Genetic Algorithms A technique for those who do not know how to solve the problem!

Selection Methods Fitness-Proportionate Selection –Roulette Wheel –Stochastic Universal Sampling Rank Selection Tournament Selection Steady-State Selection Sigma Scaling Elitism

Fitness Proportionate Selection (Roulette Wheel) Used by Holland’s original GA Here the number of times an individual is expected to reproduce is equal to individual_Fitness/AveFitness= N* P i Method 1. Sum the total expected values of individuals in Pop. Call this sum T 2. Repeat N times where N is the number of individuals in Pop. Choose a random integer r between 0 and T Loop through the individuals in the pop., summing the expected values, until the sum is greater than or equal to r. The individual whose expected value puts the sum over r is the one selected. Faults: Selection in small populations is often far from the expected values.

Stochastic Universal In order to generate individuals that better follow their expected values we can use Stochastic Universal Sampling (SAS) Here we make one call to rand() and select N equally spaced individuals from the population The Roulette Wheel method made N random calls to select N individuals Start with a random number between 1 and 1/N

Sigma Scaling Fitness proportionate selection suffers from premature convergence. More emphasis is on exploitation as apposed to exploration. Sigma Scaling addresses this problem. This keeps the selection pressure relatively constant over a run. ExpVal(i,t) = 1+ (fitness(i)-mean(t))/2*SD(t) If SD(t) <>0 otherwise ExpVal(i,t)=1.0; This, for example, gives an individual whose fitness is one SD above the mean 1.5 offspring out of N.

Elitism First used by Kenneth De Jong(1975) Here the best individuals are carried over to the new population. This often significantly improves the GA’s performance. In GAlib we set this with the command ga.elitist(gaTrue); The best individual is copied over.

Rank Selection This scheme is designed to prevent to- quick convergence. Here the rank of an individual is used, instead of its absolute fitness, in the selection process. The probability of selection is Pi = (N-i+1)/( N) Select by generating the array of partial sums and displace into it.

Tournament Selection Fitness proportionate methods require two passes through the population Rank scaling requires sorting. Here k (often two) individuals are chosen from a population. The best is selected and inserted into the population. Do this N times

Steady-State Selection This scheme is used when we would like the two populations to overlap. Here a percentage of the old population is first copied to the new population. The remainder of the population is then filled using crossover etc. The fraction of the new individuals at each generation is called the “generation gap”

GAlib’s Selection Scheme Constructors GARankSelector –The rank selector picks the best member of the population every time. GARouletteWheelSelector –This selection method picks an individual based on the magnitude of the fitness score relative to the rest of the population. The higher the score, the more likely an individual will be selected. Any individual has a probability p of being chosen where p is equal to the fitness of the individual divided by the sum of the fitnesses of each individual in the population.

More Selection Schemes GATournamentSelector –The tournament selector uses the roulette wheel method to select two individuals then picks the one with the higher score. The tournament selector typically chooses higher valued individuals more often than the RouletteWheelSelector. GADSSelector –The deterministic sampling selector (DS) uses a two-staged selection procedure. In the first stage, each individual's expected representation is calculated. A temporary population is filled using the individuals with the highest expected numbers. Any remaining positions are filled by first sorting the original individuals according to the decimal part of their expected representation, then selecting those highest in the list. The second stage of selection is uniform random selection from the temporary population.

More Selection Schemes GASRSSelector –The stochastic remainder sampling selector (SRS) uses a two- staged selection procedure. In the first stage, each individual's expected representation is calculated. A temporary population is filled using the individuals with the highest expected numbers. Any fractional expected representations are used to give the individual more likelihood of filling a space. For example, an individual with e of 1.4 will have 1 position then a 40% chance of a second position. The second stage of selection is uniform random selection from the temporary population. GAUniformSelector –The stochastic uniform sampling selector picks randomly from the population. Any individual in the population has a probability p of being chosen where p is equal to 1 divided by the population size.

