IEEE Computational Intelligence Magazine - November 2023 - 63

TABLE I Statistical comparison of test mean square error for different dimensionality reduction techniques (x/y/z mean that a method
in a row is significantly better than, similar to, or worse than the method in the column on x/y/z datasets).
PCA
KPCA(COSINE)
KPCA(COSINE)+Reference Point
44/62/0
61/45/0
Based on the transformed feature space,
a linear model is trained to make predictions.
To ensure that the constructed features
generalize well to unseen data, the
model makes predictions on the training
data by using cross-validation. The predictions
made by themodels are referred to as
semantics in GP literature, and the target
labels are known as target semantics. The
semantics of all GP individuals together
form a semantic space, where the objective
ofGP is to discover an individual that can
outputtargetsemantics based ongiven
inputs.
C.Archive Maintenance
The archive maintenance step is crucial
for selecting individuals to form the final
ensemble model. In this paper, MAPElites
is used to select a set ofhigh-quality
and complementary solutions, which
involves four stages:
❏ Reference Point Synthesis: For a
learning dataset ðX; YÞ of a supervised
learning task, where the target
label Y is known, the semantics of
ideal individuals F can be synthesized
by ð1 aÞ FðXÞþ aY.
a is a hyperparameter set to 0.1 and
1.1 in this paper, which corresponds
to the ideal points of high-quality
individuals and symmetrical highquality
individuals, respectively.
❏ Dimensionality Reduction: The ideal
semantic space is high-dimensional,
making it challenging to define a niche
in such a space. To address this issue,
MAP-Elites employs a dimensionality
reduction method, transforming the
high-dimensional space into a lowerdimensional
space that can be discretized
more easily. Many methods can
be used for dimensionality reduction in
MAP-Elites. In thiswork, kernel principal
component analysis (KPCA) with a
cosine kernel is adopted because it demonstrated
superior performance in the
experiments, as discussed inSection IV.
❏ Space Discretization: Based on the
reduced space, MAP-Elites discretizes
KPCA(RBF) KPCA(POLY)
61/44/1
71/34/1
73/33/0
74/32/0
t-SNE
58/48/0
67/39/0
the semantic space into a k k grid,
where k is a hyperparameter that
determines the granularity of MAPElites.
Each grid cell represents a
niche, containing individuals with
similar behaviors.
❏ Elites Selection: Finally, the best individual
in each grid cell is preserved
based on the discrete behavior space.
D. Solution Selection and Generation
Once obtained a set of diverse and
high-quality individuals in the external
archive A, promising individuals are
selected from the external archive A
using random selection. Based on the
selected parent individuals, offspring are
generated using random subtree crossover
and mutation. For multi-tree GP,
crossover and mutation are applied to
randomly select GP trees.
IV. Experimental Results
The experimental results in Table I show
the impact of dimensionality reduction
methods on the test mean squared errors
of ensemble models across 106 datasets.
The results indicate that incorporating
KPCA (COSINE) as the dimensionality
reduction technique within MAP-Elites
leads to a significant improvement in the
performance ofensemble models. Specifically,
KPCA (COSINE) outperforms
PCA in MAP-Elites on 44 datasets and
does not perform worse on any dataset.
Furthermore, when synthesizing reference
points, KPCA (COSINE) exhibits even
better performance, surpassing PCA on 61
datasets and not being outperformed on
any dataset.
V. Conclusion
This paper provides an interactive
approach to understanding how to use
MAP-Elites for evolutionary ensemble
learning, as well as eight dimensionality
reduction methods for automatically
inducing a behavior space based on
semantics ofGP. The experimental results
from interactive examples and large-scale
experiments show that the dimensionality
Beta-VAE
70/36/0
Isomap
63/43/0
73/33/0 70/36/0
SpectralEmbedding
64/42/0
71/35/0
reduction method significantly impacts
thepredictiveperformanceof theensemblemodel
within theMAP-Elitesframework,
with cosine-kernel-based PCA
outperforming other methods.
While this paper focuses on MAPElites,
the idea of using cosine similarity
for defining a behavior space could
potentially be extended to other QD
optimization algorithms for evolutionary
ensemble learning. In the future, it would
be interesting to investigate the impact of
different distance metrics in other QD
optimization algorithms.
Acknowledgment
This work was supported in part by theMarsden
Fund of New Zealand Government
under ContractsVUW1913, VUW1914, and
VUW2016, in part by the Science for Technological
Innovation Challenge (SfTI) fund
under Grant E3603/2903, in part by MBIE
Data Science SSIF Fund under Contract
RTVU1914, in part by Huayin Medical
under Grant E3791/4165, and in part by
MBIE Endeavor Research Programme under
ContractsC11X2001 andUOCX2104.
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NOVEMBER 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 63

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