Download PPT

Utilizing Population Distribution Information in
Differential Evolution
Yong Wang
School of Information Science and Engineering,
Central South University
[email protected]
http://ist.csu.edu.cn/YongWang.htm
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
2
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
3
Differential Evolution (1/4)
• Differential evolution (DE), proposed by Storn and Price in
1995, is a very popular evolutionary algorithm paradigm.
• DE includes three main operators, i.e., mutation,
crossover, and selection.
R. Storn and K. Price. Differential evolution—a simple and efficient adaptive
scheme for global optimization over continuous spaces, Berkeley, CA, Tech.
Rep. TR-95-012, 1995.
K. Price, R. Storn, and J. Lampinen. Differential Evolution—A Practical Approach
to Global Optimization. Berlin, Germany: Springer-Verlag, 2005.
4
Differential Evolution (2/4)
• Framework
three main
operators
mutation + crossover = trial vector generation strategy
5
Differential Evolution (3/4)
• A classic DE version: DE/rand/1/bin
base vector
differential vector
mutation
scaling factor
crossover control parameter
crossover
selection
6
Differential Evolution (4/4)
• Schematic diagram to illustrate DE/rand/1/bin
the triangle denotes the trial vector
7
ui ,G
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
8
CMA-ES (1/4)
• Principle
N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution
strategies. Evolutionary Computation, vol. 9, no. 2, pp. 159-195, 2001.
9
CMA-ES (2/4)
• Covariance matrices have an appealing geometrical
interpretation: they can be uniquely identified with the
(hyper-)ellipsoid
10
CMA-ES (3/4)
• The Eigen decomposition is denoted by C  BD 2 B T
D I
a positive
number
BI
the identity
matrix
B
C   2I
the ellipsoid is isotropic
the identity
matrix
C  D2
the ellipsoid is axis parallel oriented
a diagonal
matrix
the ellipsoid can be adapted to suit the
contour lines of the objective function
C
the new principal axes of the ellipsoid
correspond to the columns of B
11
CMA-ES (4/4)
• Rank-μ-Update
the λ offspring
produced from
μ parents
the center of
the μ parents
12
the center of the μ
best individuals from
the λ offspring
The shortcoming of CMA-ES
A
B (the basin of attraction
including the optimal solution)
C
D (a large potential basin of
attraction including some
high-quality individuals )
search space
Remark: CMA-ES is very likely to converge into a large potential
basin of attraction (such as D) since it contains much more highquality individuals!
13
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
14
Motivation
• The commonly used crossover operators of DE are
dependent mainly on the coordinate system
x2
vi ,G
xi ,G
x1
Y. Wang, H.-X. Li, T. Huang, and L Li. Differential evolution based on
covariance matrix learning and bimodal distribution parameter setting.
Applied Soft Computing, vol. 18, pp. 232-347, 2014. (CoBiDE, ESI Highly
Cited Paper top 1%)
15
CoBiDE (1/4)
• Covariance matrix learning
16
CoBiDE (2/4)
• An explanation
x2
vi ,G
covariance matrix learning
xi ,G
x2
vi ,G
xi ,G
x1
x1
x2
x1
17
CoBiDE (3/4)
The first issue: Which individuals should be chosen for computing
the covariance matrix
The second issue: How to determine the probability that the
crossover is implemented in the Eigen coordinate system
18
CoBiDE (4/4)
the center of the μ
best individuals from
the λ offspring
the λ offspring
produced from
μ parents
The third issue: the variance will decease significantly
N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution
strategies. Evolutionary Computation, vol. 9, no. 2, pp. 159-195, 2001.
19
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
20
Motivation
• Single population fails to contain enough information to
reliably estimate the covariance matrix.
• Moreover, some extra parameters have been introduced.
Y. Wang, H.-X. Li, T. Huang, and L Li. Differential evolution based on covariance
matrix learning and bimodal distribution parameter setting. Applied Soft Computing,
vol. 18, pp. 232-347, 2014.
S. Guo and C. Yang. Enhancing differential evolution utilizing Eigenvector-based
crossover operator. IEEE Transactions on Evolutionary Computation, vol. 19, no. 1,
pp. 31-49, 2015.
21
CPI-DE (1/5)
• We make use of the cumulative distribution information of
the population to establish an appropriate coordinate
system for DE’s crossover
• The algorithmic framework
Y. Wang, Z.-Z. Liu, J. Li, H.-X. Li, and G. G. Yen. Utilizing cumulative
population distribution information in differential evolution. Applied Soft
Computing, vol. 48, pp. 329-346, 2016. (CPI-DE)
22
CPI-DE (2/5)
• Rank-NP-update of the covariance matrix in DE
C
NP
( g 1)
NP
g 1)
(g)
g 1)
(g) T
  wi ( xi(:2*
)( xi(:2*
)
NP  m
NP  m
( g 1)
 (1  cNP )C
C
23
i 1
(g)
 cNP (
( g ) 2 1
cumulative population
distribution information
( g 1)
) CNP
CPI-DE (3/5)
• The relationship between rank-NP-update in CPI-DE and
rank-μ-update in CMA-ES
rank-NP-update in CPI-DE
rank-μ-Update in CMA-ES
rank-NP-update in CPI-DE is a natural extension of
rank-μ-update in CMA-ES
24
CPI-DE (4/5)
• Crossover in the Eigen coordination system
25
CPI-DE (5/5)
• The advantages of CPI-DE
– CPI-DE provides a simple yet efficient synergy of two kinds
of crossover: the crossover in the Eigen coordinate system
and the crossover in the original coordinate system.
– The crossover in the Eigen coordinate system aims at
identifying the properties of the fitness landscape and
improving the efficiency and effectiveness of DE by
producing the offspring toward the promising directions.
– The purpose of the crossover in the original coordinate
system is to maintain the superiority of the original DE.
– Moreover, no extra parameters are required in CPI-DE.
26
Outline
 Differential Evolution (DE)
 Covariance Matrix Adaptation Evolution Strategy
(CMA-ES)
 DE with Single Population Distribution Information
 DE with Cumulative Population Distribution
Information
 Conclusion
27
Conclusion
• DE are population-based optimization algorithms;
however, population distribution information has not yet
been widely utilized in the DE community, which makes
DE inefficient.
• Population distribution information is an effective tool to
enhance the performance of DE.
• Cumulative population distribution information can provide
a more reasonable estimation to the covariance matrix
than single population distribution information.
28