[PPT] - Run Time Complexity In typical application the total run time of a PowerPoint Presentation

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Run Time Complexity

determined by the number of fitness function evaluations.

ignored.

number of generations run and the population size: # Fitness Fct. Evals = # Generations . Population Size

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Convergence speed

selection pressure: – high selection pressure: fast convergence – low selection pressure: slow convergence

– large population size: more reliable convergence – small population size: less reliable convergence

pressure and population size

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Key questions

before it converges ?

convergence ? → ie. problem dependent, but:

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Models

behavior, assuming large enough (∞) populations,

population size,

labeling).

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Selection Intensity

pressure we need a measure,

measure: the selection intensity I.

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Quantitative Genetics

between individuals that are quantitative rather than qualitative.

height or the weight of the human body, whereas qualitative variation is measured in discrete units or categories such as eye color or blood type.

following concepts are defined.

defined as the difference between the mean fitness of the population at generation t + 1 and the mean fitness of the population at generation t.

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mean fitness of the parent population at generation t and the population mean fitness at generation t. The parent population is the pool of individuals remaining after selection has been applied but before the oofspring has been generated by the variation operators: S(t) = f(ts) − f(t).

N(f, σ2) we can scale the selection differential by the standard deviation σ(t).

intensity I(t). This is a dimensionless number since the standard deviation has the same units as the selection response: I(t) = S(t) σ(t) = f(ts) − f(t) σ(t) .

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the selection intensity I is simply the expected average fitness of the population after applying the selection scheme to a population with standardized normal distributed fitness (N(0, 1)).

selection differential S is given by the heritability h2: R(t) = h2S(t),

R(t) = h2σ(t)I(t).

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Proportionate selection

population at generation t,

at generation t is f(t),

applying proportionate selection: Pi(ts) = Pi(t) fi f(t) .

S(t) = f(ts) − f(t) =

Pi(ts)fi − f(t)

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=

Pi(t) f 2

f(t) − f(t) = 1 f(t) (f 2(t) − (f(t))2) = σ2(t) f(t) (i.e. Fisher’s Fundamental Theorem of Natural Selection)

N

σ2 =

(Xi − X)2 N = N

N − 2X N

N + NX

N

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= X2 − X

thus equal to the ratio of the standard deviation of the fitness and the population mean fitness: I(t) = σ(t)

the fitness variance between the individuals in the population reduces and/or if the mean fitness increases. Both typically happen at later generations when the population starts to loose its diversity. The selection pressure basically disappears.

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i i1 i2 i1 i1 i2 i3 i4 i2 i3 i1 fi 105 101 105 105 101 93 91 101 93 105 Mean fitness f(t) = 100, proportions: Pi(t), expected proportions after selection: Pi(ts), most likely number of copies: n(ts). i Pi(t) Pi(ts) n(ts) i1 0.4 0.420 4 i2 0.3 0.303 3 i3 0.2 0.186 2 i4 0.1 0.091 1 ⇒ most likely no change in number of copies !

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Selection intensity I(t) = σ(t)

Expected mean fitness after selection (i.e. the parent pool): f(ts) = f(t) + σ(t)I(t) = 100 + (5.31)(0.0531) = 100.28 .

when the fitness values are transformed with a constant term, for instance when changing temperature values from Celsius to Fahrenheit (F = 1.8 C + 32).

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Truncation Selection

fitness increase that is proportional to the standard deviation: f(ts) − f(t) = C(τ)σ,

follows that I(τ) = C(τ),

constant equal to the mean value of the right part of a standard normal distribution (f(t) = 0, σ(t) = 1), truncated at the top τ%. Values can be calculated or looked up in tables:

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τ 1% 10% 20% 40% 50% 80% I(τ) 2.66 1.76 1.2 0.97 0.8 0.34

truncation selection remains constant and is independent of the population mean fitness and variance (this is true for all ranked-based selection methods).

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Tournament Selection

random from the population, and select the solution with the best fitness.

expected value of the best ranked individual of a sample from s individuals taken from the standard normal distribution (f(t) = 0 and σ(t) = 1).

→ order statistics

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Order Statistics

random variables that are ordered (or ranked) according to their value.

certain distribution probability, and we sort the sample in increasing order of magnitude: x1:s ≤ x2:s ≤ . . . ≤ xs−1:s ≤ xs:s

the distribution of the corresponding value xi:s.

Xi:s gives the probability that the ith ranked individual of a sample of size s will have a value equal to x.

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φ(x) = 1 √ 2π e− x2

2

and the cumulative distribution is Φ(x) = x

φ(x)dx.

pi:s(x) = s s − 1 i − 1

ui:s = +∞

x pi:s(x) dx

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= s s − 1 i − 1 +∞

xφ(x)Φ(x)i−1(1 − Φ(x))s−idx

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Tournament Selection

best ranked individual of a sample of s individuals taken from the standard normal distribution: I(s) = us:s s 2 3 4 5 6 7 I(s) 0.56 (=

0.85 1.03 1.16 1.27 1.35

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Binomial/Normal Distributed Fitness Function

⇒ lower bound on convergence complexity

⇒ simple, yet accurate convergence models

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Counting Ones fitness function

X = x1 . . . xℓ, xi ∈ {0, 1} CO(X) =

xi

f(t) = l.p(t)

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Proportionate Selection: Counting Ones

p(t + 1) − p(t) = 1 l (1 − p(t)) dp(t) dt ≈ 1 l (1 − p(t))

p(t) = 1 − 0.5e−t/l

tconv = ℓ ln (ℓ)

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Figure 1: Convergence model and experimental results (ℓ = 100, N = 200) for the Bit Counting problem using proportionate selection (Stochastic Universal Sampling) and uniform crossover.

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Truncation Selection: counting ones

f(t + 1) − f(t) = σ(t)I(τ)

p(t + 1) − p(t) = I(τ) √ l

dp(t) dt ≈ I(τ) √ l

p(t) = 0.5(1 + sin ( I(τ)

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tconv = π

√ l) convergence time for truncation selection versus O(l ln (l)) for proportionate selection.

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Figure 2: Convergence model and experimental results of the propor- tion of optimal bit values for the Bit Counting problem using trunca- tion selection (pick 50 % best) and uniform crossover.

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Tournament Selection: counting ones

f(t + 1) − f(t) = σ(t)I(s),

p(t) = 0.5(1 + sin ( I(s)

tconv = π

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p(t) = 0.5(1 + sin (

tconv = π

√ πl

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Figure 3: Convergence model and experimental results of the propor- tion of optimal bit values for the Bit Counting problem using tourna- ment selection and uniform crossover.