the dynamic behavior of an ANN. The key
average of the training results from ANN’s with different
issue here is how to encode the dynamic behavior of a
initial connection weights in the ﬁtness function. If the
learning rule into static chromosomes. Trying to develop a
ANN architecture is predeﬁned and ﬁxed, the evolved
universal representation scheme which can specify any kind
learning rule would be optimized toward this architecture. If
of dynamic behaviors is clearly impractical, let alone the
a near-optimal learning rule for different ANN architectures
prohibitive long computation time required to search such
is to be evolved, the ﬁtness evaluation should be based on
a learning rule space. Constraints have to be set on the
the average training result from different ANN architectures
type of dynamic behaviors, i.e., the basic form of learning
in order to avoid overﬁtting a particular architecture.
rules being evolved in order to reduce the representation
complexity and the search space.
A. The Evolution of Algorithmic Parameters
Two basic assumptions which have often been made on
learning rules are: 1) weight updating depends only on
The adaptive adjustment of BP parameters (such as
local information such as the activation of the input node,
the learning rate and momentum) through evolution could
the activation of the output node, the current connection
be considered as the ﬁrst attempt of the evolution of
weight, etc., and 2) the learning rule is the same for all
learning rules [32], [152], [272]. Harp et al. [152] encoded
connections in an ANN. A learning rule is assumed to be a
BP’s parameters in chromosomes together with ANN’s
linear function of these local variables and their products.
architecture. This evolutionary approach is different from
That is, a learning rule can be described by the function [5]
the nonevolutionary one such as offered by Jacobs [273]
because the simultaneous evolution of both algorithmic
parameters and architectures facilitates exploration of in-
(4)
teractions between the learning algorithm and architectures
such that a near-optimal combination of BP with an archi-
tecture can be found.
where
is time,
is the weight change,
Other researchers [32], [139], [213], [272] also used an
are local variables, and the ’s are real-valued coefﬁcients
evolutionary process to ﬁnd parameters for BP but ANN’s
which will be determined by evolution. In other words,
architecture was predeﬁned. The parameters evolved in this
the evolution of learning rules in this case is equivalent to
case tend to be optimized toward the architecture rather
the evolution of real-valued vectors of
’s. Different
’s
than being generally applicable to learning. There are a
determine different learning rules. Due to a large number
number of BP algorithms with an adaptive learning rate
of possible terms in (4), which would make evolution
and momentum where a nonevolutionary approach is used.
very slow and impractical, only a few terms have been
Further comparison between the two approaches would be
used in practice according to some biological or heuristic
quite useful.
knowledge.
There are three major issues involved in the evolution
B. The Evolution of Learning Rules
of learning rules: 1) determination of a subset of terms
described in (4); 2) representation of their real-valued
The evolution of algorithmic parameters is certainly
coefﬁcients as chromosomes; and 3) the EA used to evolve
interesting but it hardly touches the fundamental part of
these chromosomes. Chalmers [264] deﬁned a learning rule
a training algorithm, i.e., its learning rule or weight up-
as a linear combination of four local variables and their
dating rule. Adapting a learning rule through evolution is
six pairwise products. No third- or fourth-order3 terms
expected to enhance ANN’s adaptivity greatly in a dynamic
environment.
3 The order is deﬁned as the number of variables in a product.
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PROCEEDINGS OF THE IEEE, VOL. 87, NO. 9, SEPTEMBER 1999

were used. Ten coefﬁcients and a scale parameter were
be very high, i.e., many different architectures and learning
encoded in a binary string via exponential encoding. The
tasks have to be used in the ﬁtness evaluation.
architecture used in the ﬁtness evaluation was ﬁxed because
only single-layer ANN’s were considered and the number
V.
OTHER COMBINATIONS BETWEEN ANN’S AND EA’S
of inputs and outputs were ﬁxed by the learning task at
hand. After 1000 generations, starting from a population of
A. The Evolution of Input Features
randomly generated learning rules, the evolution discovered
the well-known delta rule [7], [274] and some of its vari-
For many practical problems, the possible inputs to an
ants. These experiments, although simple and preliminary,
ANN can be quite large. There may be some redundancy
demonstrated the potential of evolution in discovering novel
among different inputs. A large number of inputs to an
learning rules from a set of randomly generated rules.
