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Evolutionary Algorithms EVO Differential Evolution and Evolutionary Strategies L9

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Title: Evolutionary Algorithms EVO Differential Evolution and Evolutionary Strategies L9


1
Evolutionary Algorithms (EVO) Differential
Evolution and Evolutionary Strategies (L9)
  • John A Clark
  • Professor of Critical Systems
  • Non-Standard Computation Group Dept. of Computer
    Science
  • University of York, UK

2
Differential Evolution and Evolutionary Strategies
  • Two techniques that have found application in
    numerical optimisation and mixed-type
    optimisations.
  • Some interesting deviations from previous
    techniques.

3
Differential Evolution
4
Representation
5
Maintain Population - NP
We will maintain a population of size NP
6
Initialisation
Xi,2
X i,D-1
Xi,D
Different Uniform U(0,1) variable value for each
i and j.
7
Recombination
  • For each member i of the population G pick
    randomly three other different members r1, r2,
    and r3

8
Recombination and Mutation
  • We progressively define the elements of the
    temporary candidate Z

9
Recombination in Pictures
ki2
Zi
10
Selection for Next Generation
11
Evolution
12
Practical Advice
  • Set NP to 10 times the number of parameters
  • Set weighting factor F0.8
  • Set crossover constant CR0.9.
  • Initialize your parameter vectors by exploiting
    their full numerical range, i.e. if a parameter
    is allowed to exhibit values in the range -100,
    100 it's a good idea to pick the initial values
    from this range instead of unnecessarily
    restricting diversity.
  • If you experience mis-convergence you usually
    have to increase the value for NP, but often you
    only have to adjust F.
  • If you increase NP and simultaneously lower F a
    little, convergence is more likely to occur but
    generally takes longer, i.e. DE is getting more
    robust (there is always a convergence
    speed/robustness tradeoff).

http//www.icsi.berkeley.edu/storn/code.html
13
Practical Advice
  • DE is much more sensitive to the choice of F than
    it is to the choice of CR.
  • CR is more like a fine tuning element.
  • High values of CR like CR1 give faster
    convergence if convergence occurs. Sometimes,
    however, you have to go down as much as CR0 to
    make DE robust enough for a particular problem.
  • F and CR are both generally in the range 0.5,
    1. for most problems (F may be greater than 1).
  • Keep in mind that different problems usually
    require different settings for NP, F and CR (have
    a look into the different papers to get a feeling
    for the settings).

http//www.icsi.berkeley.edu/storn/code.html
14
Resources
  • Quite a few differential evolution papers (and
    other resources, e.g. applets and code) around on
    the web.
  • See the Differential Evolution Homepage
  • http//www.icsi.berkeley.edu/storn/code.html

15
Evolution strategies
16
Evolutionary Strategies
  • Developed in 60s and 70s in Germany by Bienert,
    Rechenberg, and Schwefel
  • As with DE the technique has its origins in
    numerical optimisation problems.
  • Some features
  • Typically work with real valued vectors
  • Mutation taken from a Gaussian (normal)
    distribution
  • Evolution of evolutionary parameters (e.g.
    mutation rate)
  • A wide variety of evolutionary strategies are
    available, in some cases a population size of 1
    (!!!) is adopted.

17
Basic Representation
  • To minimise f Rn-gtR

x1
xn
p1
pk
Vector of real-valued fields
Real-valued evolutionary parameters
x1
xn
s1
x1
xn
s1
sn
Can have additional parameters if we want
correlated mutations
18
General Mutation
x1
xn
s1
sn
mutate
x1
xn
s1
sn
mutate
x1
xn
s1
sn
Parameters do not need to mutate at the same rate
as the xs.
19
Basic Solution Mutation
  • Each field is mutated according to a Gaussian or
    normal distribution (mean 0) with standard
    deviation s.

t
x1
xn
s
N(0,s)
N(0,s)
t1
x1
xn
s
20
1/5 Success Rule for updating s
  • Determine percentage ps of successful mutations
    in past k iterations.
  • Update ? after every k iterations by
  • ? ? / c if ps gt 1/5
  • ? ? c if ps lt 1/5
  • ? ? if ps 1/5where 0.8 ? c ? 1

(More complex ways to update in multi-parameter
case)
21
Early ES
  • This had a single parent which gave rise to a
    single child by mutation.
  • Child compared with parent and the better
    survives.
  • This is a local search procedure which is
    essentially hill-climbing.
  • However, the mutation rate is part of the
    chromosome and the update strategy for the
    standard deviation of the mutation distribution
    is updated too.
  • Can allow bigger jumps to be hypothesised as
    progress is impeded.
  • Note in some forms of annealing the move
    function can be dynamic too.

22
General Strategy
  • Have m parents
  • Breed l children (l 7 m is common, high degree
    of pressure)
  • Breeding can take many forms
  • Selection of two parents most typically using
    uniform distribution - field is average of
    parental field values or the value of one of the
    parents field values, or
  • Select two parents for each field of the child..
  • Mutation
  • Select best m from the resulting expanded pool (
    a deterministic cull). Basis of selection is
    either
  • The set of children only (?,?)-selection.
  • The set of parents and children (??)-selection
    (elitist)
  • Which would you prefer and why?

23
Example Applications of Evolutionary Strategies
  • See
  • http//www.bionik.tu-berlin.de/user/giani/esdemos/
    evo.html
  • Technical Report by Michael Herdy and Giannino
    Patone

24
Polynomial Approximation with an Evolution
Strategy
  • To a set of datapoints in the two-dimensional
    plane a polynomial is to be fitted in such a way,
    that all points are described by the polynomial
    with minimal error.

Data
25
Polynomial Approximation with an Evolution
Strategy
http//www.bionik.tu-berlin.de/user/giani/esdemos/
evo.html
26
Polynomial Approximation with an Evolution
Strategy
  • (1, 10)-ES a fractional rational polynomial of
    degree six has been fitted to a noisy dataset

http//www.bionik.tu-berlin.de/user/giani/esdemos/
evo.html
27
Magic Square
  • An age old problem.
  • Example. Arrange the numbers 1 to 64 in an 8 by 8
    grid such that all row sums, column sums, and the
    two diagonal sums are equal
  • Cost function measures deviation from desired
    row, column and diagonal sum value (which is
    S260).

28
Magic Square
http//www.bionik.tu-berlin.de/user/giani/esdemos/
evo.html
29
Brachystrochrone- Johann Bernoulli
  • Find path between two points between the two
    points A and B, on which a movable point P slides
    without friction under gravity in the shortest
    time.
  • Often the first problem studied by UG
    mathematicians in calculus of variations.

30
Brachystrochrone - Johann Bernoulli
  • The track is realized by a polygon. The location
    of the two points A (Start) and B (Finish)
    depends on the difference in height and width of
    the track chosen by the operator, they are fixed
    during the optimization.
  • They may be changed between 0 and 5000 mm. As a
    default width and height of the track are set
    equal to 5000 mm.
  • Between A and B a number of n pillars is given.
    This number may be changed between 1 and 50. The
    position of these pillars is changed to optimal
    values during the optimization.
  • According to the chose type of pillar
    (equidistant x, equidistant y, variable x and y)
    there are n or 2n variables.

31
Brachystrochrone
http//www.bionik.tu-berlin.de/user/giani/esdemos/
evo.html
32
Conclusions
  • Have seen two different types of techniques.
  • Both with roots in numerical optimisation.
  • Some important deviations, e.g. the evolution of
    evolutionary parameters
  • Idea also adopted in other variants of
    techniques we have seen.
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