Sufficient Conditions for Coarsegraining Evolutionary Dynamics
1. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Sufficient Conditions for Coarse-Graining
Evolutionary Dynamics
Keki Burjorjee
DEMO Lab
Computer Science Department
Brandeis University
Waltham, MA, USA
FOGA IX (2007)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
2. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
3. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
4. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
5. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
6. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
7. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Genetic Algorithms and the Building Block Hypothesis
In 1975 Holland published his seminal work on GAs
Description of the Genetic Algorithm
A theory of adaptation for GAs
which later came to be known as BBH
GAs useful for adapting solutions to difficult real world
problems
However BBH has drawn considerable skepticism amongst
many researchers (e.g. Vose, Wright, Rowe)
Despite criticism of BBH, no alternate full-fledged theories of
adaptation have been proposed
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
8. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
For genomes of non-trivial length, current theoretical results do
not permit the formulation of principled theories of adaptation
Schema theories only permit a tractable analysis of
evolutionary dynamics over a single generation
Markov Chain approaches only yield a qualitative description of
evolutionary dynamics in the asymptote of time
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
9. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
For genomes of non-trivial length, current theoretical results do
not permit the formulation of principled theories of adaptation
Schema theories only permit a tractable analysis of
evolutionary dynamics over a single generation
Markov Chain approaches only yield a qualitative description of
evolutionary dynamics in the asymptote of time
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
10. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
For genomes of non-trivial length, current theoretical results do
not permit the formulation of principled theories of adaptation
Schema theories only permit a tractable analysis of
evolutionary dynamics over a single generation
Markov Chain approaches only yield a qualitative description of
evolutionary dynamics in the asymptote of time
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
11. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
For genomes of non-trivial length, current theoretical results do
not permit the formulation of principled theories of adaptation
Schema theories only permit a tractable analysis of
evolutionary dynamics over a single generation
Markov Chain approaches only yield a qualitative description of
evolutionary dynamics in the asymptote of time
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
12. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
The infinite population assumption is often used to make
mathematical models of GA dynamics tractable
Even with this assumption there are currently no theoretical
results which permit a principled analysis of any aspect of GA
behavior over the “short-term”
“short-term” = small number of generations
No theoretical results that Accurate theories of
permit principled short-term ⇒ adaptation for GAs will
analyses of evolutionary evade discovery
dynamics
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
13. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
The infinite population assumption is often used to make
mathematical models of GA dynamics tractable
Even with this assumption there are currently no theoretical
results which permit a principled analysis of any aspect of GA
behavior over the “short-term”
“short-term” = small number of generations
No theoretical results that Accurate theories of
permit principled short-term ⇒ adaptation for GAs will
analyses of evolutionary evade discovery
dynamics
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
14. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
The infinite population assumption is often used to make
mathematical models of GA dynamics tractable
Even with this assumption there are currently no theoretical
results which permit a principled analysis of any aspect of GA
behavior over the “short-term”
“short-term” = small number of generations
No theoretical results that Accurate theories of
permit principled short-term ⇒ adaptation for GAs will
analyses of evolutionary evade discovery
dynamics
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
15. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
No Alternate Theories of Adaptation for GAs — Why?
The infinite population assumption is often used to make
mathematical models of GA dynamics tractable
Even with this assumption there are currently no theoretical
results which permit a principled analysis of any aspect of GA
behavior over the “short-term”
“short-term” = small number of generations
No theoretical results that Accurate theories of
permit principled short-term ⇒ adaptation for GAs will
analyses of evolutionary evade discovery
dynamics
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
16. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
The Promise of Coarse-Graining
Coarse-graining a very useful technique from theoretical
Physics
If successfully applied to an IPGA it permits a principled
analysis of certain aspects of the IPGA’s dynamics over
multiple generations
Therefore coarse-graining is a promising approach to the
formulation of principled theories of evolutionary adaptation
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
17. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
The Promise of Coarse-Graining
Coarse-graining a very useful technique from theoretical
Physics
If successfully applied to an IPGA it permits a principled
analysis of certain aspects of the IPGA’s dynamics over
multiple generations
Therefore coarse-graining is a promising approach to the
formulation of principled theories of evolutionary adaptation
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
18. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
The Promise of Coarse-Graining
Coarse-graining a very useful technique from theoretical
Physics
If successfully applied to an IPGA it permits a principled
analysis of certain aspects of the IPGA’s dynamics over
multiple generations
Therefore coarse-graining is a promising approach to the
formulation of principled theories of evolutionary adaptation
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
19. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
The Promise of Coarse-Graining
Coarse-graining a very useful technique from theoretical
Physics
If successfully applied to an IPGA it permits a principled
analysis of certain aspects of the IPGA’s dynamics over
multiple generations
Therefore coarse-graining is a promising approach to the
formulation of principled theories of evolutionary adaptation
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
28. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Coarse-Graining by Depiction
Partition Function
t+1
y1 = . . .
t+1
=
y2 . . . .