ANN increase its size and thus require more training data
However, constraints set on learning rules could prevent
and longer training times in order to achieve a reasonable
some from being evolved such as those which include third-
generalization ability. Preprocessing is often needed to
or fourth-order terms.
reduce the number of inputs to an ANN. Various dimension
Similar experiments on the evolution of learning rules
reduction techniques, including the principal component
were also carried out by others [265], [266], [267], [269],
analysis, have been used for this purpose.
[270]. Fontanari and Meir [267] used Chalmers’ approach
The problem of ﬁnding a near-optimal set of input
to evolve learning rules for binary perceptrons. They also
features to an ANN can be formulated as a search problem.
considered four local variables but only seven terms were
Given a large set of potential inputs, we want to ﬁnd a near-
adopted in their learning rules, which included one ﬁrst-
optimal subset which has the fewest number of features
order, three second-order, and three third-order terms in (4).
but the performance of the ANN using this subset is no
Baxter [269] took one step further than just the evolution
worse than that of the ANN using the whole input set. EA’s
of learning rules. He tried to evolve complete ANN’s
have been used to perform such a search effectively [267],
(including connection weights, architectures, and learning
[277]–[287]. Very good results, i.e., better performance
rules) in a single level of evolution. It is clear that the search
with fewer inputs, have been reported from these studies.
space of possible ANN’s would be enormous if constraints
In the evolution of input features, each individual in the
population represents a subset of all possible inputs. This
were not set on the connection weights, architectures, and
can be implemented using a binary chromosome whose
learning rules. In his experiments, only ANN’s with binary
length is the same as the total number of input features.
threshold nodes were considered, so the weights could only
Each bit in the chromosome corresponds to a feature. “1”
be
1 or
1. The number of nodes in ANN’s was ﬁxed.
indicates presence of a feature, while “0” indicates absence
The learning rule only considered two Boolean variables.
of the feature. The evaluation of an individual is carried out
Although Baxter’s experiments were rather simple, they
by training an ANN with these inputs and using the result
conﬁrmed that complex behaviors could be learned and
to calculate its ﬁtness value. The ANN architecture is often
the ANN’s learning ability could be improved through
ﬁxed. Such evaluation is very noisy, however, because of
evolution [269].
the reason explained in Section III-D.
Bengio et al.’s approach [265], [266] is slightly different
Not only does the evolution of input features provide a
from Chalmers’ in the sense that gradient descent algo-
way to discover important features from all possible inputs
rithms and simulated annealing, rather than EA’s, were
automatically, it can also be used to discover new training
used to ﬁnd near-optimal
’s. In their experiments, four
examples. Zhang and Veenker [288] described an active
local variables and one zeroth-order, three ﬁrst-order, and
learning paradigm where a training algorithm based on
three second-order terms in (4) were used.
EA’s can self-select training examples. Cho and Cha [289]
Research related to the evolution of learning rules in-
proposed another algorithm for evolving training sets by
cludes Parisi et al.’s work on “econets,” although they
adding virtual samples.
did not evolve learning rules explicitly [260], [275]. They
emphasized the crucial role of the environment in which
the evolution occured while only using some simple neural
B. ANN as Fitness Estimator
networks. The issue of environmental diversity is closely
EA’s have been used with success to optimize various
related to the noisy ﬁtness evaluation as pointed out in
control parameters [290]–[292]. However, it is very time
Section III-D and at the beginning of Section IV. There
consuming and costly to obtain ﬁtness values for some
are two possible sources of noise. The ﬁrst is the decoding
control problems as it is impractical to run a real system
process (morphogenesis) of chromosomes. The second is
for each combination of control parameters. In order to
introduced when a decoded learning rule is evaluated by
get around this problem and make evolution more efﬁcient,
using it to train ANN’s. The environmental diversity is
ﬁtness values are often approximated rather than computed
essential in obtaining a good approximation to the ﬁtness
exactly. ANN’s are often used to model and approximate
of the decoded learning rule and thus in reducing the noise
a real control system due to their good generalization
from the second source. If a general learning rule which
abilities. The input to such ANN’s will be a set of control
is applicable to a wide range of ANN architectures and
parameters. The output will be the control system output
learning tasks is needed, the environmental diversity has to
from which an evaluation of the whole system can easily be
YAO: EVOLVING ARTIFICIAL NEURAL NETWORKS
1437

obtained. When an EA is used to search for a near-optimal
experiments which show that EA’s can be used to evolve
set of control parameters, the ANN will be used in ﬁtness
ANN ensembles [192], [193], [302]–[305].
