Parameter Control

Document technical information

Format pdf
Size 441.0 kB
First found May 22, 2018

Document content analysis

Category
Also themed
Language
English
Type
not defined
Concepts
no text concepts found

Transcript

Chapter 8
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Motivation 1
An EA has many strategy parameters, e.g.
 mutation operator and mutation rate
 crossover operator and crossover rate
 selection mechanism and selective pressure (e.g.
tournament size)
 population size
Good parameter values facilitate good performance
Q1 How to find good parameter values ?
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
2 / 23
Motivation 2
EA parameters are rigid (constant during a run)
BUT
an EA is a dynamic, adaptive process
THUS
optimal parameter values may vary during a run
Q2: How to vary parameter values?
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
3 / 23
Parameter tuning
Parameter tuning: the traditional way of testing and
comparing different values before the “real” run
Problems:
 users mistakes in settings can be sources of errors or suboptimal performance
 costs much time
 parameters interact: exhaustive search is not practicable
 good values may become bad during the run
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
4 / 23
Parameter control
Parameter control: setting values on-line, during the
actual run, e.g.
 predetermined time-varying schedule p = p(t)
 using feedback from the search process
 encoding parameters in chromosomes and rely on natural selection
Problems:
 finding optimal p is hard, finding optimal p(t) is harder
 still user-defined feedback mechanism, how to ``optimize"?
 when would natural selection work for strategy parameters?
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
5 / 23
Example
 min f(x1,…,xn)
 Li  xi  Ui for i = 1,…,n
 gi (x)  0
 hi (x) = 0
bounds
for i = 1,…,q
for i = q+1,…,m
inequality constraints
equality constraints
Algorithm:
 EA with real-valued representation (x1,…,xn)
 arithmetic averaging crossover
 Gaussian mutation: x‟ i = xi + N(0, )
standard deviation  is called mutation step size
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
6 / 23
Varying mutation step size: option1
Replace the constant  by a function (t)
 (t )  1 - 0.9  Tt
0  t  T is the current generation number
 Features:
 changes in  are independent from the search progress
 strong user control of  by the above formula
  is fully predictable
 a given  acts on all individuals of the population
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
7 / 23
Varying mutation step size: option2
Replace the constant  by a function (t) updated after every
n steps by the 1/5 success rule (cf. ES chapter):
 (t  n ) / c if ps  1/5

 (t )   (t  n )  c if ps  1/5
 (t  n )
otherwise

 Features:
 changes in  are based on feedback from the search progress
 some user control of  by the above formula
  is not predictable
 a given  acts on all individuals of the population
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
8 / 23
Varying mutation step size: option3
 Assign a personal  to each individual
 Incorporate this  into the chromosome: (x1, …, xn, )
 Apply variation operators to xi„s and 
     e N ( 0, )
xi  xi  N (0, )
 Features:
 changes in  are results of natural selection
 (almost) no user control of 
  is not predictable
 a given  acts on one individual
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
9 / 23
Varying mutation step size: option4
Assign a personal  to each variable in each individual
Incorporate ‟s into the chromosomes: (x1, …, xn, 1, …,  n)
Apply variation operators to xi„s and i„s
N ( 0, )

 i  i  e
xi  xi  N (0, i )
 Features:
 changes in i are results of natural selection
 (almost) no user control of i
 i is not predictable
 a given i acts on 1 gene of one individual
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
10 / 23
Example cont‟d
Constraints
 gi (x)  0
 hi (x) = 0
for i = 1,…,q
for i = q+1,…,m
inequality constraints
equality constraints
are handled by penalties:
eval(x) = f(x) + W × penalty(x)
where
1 for violated constraint
penalty ( x )   
j 1 0 for satisfied constraint
m
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
11 / 23
Varying penalty: option 1
Replace the constant W by a function W(t)
W (t )  (C  t )
α
0  t  T is the current generation number
 Features:
 changes in W independent from the search progress
 strong user control of W by the above formula
 W is fully predictable
 a given W acts on all individuals of the population
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
12 / 23
Varying penalty: option 2
Replace the constant W by W(t) updated in each generation
 β  W(t) if last k champions all feasible

W (t  1)  γ  W(t) if last k champions all infeasible
W(t)
otherwise

 < 1,  > 1,     1 champion: best of its generation
 Features:
 changes in W are based on feedback from the search progress
 some user control of W by the above formula
 W is not predictable
 a given W acts on all individuals of the population
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
13 / 23
Varying penalty: option 3
Assign a personal W to each individual
Incorporate this W into the chromosome: (x1, …, xn, W)
Apply variation operators to xi„s and W
eval ((x, W)) = f (x) + W × penalty(x)
while for mutation step sizes we had
eval ((x, )) = f (x)
this option is thus sensitive “cheating”  makes no sense
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
14 / 23
Lessons learned from examples
Various forms of parameter control can be distinguished by:
 primary features:
 what component of the EA is changed
 how the change is made
 secondary features:
 evidence/data backing up changes
 level/scope of change
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
15 / 23
What
Practically any EA component can be parameterized and
thus controlled on-the-fly:
 representation
 evaluation function
 variation operators
 selection operator (parent or mating selection)
 replacement operator (survival or environmental selection)
 population (size, topology)
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
16 / 23
How
Three major types of parameter control:
 deterministic: some rule modifies strategy parameter
without feedback from the search (based on some
counter)
 adaptive: feedback rule based on some measure
monitoring search progress
 self-adaptative: parameter values evolve along with
solutions; encoded onto chromosomes they undergo
variation and selection
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
17 / 23
Global taxonomy
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
18 / 23
Evidence informing the change
The parameter changes may be based on:
 time or nr. of evaluations (deterministic control)
 population diversity
 gene distribution, etc.
 relative fitness of individuals creeated with given values
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
19 / 23
Evidence informing the change
 Absolute evidence: predefined event triggers change, e.g.
increase pm by 10% if population diversity falls under
threshold x
 Direction and magnitude of change is fixed
 Relative evidence: compare values through solutions
created with them, e.g. increase pm if top quality offspring
came by high mut. Rates
 Direction and magnitude of change is not fixed
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
20 / 23
Scope/level
The parameter may take effect on different levels:
 environment (fitness function)
 population
 individual
 sub-individual
Note: given component (parameter) determines possibilities
Thus: scope/level is a derived or secondary feature in the
classification scheme
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
21 / 23
Refined taxonomy
 Combinations
of types and evidences
 Possible:
+
 Impossible: -
Parameter Control
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
22 / 23
Evaluation / Summary
 Parameter control offers the possibility to use appropriate
values in various stages of the search