.
.
t+1
y10 = . . .
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
29. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Previous Coarse-Graining Results
Wright, Vose, and Rowe (Wright et al. 2003) show that any
mask based recombination operation of an IPGA can be
coarse-grained.
However they argue that the selecto-recombinative dynamics
of an IPGA with an arbitrary initial population cannot be
coarse-grained unless the fitness function satisfies a very
strong constraint
I call this constraint schematic fitness invariance
for some schema partition, and for any schema in the partition,
all the genomes in the schema have exactly the same fitness
Wright et al. argue that schematic fitness invariance is so
strong that it renders a coarse-graining of
selecto-recombinative dynamics useless
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
30. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Previous Coarse-Graining Results
Wright, Vose, and Rowe (Wright et al. 2003) show that any
mask based recombination operation of an IPGA can be
coarse-grained.
However they argue that the selecto-recombinative dynamics
of an IPGA with an arbitrary initial population cannot be
coarse-grained unless the fitness function satisfies a very
strong constraint
I call this constraint schematic fitness invariance
for some schema partition, and for any schema in the partition,
all the genomes in the schema have exactly the same fitness
Wright et al. argue that schematic fitness invariance is so
strong that it renders a coarse-graining of
selecto-recombinative dynamics useless
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
31. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Previous Coarse-Graining Results
Wright, Vose, and Rowe (Wright et al. 2003) show that any
mask based recombination operation of an IPGA can be
coarse-grained.
However they argue that the selecto-recombinative dynamics
of an IPGA with an arbitrary initial population cannot be
coarse-grained unless the fitness function satisfies a very
strong constraint
I call this constraint schematic fitness invariance
for some schema partition, and for any schema in the partition,
all the genomes in the schema have exactly the same fitness
Wright et al. argue that schematic fitness invariance is so
strong that it renders a coarse-graining of
selecto-recombinative dynamics useless
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
32. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Previous Coarse-Graining Results
Wright, Vose, and Rowe (Wright et al. 2003) show that any
mask based recombination operation of an IPGA can be
coarse-grained.
However they argue that the selecto-recombinative dynamics
of an IPGA with an arbitrary initial population cannot be
coarse-grained unless the fitness function satisfies a very
strong constraint
I call this constraint schematic fitness invariance
for some schema partition, and for any schema in the partition,
all the genomes in the schema have exactly the same fitness
Wright et al. argue that schematic fitness invariance is so
strong that it renders a coarse-graining of
selecto-recombinative dynamics useless
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
33. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Previous Coarse-Graining Results
Wright, Vose, and Rowe (Wright et al. 2003) show that any
mask based recombination operation of an IPGA can be
coarse-grained.
However they argue that the selecto-recombinative dynamics
of an IPGA with an arbitrary initial population cannot be
coarse-grained unless the fitness function satisfies a very
strong constraint
I call this constraint schematic fitness invariance
for some schema partition, and for any schema in the partition,
all the genomes in the schema have exactly the same fitness
Wright et al. argue that schematic fitness invariance is so
strong that it renders a coarse-graining of
selecto-recombinative dynamics useless
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
34. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
35. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
36. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
37. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
38. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
39. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
40. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Comparison between Coarse-Graining Results
I show that if the class of initial populations is appropriately
constrained then it is possible to coarse-grain the
selecto-recombinative dynamics of an IPGA for a much weaker
constraint on the fitness function
The constraint on the class of initial populations is not
onerous
A uniformly distributed population satisfies this constraint
The constraint on the fitness function is weak enough that it
makes the coarse-graining result potentially useful in a theory
of adaptation
Constraint on Constraint on
Initial Population Fitness Function
Wright et al. 2003 None Severe
Burjorjee 2007 Non-onerous Weak
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
41. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
42. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
43. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
44. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
45. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
46. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Structure of this Talk
1. Describe an abstract framework for analyzing the dynamics of
a selecto-recombinative infinite population EA (IPEA)
2. Describe the theoretical technique used to Coarse-Grain the
dynamics of an IPEA
3. Present results that show that these dynamics can be
coarse-grained provided that the IPEA satisfies certain
abstract conditions
4. Describe how the coarse-graining results can be used to
coarse-grain the dynamics of an IPGA with long genomes and
a non-trivial fitness functions
5. Present experimental validation of this theory
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
47. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling Populations and Operations on Populations
Modeling scheme based on the one used in (Toussaint 2003)
Populations modeled as distributions over the genome set
Distribution values sum to 1
Population-level effect of evolutionary operations modeled as
the application of parameterized mathematical operators to
genomic distributions
Parameter objects used by the operators (fitness function,
transmission function) give individual-level information about
the genomes
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
48. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling Populations and Operations on Populations
Modeling scheme based on the one used in (Toussaint 2003)
Populations modeled as distributions over the genome set
Distribution values sum to 1
Population-level effect of evolutionary operations modeled as
the application of parameterized mathematical operators to
genomic distributions
Parameter objects used by the operators (fitness function,
transmission function) give individual-level information about
the genomes
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
49. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling Populations and Operations on Populations
Modeling scheme based on the one used in (Toussaint 2003)
Populations modeled as distributions over the genome set
Distribution values sum to 1
Population-level effect of evolutionary operations modeled as
the application of parameterized mathematical operators to
