Evolutionary computing:5. Evolutionary strategies

Resource:evolutionary Computing, A.e.eiben

1. What is Evolution strategies (ES)

Evolution Strategies (ES) is another member of the evolutionary algorithm family.

ES Technical Summary Tableau

2. Introductory Example2.1 Task

Minimimise F:rn R

2.2 Original Algorithm

"Two-membered ES" using

• Vectors from Rn directly as chromosomes
• Population Size 1
• Only mutation creating one child
• Greedy selection

2.3 PSEUDOCDE

------------------------------------------------------------

Set t = 0

Create Initial Point XT =〈x1t,..., xnt〉

REPEAT UNTIL (termin. COND satisfied) do

Draw Zi from a normal distr. For all i = 1,..., n

YIT = xit + Zi

IF F (XT) < F (YT) then xt+1 = XT

ELSE xt+1 = YT

Set T = t+1

Od

------------------------------------------------------------

2.4 Explanation

as is shown on the pseudocode above, given a current solution xT in the form of a vector of length n, a new C Andidate xt+1 is created by adding a random number zi for i =1,..., n to each of the N components.

(Let's talk about the random number Zi)

The random number Zi:

A Gaussian, or normal, distribution is used with zero mean and standard deviationσfor drawing the random numbers.

This distribution are symmetric about zero and have the feature that the probability of drawing a random number with any giv En magnitude is a rapidly decreasing function of the standard deviationσ. (More information about Gaussian distribution)

(Let's talk about the random numberσ)

Thus Theσvalue is a parameter of the algorithm that determines the extent to which given values XI be perturbed by the Mutation operator.

For this reasonσis often called the mutation step size.

Theoretical studies motivated an on-line adjustment of step sizes by the famous 1/5 success Rule. This rule states, the ratio of successful mutaions (those in which, the child was fitter than the parent) to all Mutatio NS should be 1/5.

• If the ratio is greater than 1/5, the step size should being increased to make a wider search of the space.
• If the ratio is less than 1/5 then it should being decreased to concentrate the search more around the current solution.

The rule is executed at periodic intervals.

For instance, after K iterations Eachσis reset by

• σ=σ/c if PS > 1/5
• Σ=σ C if PS < 1/5
• Σ=σif PS = 1/5

Where Ps is the relative frequency of successful mutations measured over a number of trials, and the parameter C is In the range [0.817,1]

As is apparent, using the-mechanism the step sizes change based in feedback from the search process.

2.5 Conclusion

This example illuminiates some essential characteristics of evolution strategies:

1. Evolution strategies is typically used for continuousparameter optimisation.
2. There is a strong emphasis on mutation for creating offspring.
3. Mutation is implemented by adding some random noise drawn from a Gaussiandistribution.
4. Mutation parameters is changed during a run of the algorithm

3. Representation

Chromosomes consist of three parts:

• Object variables:x1,..., xn
• Strategy Parameters:
  • Mutation Step sizes:σ1,..., σnσ
  • Rotation angles:α1,..., Αnα

Full size:〈x1,..., xn,σ1,..., σn, α1,..., αk〉,where k = n (n-1)/2 (No. of i,j Pairs)---this was the generalform of individuals in ES

theσvalues represent the mutation step sizes, and their number nσis usually either 1 or N. For any easonable self-adaptation mechanism at least oneσmust is present.

theαvalues, which represent interactions between the step sizes used for different variables, is not always us Ed.

4. Mutation4.1 Main mechanism

Changing value by adding the random noise drawn from normal distribution

The mutation operator in ES are based on a normal (Gaussian) distribution requiring the parameters: the meanξ and The standard deviationσ.

Mutations then is realised by adding someδxi to each XI, where theδxi values is randomly drawn using the give n Gaussian N (ξ,σ), with the corresponding probability density function.

Xi ' = xi + N (0,σ)

Xi ' can be seen as a new XI.

N (0,σ) here denotes a random number drawn from a Gaussian distribution with Zero mean and standard D Eviationσ.

4.2 Key Ideas

• Σis part of the chromosome〈x1,..., xn,σ〉
• Σis also mutated intoσ ' (see later how)
• Self-adaption

4.3 A simplest case

In the simplest case we would has one step size, applied to all the Components XI and candidate solutions of the form <x1, ..., xn,σ>

Mutations was then realised by replacing <x1, ..., xn,σ> by <x1 ', ..., xn ', σ ';, whereσ ' is the mutated Value ofσ and Xi ' = XI + N (0,σ)

4.4 Mutate The value ofσ

the mutation step sizes is not set by the user; rather Theσis coevolving with the solutions.

In order to achieve this behaviour:

1. Modify the value Ofσfirst
2. Mutate the XI values with the Newσvalue.

The rationale behind a new individual <x ', Σ ' > is effectively evaluated twice:

1. primarily, it is evaluated directly for its viability during survivor selection based on F (X ').
2. Second, it is evaluated for its ability to create good offspring.

This happens indirectly:a given step size (σ) evaluates favourably if the offspring generated by using it prove Viable (in the first sense).

To sumup, an individual <x ', Σ ' > represents both a good X ' This survived selection and a goodσ ' that Prov Ed successful in generating the Good x ' from X.

4.5 uncorrelated Mutation with one Step Size (σ)

In the case of uncorrelated mutation with one step size, the same distribution are used to mutate each XI , therefore we only has one strategy parameterσin each individual.

Thisσis mutated each time step by multiplying it by a term eγ, withγa random variable drawn each time from a normal di Stribution with mean 0 and standard deviationτ.

Since N (0,τ) =τ N (0,1), the mutation mechanism is thus specified by the following formulas:

• Σ ' =σ eτ N (0,1)
• Xi ' = Xi +σ ' ni (0,1)

Furthermore, since standard deviations very close to zero is unwanted (they would has on average a negligible effect), the Following boundary rule is used to force step sizes to be no smaller than a threshold:

• Σ ' <ε0⇒σ ' =ε0

Tips:

• N (0,1) denotes a draw from the standard normal distribution
• Ni (0,1) denotes a separate draw from the standard normal distribution for each variable I.

The proportionality constantτis an external parameter to is set by the user.

It is usually inversely proportional to the square root of the problem size:

• Tau ∝ 1/n½

The Parameterτcan is interpreted as a kind of learning, as in neural networks.

Evolutionary computing:5. Evolutionary strategies