GAlib Scaling Constructors GANoScaling() –The fitness scores are identical to the objective scores. No scaling takes place. GALinearScaling(float c = gaDefLinearScalingMultiplier) –The fitness scores are derived from the objective scores using the linear scaling method described in Goldberg's book. You can specify the scaling coefficient. Negative objective scores are not allowed with this method. Objective scores are converted to fitness scores using the relation f = a * obj + b where a and b are calculated based upon the objective scores of the individuals in the population as described in Goldberg's book. f obj a*obj+b

More Scalings GASigmaTruncationScaling(float c = gaDefSigmaTruncationMultiplier) Use this scaling method if your objective scores will be negative. It scales based on the variation from the population average and truncates arbitrarily at 0. The mapping from objective to fitness score for each individual is given by f = obj - (obj_ave - c * obj_dev) GAlib Usage GASigmaTruncationScaling sigmaTruncation; //Declare object ga.scaling(sigmaTruncation);

More Scalings GAPowerLawScaling(int k = gaDefPowerScalingFactor) –Power law scaling maps objective scores to fitness scores using an exponential relationship defined as f = obj ^ k GASharing(GAGenomeComparator func = 0, float cutoff = gaDefSharingCutoff, float alpha = 1) This scaling method is used to do speciation. The fitness score is derived from its objective score by comparing the individual against the other individuals in the population. If there are other similar individuals then the fitness is derated. The distance function is used to specify how similar to each other two individuals are. A distance function must return a value of 0 or higher, where 0 means that the two individuals are identical (no diversity).

Crossover Schemes available for GA1DArrayGenome There are many crossover methods built into GAlib. Generally they are genome specific. 1.Single point crossover 2.Two point crossover 3.Uniform Crossover 4.EvenOdd Crossover 5.Partial Match Crossover 6.Order Crossover 7.CycleCrossover.

Two Point Crossover Chromosome 1: 11011*0010* Chromosome 2: 01011*1100* Offspring 1: 11011*1100* Offspring 2: 01011*0010* =+

Uniform Crossover In this method each gene of the offspring is selected randomly from the corresponding genes of the parents. One-point and two-point crossover produce two offspring, whilst uniform crossover produces only one.

Creating your own Crossover You can write your own crossover that is specific to your genome. In your GA you announce to the genome that you have done this by using the following command Genome.crossover(MyCrossover); You then must write the code for MyCrossover.

GAlib Sexual Crossover Sexual crossover takes four arguments: two parents and two children. If one child is nil, the operator should be able to generate a single child. The genomes have already been allocated, so the crossover operator should simply modify the contents of the child genome as appropriate. The crossover function should return the number of crossovers that occurred. Your crossover function should be able to operate on one or two children, so be sure to test the child pointers to see if the genetic algorithm is asking you to create one or two children.

Example Crossover int MyCrossover(const GAGenome& p1, const GAGenome& p2, GAGenome* c1, GAGenome* c2){ GA1DBinaryStringGenome &mom=(GA1DBinaryStringGenome &)p1; GA1DBinaryStringGenome &dad=(GA1DBinaryStringGenome &)p2; int n=0; unsigned int site = GARandomInt(0, mom.length()); unsigned int len = mom.length() - site; if(c1){ GA1DBinaryStringGenome &sis=(GA1DBinaryStringGenome &)*c1; sis.copy(mom, 0, 0, site); sis.copy(dad, site, site, len); n++; } if(c2){ GA1DBinaryStringGenome &bro=(GA1DBinaryStringGenome &)*c2; bro.copy(dad, 0, 0, site); bro.copy(mom, site, site, len); n++; } return n; }

Permutation Crossovers Required for TSP Required for Decoding messages etc Random crossover of two permutations seldom result in another permutation A permutation space is N! in size. SO!

Categories of Perm. Crossovers Disqualification –Just kill the bad chromosomes. Why is this bad? Repairing –Invalid chromosomes are fixed. Inventing Specialized Operators –Crossovers generate only legal permutations Transformation –Transform permutation space into a vector space and cross in vector space.