genomic distributions
Parameter objects used by the operators (fitness function,
transmission function) give individual-level information about
the genomes
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
50. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling Populations and Operations on Populations
Modeling scheme based on the one used in (Toussaint 2003)
Populations modeled as distributions over the genome set
Distribution values sum to 1
Population-level effect of evolutionary operations modeled as
the application of parameterized mathematical operators to
genomic distributions
Parameter objects used by the operators (fitness function,
transmission function) give individual-level information about
the genomes
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
51. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling Populations and Operations on Populations
Modeling scheme based on the one used in (Toussaint 2003)
Populations modeled as distributions over the genome set
Distribution values sum to 1
Population-level effect of evolutionary operations modeled as
the application of parameterized mathematical operators to
genomic distributions
Parameter objects used by the operators (fitness function,
transmission function) give individual-level information about
the genomes
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
52. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Transmission Functions
Use transmission functions (Altenberg 1994) to represent the
variational information at the individual-level
Example: T (g |g1 , . . . gn ) is the probability that parents
g1 , . . . , gn will yield a child g when recombined
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
53. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Transmission Functions
Use transmission functions (Altenberg 1994) to represent the
variational information at the individual-level
Example: T (g |g1 , . . . gn ) is the probability that parents
g1 , . . . , gn will yield a child g when recombined
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
54. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Transmission Functions
Use transmission functions (Altenberg 1994) to represent the
variational information at the individual-level
Example: T (g |g1 , . . . gn ) is the probability that parents
g1 , . . . , gn will yield a child g when recombined
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
55. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Variation on Populations
Given some transmission function T , the effect of variation at
the population-level is modeled by the variation operator VT
For some population p, if p = VT (p), then, p is as follows:
m
p (g ) = T (g |g1 , . . . , gm ) p(gi )
(g1 ,...,gm ) i=1
∈ mG 1
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
56. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Variation on Populations
Given some transmission function T , the effect of variation at
the population-level is modeled by the variation operator VT
For some population p, if p = VT (p), then, p is as follows:
m
p (g ) = T (g |g1 , . . . , gm ) p(gi )
(g1 ,...,gm ) i=1
∈ mG 1
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
57. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Variation on Populations
Given some transmission function T , the effect of variation at
the population-level is modeled by the variation operator VT
For some population p, if p = VT (p), then, p is as follows:
m
p (g ) = T (g |g1 , . . . , gm ) p(gi )
(g1 ,...,gm ) i=1
∈ mG 1
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
58. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Selection on Populations
Given some fitness function f : G → R+ , the effect of fitness
proportional selection is modeled by the selection operator Sf
For some population p, if p = Sf (p), then p is as follows: For
any genotype g ,
f (g )p(g )
p (g ) =
Ef (p)
where Ef is the weighted average fitness of p
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
59. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Selection on Populations
Given some fitness function f : G → R+ , the effect of fitness
proportional selection is modeled by the selection operator Sf
For some population p, if p = Sf (p), then p is as follows: For
any genotype g ,
f (g )p(g )
p (g ) =
Ef (p)
where Ef is the weighted average fitness of p
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
60. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Modeling the Effect of Selection on Populations
Given some fitness function f : G → R+ , the effect of fitness
proportional selection is modeled by the selection operator Sf
For some population p, if p = Sf (p), then p is as follows: For
any genotype g ,
f (g )p(g )
p (g ) =
Ef (p)
where Ef is the weighted average fitness of p
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
61. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
62. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
63. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
64. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
65. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
66. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
67. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Some Terminology and Notation
β : G → K a surjective function
Call β a partitioning
Call co-domain K the theme set
Call the elements of K themes
k β denotes the set of all g ∈ G such that β(g ) = k
Call k β the theme class of k under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
68. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Projection Operator
Let β : G → K be a partitioning
A projection operator Ξβ ‘projects’ a distribution pG over G
‘through’ β to create a distribution pK = Ξβ (pG ) over the
theme set
β
K
G
For any k ∈ K ,
pK (k) = p(g )
g∈ k β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
69. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Projection Operator
Let β : G → K be a partitioning
A projection operator Ξβ ‘projects’ a distribution pG over G
‘through’ β to create a distribution pK = Ξβ (pG ) over the
theme set
Ξβ
β
K
G
For any k ∈ K ,
pK (k) = p(g )
g∈ k β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
70. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Projection Operator
Let β : G → K be a partitioning
A projection operator Ξβ ‘projects’ a distribution pG over G
‘through’ β to create a distribution pK = Ξβ (pG ) over the
theme set
Ξβ
β
K
G
For any k ∈ K ,
pK (k) = p(g )
g∈ k β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
71. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
72. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
73. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
74. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
75. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
76. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
77. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
78. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Semi-Concordance, Concordance, Global Concordance
β : G → K some partitioning (i.e. surjective function)
W : ΛG → ΛG some operator
U ⊆ ΛG some subset of populations
Say that W is semi-concordant with β on U if
U
W / ΛG
Ξβ Ξβ
ΛK / ΛK
Q
If W(U) ⊆ U then W concordant with β on U
If U = ΛG then W globally concordant with β on U
Global concordance ⇔ compatibility (Vose 1999)
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
79. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Concordance
Suppose W concordant with β on U , i.e.