evaluation rather than the real control system [293]–[297].
This combination of ANN’s and EA’s has a couple of
D. Others
advantages in evolving control systems. First, the time-
consuming ﬁtness evaluation based on real control systems
There are some other novel combinations between
is replaced by fast ﬁtness evaluation based on ANN’s. Sec-
ANN’s and EA’s. For example, Zitar and Hassoun [306]
ond, this combination provides safer evolution of control
used EA’s to extract rules in a reinforcement learning
systems. EA’s are stochastic algorithms. It is possible that
system and then used them to train ANN’s. Sziranyi
some poor control parameters may be generated in the
[99] and Pal and Bhandari [98] used EA’s to tune circuit
evolutionary process. These parameters could damage a
parameters and templates in cellular ANN’s. Olmez [97]
real control system. If we use ANN’s to estimate ﬁtness,
used EA’s to optimize a modiﬁed restricted Coulomb
we do not need to use the real system and thus can avoid
energy (RCE) ANN. Imada and Araki [307] used EA’s
damages to the real system. However, how successful this
to evolve connection weights for Hopﬁeld ANN’s. Many
combination approach will be depends largely on how well
others used EA’s and ANN’s for combinatorial or global
ANN’s learn and generalize.
(numerical) optimization in order to combine EA’s global
search capability with ANN’s fast convergence to local
optima [308]–[318].
C. Evolving ANN Ensembles
Learning is often formulated as an optimization problem
VI. CONCLUDING REMARKS
in the machine learning ﬁeld. However, learning is different
Although evolution has been introduced into ANN’s at
from optimization in practice because we want the learned
various levels, they can roughly be divided into three: the
system to have best generalization, which is different from
evolution of connection weights, architectures, and learning
minimizing an error function on a training data set. The
rules. This section ﬁrst describes a general framework for
ANN with the minimum error on a training data set may
ANN’s and then draws some conclusions.
not have best generalization unless there is an equivalence
between generalization and the error on the training data.
Unfortunately, measuring generalization quantitatively and
A. A General Framework for EANN’s
accurately is almost impossible in practice [298] although
A general framework for EANN’s can be described
there are many theories and criteria on generalization,
by Fig. 13 [3]–[5]. The evolution of connection weights
such as the minimum description length (MDL) [299],
proceeds at the lowest level on the fastest time scale in an
Akaike information criteria (AIC) [300], and minimum
environment determined by an architecture, a learning rule,
message length (MML) [301]. In practice, these criteria
and learning tasks. There are, however, two alternatives to
are often used to deﬁne better error functions in the hope
decide the level of the evolution of architectures and that of
that minimizing the functions will maximize generalization.
learning rules: either the evolution of architectures is at the
While these functions often lead to better generalization of
highest level and that of learning rules at the lower level
learned systems, there is no guarantee.
or vice versa. The lower the level of evolution, the faster
EA’s are often used to maximize a ﬁtness function or
the time scale it is on.
minimize an error function, and thus they face the same
From the engineering perspective, the decision on the
problem as described above: maximizing a ﬁtness function
level of evolution depends on what kind of prior knowledge
is different from maximizing generalization. The EA is
is available. If there is more prior knowledge about an
actually used as an optimization, not learning, algorithm.
