Lecture 3

Question 1

Why is (standard) crossover not a good idea for real-valued problems?

Answer 1

The range of possible values for a single gene is way too large to keep selecting genes from the initialised genotypes. Hence we need to create new gene values based on the genotypes in each generation.

Question 2

Explain how Classic Differential Evolution creates new solutions

Answer 2

1. You start by initialising values within the range for each gene.
2. In order to generate new values you do the following steps:
3. You pick a main individual as a base.
4. Then you randomly select 3 other individuals from the same population.
5. You perform v = x0 + F(x1 - x2) in order to create a new temporary solution.
6. Then you can perform uniform crossover between genes of the base individual and the temporary solution.
   ( 7. Now it is selection time )

Question 3

In Classic Differential Evolution, what is the variable ‘Cr’ and how does it operate?

Answer 3

The crossover possibility.
The chance that the newly generated solution is picked over the base solution

Question 4

What type (of structures) are the genotype of real valued solutions?

Answer 4

Vectors

Question 5

In Classic Differential Evolution, does the increase of parameter F mean an increase in diversity?

Answer 5

Generally yes.

Question 6

What are the most affecting parameters in Classic Differential Evolution?

Answer 6

• population size
• how the individuals are samples.
• scale factor F

Question 7

What does topology mean for CPSO?

Answer 7

The way that the neighbourhoods are set up; which individual can see which other individual.

Question 8

What does CPSO mean?

Answer 8

Classic Particle Swarm Optimalisation

Question 9

For what type of problem is the CPSO very efficient?

Answer 9

Many local optima

Question 10

What is one of the biggest problems with CPSO?

Answer 10

Due to their parameters, they can either oscilate very heavily, or be too rigid (which both result in more generations and more computation time.)

Question 11

What are the three components of CPSO?

Answer 11

inertia-, cognitive-, and social component

Question 12

Which component(s) of CPSO are influenced by randomness and why?

Answer 12

cognitive and social, the randomness is introduced in how heavy these components weight. This leads to more diversity in the next generation.

Question 13

In CPSO what does the z_{i,g}_ variable mean and in which component is it ?

Answer 13

It means the best ever found solution in the current population. it can be found in the social component

Question 14

In CPSO, how are new solutions generated?

Answer 14

It takes the inertia from its parent, and adds the (weighted) cognitive and social component. Then it adds this new velocity (or inertia) and adds this to the genotype of it’s parent.

Question 15

How many parents does offspring have in CPSO?

Answer 15

1

Question 16

Which parameters for CPSO have the most influence?

Answer 16

• swarm size
• neighborhood topology
• Coefficients (i.e. inertia weight and acceleration constants)

Question 17

What does convex combinations mean?

Answer 17

If you imagine a triangle, then the convex combination lays within the surface of that triange.

Or in case of EA:
A new solution does not ponder outside of the solution space.

Question 18

Does CSPO exist of convex combinations and why so?

Answer 18

Yes, the new solution is created from the three components of it’s parent.
These components are weighted between [0,1) and thus stay within the search space of the problem.

Question 19

What is the problem with CSPO?

Answer 19

The algorithm really aims to convolute its population fast. Although** this increases diversity,** this means that it is very likely that the algorithm convolutes on local optima, as the real solution gets out of reach of the search space.

Question 20

How can you tell that variation conciders covariance when looking at the Gaussian plot of that variation?

Answer 20

When the gaussian shows an angled eclipse.

Question 21

What does a (standing) ecplise mean in gaussian variation?

Answer 21

It represents how much each individual is allowed to vary when sampled with it. : It basically shows the bounds of variation for each gene.

Question 22

In what EA branch/variant is normal distribution varation central?

Answer 22

Evolutionary Strategies (ES)

Question 23

What is the similarity and difference between mutation and crossover?

Answer 23

They are both forms of variation. Crossover uses multiple ‘parents’ to draw new solutions.
Mutation uses a probability model on a singular parent’s genotype in order to create a new sample.

Question 24

What are the properties of ES genotypes?

Answer 24

They do not only contain a solution,
But more importantly it includes parameters that encode its normal distribution (for mutation)

Question 25

What are the three variants in ES?