U
W /U
Ξβ Ξβ
ΛK / ΛK
Q
For initial population pG ∈ U, let pK = Ξβ (pG )
Can observe the “shadow” of the dynamics induced by W by
studying the effect of the repeated application of Q to pK
If the size of K is small then such a study becomes
computationally feasible
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
80. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Concordance
Suppose W concordant with β on U , i.e.
U
W /U
Ξβ Ξβ
ΛK / ΛK
Q
For initial population pG ∈ U, let pK = Ξβ (pG )
Can observe the “shadow” of the dynamics induced by W by
studying the effect of the repeated application of Q to pK
If the size of K is small then such a study becomes
computationally feasible
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
81. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Concordance
Suppose W concordant with β on U , i.e.
U
W /U
Ξβ Ξβ
ΛK / ΛK
Q
For initial population pG ∈ U, let pK = Ξβ (pG )
Can observe the “shadow” of the dynamics induced by W by
studying the effect of the repeated application of Q to pK
If the size of K is small then such a study becomes
computationally feasible
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
82. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Concordance
Suppose W concordant with β on U , i.e.
U
W /U
Ξβ Ξβ
ΛK / ΛK
Q
For initial population pG ∈ U, let pK = Ξβ (pG )
Can observe the “shadow” of the dynamics induced by W by
studying the effect of the repeated application of Q to pK
If the size of K is small then such a study becomes
computationally feasible
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
83. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Concordance
Suppose W concordant with β on U , i.e.
U
W /U
Ξβ Ξβ
ΛK / ΛK
Q
For initial population pG ∈ U, let pK = Ξβ (pG )
Can observe the “shadow” of the dynamics induced by W by
studying the effect of the repeated application of Q to pK
If the size of K is small then such a study becomes
computationally feasible
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
84. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Theoretical Modus Operandi
Let G = VT ◦ Sf
I give sufficient conditions on T and f under which a
concordance result can be proved for G
One of these conditions is called Ambivalence.
It is defined with respect to the transmission function T and a
partitioning
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
85. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Theoretical Modus Operandi
Let G = VT ◦ Sf
I give sufficient conditions on T and f under which a
concordance result can be proved for G
One of these conditions is called Ambivalence.
It is defined with respect to the transmission function T and a
partitioning
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
86. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Theoretical Modus Operandi
Let G = VT ◦ Sf
I give sufficient conditions on T and f under which a
concordance result can be proved for G
One of these conditions is called Ambivalence.
It is defined with respect to the transmission function T and a
partitioning
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
87. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Theoretical Modus Operandi
Let G = VT ◦ Sf
I give sufficient conditions on T and f under which a
concordance result can be proved for G
One of these conditions is called Ambivalence.
It is defined with respect to the transmission function T and a
partitioning
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
88. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
β
11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
89. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
90. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
91. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
c
b
a
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
92. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
c
b
a
1
0
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
1
0 11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
93. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
c
b
a
1
0
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
1
0 11
00 11
00 11
00
11
00
G K
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
94. Introduction Abstract Framework Coarse-Graining Results Application to GAs Experimental Validation Conclusion
Ambivalence (By Example)
An Ambivalent 2-parent transmission function T
c
b
a
1
0
1
0 β
1
0 11
00 11
00 11
00
11
00
11
00 11
00
11
00 11
00
1
0 11
00 11
00 11
00
11
00
G K
b a
c
Say that T is ambivalent under β
Keki Burjorjee Suff. Conditions for Coarse-Graining Evolutionary Dynamics