ANN’s architecture than that about their learning rules, or
While little can be done for traditional nonpopulation-based
if a particular class of architectures is pursued, it is better
learning, there are opportunities for improving population-
to put the evolution of architectures at the highest level
based learning, e.g., evolutionary learning.
because such knowledge can be encoded in an architecture’s
Since the maximum ﬁtness may not be equivalent to best
genotypic representation to reduce the (architecture) search
generalization in evolutionary learning, the best individual
space and the lower-level evolution of learning rules can
with the maximum ﬁtness in a population may not be
be biased toward this type of architectures. On the other
the one we want. Other individuals in the population may
hand, the evolution of learning rules should be at the
contain some useful information that will help to improve
highest level if there is more prior knowledge about them
generalization of learned systems. It is thus beneﬁcial
available or if there is a special interest in certain type of
to make use of the whole population rather than any
learning rules. Unfortunately, there is usually little prior
single individual. A population always contains at least
knowledge available about either architectures or learning
as much information as any single individual. Hence,
rules in practice except for some very vague statements
combining different individuals in the population to form
[319]. In this case, it is more appropriate to put the evolution
an integrated system is expected to produce better results.
of architectures at the highest level since the optimality
Such a population of ANN’s is called an ANN ensemble
of a learning rule makes more sense when evaluated in
in this section. There have been some very successful
an environment including the architecture to which the
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PROCEEDINGS OF THE IEEE, VOL. 87, NO. 9, SEPTEMBER 1999

characteristics of the design problem given in the begin-
ning of Section III. The direct encoding scheme of ANN
architectures is very good at ﬁne tuning and generating
a compact architecture. The indirect encoding scheme is
suitable for ﬁnding a particular type of ANN architecture
quickly. Separating the evolution of architectures and that
of connection weights can make ﬁtness evaluation inaccu-
rate and mislead evolution. Simultaneous evolution of ANN
architectures and connection weights generally produces
better results. It is argued that crossover produces more
harm than beneﬁt in evolving ANN’s using distributed
representation (e.g., multilayer perceptrons) because it de-
stroys knowledge learned and distributed among different
connections easily. Crossover would be more suitable for
localized ANN’s, such as RBF networks.
Evolution can also be used to allow an ANN to adapt its
learning rule to its environment. In a sense, the evolution
provides ANN’s with the ability of learning to learn. It also
helps to model the relationship between learning and evo-
lution. Preliminary experiments have shown that efﬁcient
learning rules can be evolved from randomly generated
rules. Current research on the evolution of learning rules
normally assumes that learning rules can be speciﬁed by
(4). While constraints on learning rules are necessary to
Fig. 13.
A general framework for EANN’s.
reduce the search space in the evolution, they might prevent
some interesting learning rules from being discovered.
Global search procedures such as EA’s are usually com-
learning rule is applied. Fig. 13 summarizes different levels
putationally expensive. It would be better not to employ
of evolution in ANN’s.
EA’s at all three levels of evolution. It is, however, beneﬁ-
Fig. 13 can also be viewed as a general framework of
cial to introduce global search at some levels of evolution,
adaptive systems if we do not restrict ourselves to EA’s
especially when there is little prior knowledge available
and three levels. Simulated annealing, gradient descent,
at that level and the performance of the ANN is required
and even exhaustive search could be considered as special
to be high, because the trial-and-error and other heuristic
cases of EA’s. For example, the traditional BP network
methods are very ineffective in such circumstances.
can be considered as a special case of our general frame-
With the increasing power of parallel computers, the
work with one-shot (only-one-candidate) search used in
evolution of large ANN’s becomes feasible. Not only can
the evolution of architectures and learning rules and BP
such evolution discover possible new ANN architectures
used in the evolution of connection weights. In fact, the
and learning rules, but it also offers a way to model the
general framework provides a basis for comparing various
creative process as a result of ANN’s adaptation to a
speciﬁc EANN models according to the search procedures
dynamic environment.
they used at three different levels since it deﬁnes a three-
dimensional space where zero represents one-shot search
and
represents exhaustive search along each axis. Each
ACKNOWLEDGMENT
EANN model corresponds to a point in this space.
The author is grateful to D. B. Fogel and an anonymous
referee for their constructive comment on earlier versions
of this paper.
B. Conclusions
Evolution can be introduced into ANN’s at many differ-
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Programming Society.
YAO: EVOLVING ARTIFICIAL NEURAL NETWORKS
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