Permutation Operators Partially mapped crossover (PMX) Order crossover (X) Uniform order crossover Edge recombination There are many other that we will not discuss.

Partially Mapped Crossover (Goldbert & Lingle, 1985) Given two parents s and t, PMX randomly picks two crossover points. The child is constructed in the following way. Starting with a copy of s, the positions between the crossover points are, one by one, set to the values of t in these positions. This is performs by applying a swap to s. The swap is defined by the corresponding values in s and t within the selected region.

PMX example No change For the second offspring just swap the parents and apply the same operation First offspring

Order Crossover (Davis 1985) This crossover first determines to crossover points. It then copies the segment between them from one of the parents into the child. The remaining alleles are copied into the child (l to r) in the order that they occur in the other parents. Switching the roles of the parents will generate the other child.

Order Crossover Example The remaining alleles are Their order in the other parent is

Uniform Order Crossover (Davis 1991) Here a randomly-generated binary mask is used to define the elements to be taken from that parent. The only difference between this and order crossover is that these elements in order crossover are contiguous offspring

Edge Recombination (Whitley Starkweather Fuquay 1989 ) This operator was specially designed for the TSP problem. This scheme ensures that every edge (link) in the child was shared with one or other of its parents. This has been shown to be very effective in TSP applications. Constructs an edge map, which for each site lists the edges available to it from the two parents that involve that city. Mark edges that occur in both with a +.

Example Edge Table g d m h b j f i a k e c c e k a g b h i j f m d a: +k, g,i g: a, b, c, db: +h,g,i h: +b, i, mc: +3, d, gi: h, j, a, f d: +m, g, cj: +f, i, be: +k, +c k: +e, +af: +j, m, i m: +d, f, h

Edge Recombination Algorithm Pick a city at random Set current_city to this city. Remove reference to current_city form table. Examine list for current_city: –If there is a common entry(+) pick that –Else pick entry which has the shortest list –Split ties randomly If stuck (list is empty), start from other end, or else pick a new city at random.

Example Continued Randomly pick a, delete all a’s from table [a] Select k (common neighbor)[ak] Select e (only item in k’s list)[ake] Select c (only item in e’s list)[akec] d or g: pick d at random[akecd] Select m (common edge with d)[akecdm] f or h: pick h at random[akecdmh] Select b ( common edge)[akecdmhb] Select g (shortest list -0)[akecdmhbg] g has empty list so reverse direction [gbhmdcdka] Select i (only item in a’s list)[gbhmdcdkai] Select f at random, then j[gbhmdcekaifj]

Inversion Transformations This scheme will allow normal crossover and mutation to operate as usual. In order to accomplish this we map the permutation space to a set of contiguous vectors. Given a permutation of the set {1,2,3,…,N} let a j denote the number of integers in the permutation which precede j but are greater than j. The sequence a 1,a 2,a 3,…,a n is called the inversion sequence of the permutation. The inversion sequence of is There are 4 integers greater than 1

Inversion of Permutations The inversion sequence of a permutation is unique! Hence there is a 1-1 correspondence between permutations and their inversion sequence. Also the right most inv number is 0 so dropped x y (0 0) (0 1) (1 1) (2 1) (2 0) (1 0)

Inversions Continued What does a 4 digit permutation map to? > (0 0 0) > (1 0 0) > (3 2 1) > (2 0 1) > (0 1 1) etc Maps to a partial 3D lattice structure

Converting Perm to Inv Input perm: array of permutation Output: inv: array holding inv sequence For (i=1;i<=N;i++){ inv[i]=0; m=1; while(perm[m]<>i){ if (perm[m]>i )then inv[i]++; m++; }

Convert inv to Perm Input: inv[] Output: perm[] For(i=1;i<=N;i++){ for(m=i+1;m<=N;m++) if (pos[m]>=inv[i]+1)pos[m]++; pos[i]=omv[i]+1; } For(i=1;i<=N;i++) perm[i]=i;

So what do we do? Our population is of course a set of permutations. These permutations are each mapped to their inv to create a population of inv’s say We do normal crossovers in this mapped population as well as normal mutations. In order to determine fitness we of course must apply Fitness(Inverse(inv)) Is this all worth doing?