Answer 25

1. One variance, no covariances
   (𝑥0, 𝑥1, … , 𝑥ℓ−1, 𝜎)
2. ℓ variances, no covariances
   (𝑥0, 𝑥1, … , 𝑥ℓ−1, 𝜎0, 𝜎1, … , 𝜎ℓ−1)
3. Full covariance matrix (general ellipsoidal density contours)
   (𝑥0, 𝑥1, … , 𝑥ℓ−1, 𝜎0, 𝜎1, … , 𝜎ℓ−1, 𝛼1, 𝛼2, … , 𝛼ℓ(ℓ−1)/2)
   (note: 𝛼𝑖 are rotation angles)

Question 26

What variant of ES would create elipsoidal mutation distributrions?

Answer 26

Variant 2: ℓ variances, no covariances.
(𝑥0, 𝑥1, … , 𝑥ℓ−1, 𝜎0, 𝜎1, … , 𝜎ℓ−1)

Question 27

For the third ES variant, why do we sample angles and not covairances?

Answer 27

Because covariance matrix must remain
.positive definite.
So the variance encode the stretching and the rotation ensure the convairance matrix is always ‘proper’: we can sample it.

Question 28

For the third ES variant,
how many rotation angles do we expect in each genotype?

Answer 28

ℓ(ℓ−1)/2

Because each genotype has covariance with all genotypes apart from itself, and we dont need double definitions (hence the 2)

Question 29

How do we represent the genotype of an individual in ES?

Answer 29

(𝒙, 𝒔), where
* 𝒙 is the vector of encoded problem variables and
* 𝒔 is the decoded vector of (normal) strategy parameters

Question 30

In ES, why do the strategy parameters also need to undergo mutation?

Answer 30

This is to better control the search space: It can have a high variance in early generations, but we want low variance in later generations. “meaning mutation”

Question 31

In ES, what does 𝜓(𝒔) represent?

Answer 31

It is the encoded
covariance matrix that is generated from the strategy parameters (𝒔). Such that (𝒙, 𝒔) → (𝒙 + ∆𝒙, 𝒔) where
∆𝒙 ~ 𝒩(0, 𝜓 𝒔 )

Question 32

What is self adaption (in ES)?

Answer 32

The mechanism to control own mutability.

Question 33

In ES what does selection influence?

Answer 33

fitness directly,
distribution quality indirectly

Question 34

The mathematical definition of log-normal multiplicatively mutated variance in ES variant 1.

Answer 34

𝜎 → max {𝜎𝑒^𝑧, 𝜀_𝜎_}
where𝑧 ~ 𝒩(0, 𝜏0^2)

Question 35

The mathematical definition of log-normal multiplicatively mutated variance in ES variant 2 & 3.

Answer 35

𝜎 → max {𝜎𝑒^{𝑧+𝑧i}, 𝜀_𝜎_}
where 𝑧 ~ 𝒩(0, 𝜏0^2) and 𝑧i ~ 𝒩(0, 𝜏”0^2)

Question 36

What are the reasons to prefer lognormal mutation?

Answer 36

– Standard deviations remain positive
– Median equals one (for 𝜇 = 0)
– Smaller modifications more likely than larger ones

Question 37

In ES variant 3, how are the covariance and rotation angle mutated?

Answer 37

lognormal multiplcatively &
normal additvely
(𝛼𝑗 ← 𝛼𝑗 + 𝑧𝑗 where 𝑧𝑗 ~ 𝒩(0, 𝛽2))

Question 38

What is recombination in ES?

Answer 38

The act of creating a singular offspring by selectiong components from the genotype of parents.

Question 39

What is Global intermediary recombination?

Answer 39

Average genotype over all parents

Question 40

What is Local intermediary recombination?

Answer 40

A Convex combination of 2 parents

Question 41

What is discrete recombination?

Answer 41

Pick one of the parents.

Question 42

In ES, what symbol do they use for population size and offspring size?

Answer 42

𝜇 and 𝜆

Question 43

Does ES have high selection presure?

Answer 43

Yes. Selection presure can be set using the parameters, but usually it is set very high. The selection pressure occurs because you take multiple samples for a single variable, of which only the fittest survives.

Question 44

In what cases do ES variant 1 and 2 excel?

Answer 44

Real value problems where there is no strong correlation between variables.

Question 45

Where does variant 3 of ES excel at?

Answer 45

It can be more thorough, but often it does not hold up because of the slow self-adaptation.

Question 46

Why does ES variant 3 have slow self adaptation?

Answer 46

Imagine if you have a gene that has a variance a million times larges than another gene. When you rotate that covariance matrix (by mutating the rotation variable), the sampling space of that variable becomes incredibly skewed. Hence it will take a very long time for the ES to adapt all these variables.