Mutations of Permutations Swap Mutation Scramble Mutation 2-Swap Insert These will maintain legal permutations

Swap Mutation Select two positions at random and swap the allele values at those positions. Sometimes called the “order-based” mutation. ABCDEFGJ => AECDBFGJ

Scramble Mutation Pick a subset of positions at random and reorder their contents randomly Some research has shown swap is best and others have shown scramble is best in certain apps. Who knows? ABCDEFGH => AHFDECGB

Other Permutation Mutations 2-Swap (nice for TSP) –Pick two point and invert subtour –AB.CDEF.GH => AB.FEDC.GH Insert Mutation –Pick a value at random (say E), insert into another (rand chosen position, say B) and shift the rest over –ABCDEFG => AEBCDFG

How about code breaking Assume that we have the 26 letters of the alphabet permutated. This permutation is used to encode a normal message. How do we decode this using a GA? Is this even a good idea? What is the fitness function?

Encoding “ABCDEFGHIJKLMNOPQRSTUVWXYZ “ “TUHNIXWAVBJQCDPZ_MOSYLRKEFG” The decoding string is just a permutation of the original string. Hence “NOW IS THE TIME” encodes to “DPKGV etc “

Fitness? The textbook assumes that it know the answer during the lookup phase. What if you don’t know the answer? One possibility involves the use of a dictionary. When you attempt a decoding and get something like AVE IS HEI TIME You can use the spaces as a separators and access the dictionary for each “word”. In this case IS and TIME are found and hence increase its fitness.

Can we evolve the equation of gravity (F= m 1 m 2 /r 2 ) k g is omitted. The first question is how do we represent a function like F in a chromosome? By using trees of course. Expression trees to be precise. Expressions (A+B)*C-D and m 1 m 2 /r 2 easily represented as trees although several different trees may work. Why?

(A+B)*C-D m 1 m 2 /r 2 + AB * C - D / ** m1m1 m2m2 RR

Things to note Interior nodes are operators while leaf nodes are variables or constants. Some operators may be unary such as SQRT as well as the usual binary operators. If trees are used as a chromosome then specialized operators for mutation and crossover need to be developed. GALib has support for trees GATreeGenome

Tree Mutation GALib as well as many other uses the swap node mutator as well as a swap tree mutator. Swap node swaps the contents of the two specified nodes. Sub-trees connected to either node are not affected; only the specified nodes are swapped. Swap tree swaps the contents of the two specified nodes as well as any sub-trees connected to the specified nodes. Of course you can define anything that you want.

An Example Tree Crossover * A- SQR * A AA / A / A/ // AAAA * A- * AA A // A/ // AAA AA Children

Fitness We can calculate the fitness of a function by running it on a set of “fitness cases” These are a set of inputs for which the correct output is known. For example in the case of gravity we can build a set of triples (m1,m2,r,F) which represent the two masses, the distance between them and the resulting force between them.

John Koza’s work(1992,1994) (Genetic Programming GP) Koza has used schemes such as just discussed to evolve Lisp programs A Lisp function is really a preorder listing of the expression tree. He used 10% population overlap Kozo typically does not use mutation, instead he builds large initial populations with the (he hopes) necessary diversity. Chunking has also been addressed by some of his later research. Chunking is a mechanism for automatically chunking parts of a chromosome so they will not be split up under crossover. (IE subroutines?)

Questions about GP’s Will the technique scale up to more complex cases (bigger programs) What if the function and variable set is large? GP often finds a function that satisfies the test cases but when applied to remaining data will not work.