Lec 7 genetic algorithms

2
Evolution
 Oversimplified description of how evolution works in biology:
 Organisms (animals or plants) produce a number of offspring
which are almost, but not entirely, like themselves
 Variation may be due to mutation (random changes)
 Variation may be due to sexual reproduction (offspring have some
characteristics from each parent)
 Some of these offspring may survive to produce offspring of their
own—some won’t
 The “better adapted” offspring are more likely to survive
 Over time, later generations become better and better adapted
 Genetic algorithms use this same process to “evolve” better
programs

3
Genotypes and phenotypes
 Genes are the basic “instructions” for building an
organism
 A chromosome is a sequence of genes
 Biologists distinguish between an organism’s genotype
(the genes and chromosomes) and its phenotype (what
the organism actually is like)
 Example: You might have genes to be tall, but never grow to
be tall for other reasons (such as poor diet)
 Similarly, “genes” may describe a possible solution to a
problem, without actually being the solution

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The basic genetic algorithm
 Start with a large “population” of randomly generated
“attempted solutions” to a problem
 Repeatedly do the following:
 Evaluate each of the attempted solutions
 Keep a subset of these solutions (the “best” ones)
 Use these solutions to generate a new population
 Quit when you have a satisfactory solution (or you run out of time)

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A really simple example
 Suppose your “organisms” are 32-bit computer words
 You want a string in which all the bits are ones
 Here’s how you can do it:
 Create 100 randomly generated computer words
 Count the 1 bits in each word
 Exit if any of the words have all 32 bits set to 1
 Keep the ten words that have the most 1s (discard the rest)
 From each word, generate 9 new words as follows:
 Pick a random bit in the word and toggle (change) it
 Note that this procedure does not guarantee that the next
“generation” will have more 1 bits, but it’s likely

7
A more realistic example, part I
 Suppose you have a large number of (x, y) data points
 For example, (1.0, 4.1), (3.1, 9.5), (-5.2, 8.6), ...
 You would like to fit a polynomial (of up to degree 5) through
these data points
 That is, you want a formula y = ax5 + bx4 + cx3 + dx2 +ex + f that gives
you a reasonably good fit to the actual data
 Here’s the usual way to compute goodness of fit:
 Compute the sum of (actual y – predicted y)2 for all the data points
 The lowest sum represents the best fit
 There are some standard curve fitting techniques, but let’s assume
you don’t know about them
 You can use a genetic algorithm to find a “pretty good” solution

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A more realistic example, part II
 Your formula is y = ax5 + bx4 + cx3 + dx2 +ex + f
 Your “genes” are a, b, c, d, e, and f
 Your “chromosome” is the array [a, b, c, d, e, f]
 Your evaluation function for one array is:
 For every actual data point (x, y), ( red to mean “actual data”)
 Compute ý = ax5 + bx4 + cx3 + dx2 +ex + f
 Find the sum of (y – ý)2 over all x
 The sum is your measure of “badness” (larger numbers are worse)
 Example: For [0, 0, 0, 2, 3, 5] and the data points (1, 12) and (2, 22):
 ý = 0x5 + 0x4 + 0x3 + 2x2 +3x + 5 is 2 + 3 + 5 = 10 when x is 1
 ý = 0x5 + 0x4 + 0x3 + 2x2 +3x + 5 is 8 + 6 + 5 = 19 when x is 2
 (12 – 10)2 + (22 – 19)2 = 22 + 32 = 13
 If these are the only two data points, the “badness” of [0, 0, 0, 2, 3, 5] is 13

9
A more realistic example, part III
 Your algorithm might be as follows:
 Create 100 six-element arrays of random numbers
 Repeat 500 times (or any other number):
 For each of the 100 arrays, compute its badness (using all data
points)
 Keep the ten best arrays (discard the other 90)
 From each array you keep, generate nine new arrays as
follows:
 Pick a random element of the six
 Pick a random floating-point number between 0.0 and 2.0
 Multiply the random element of the array by the random
floating-point number
 After all 500 trials, pick the best array as your final answer

10
Asexual vs. sexual reproduction
 In the examples so far,
 Each “organism” (or “solution”) had only one parent
 Reproduction was asexual (without sex)
 The only way to introduce variation was through mutation
(random changes)
 In sexual reproduction,
 Each “organism” (or “solution”) has two parents
 Assuming that each organism has just one chromosome, new
offspring are produced by forming a new chromosome from
parts of the chromosomes of each parent

11
The really simple example again
 Suppose your “organisms” are 32-bit computer words,
and you want a string in which all the bits are ones
 Here’s how you can do it:
 Create 100 randomly generated computer words
 Count the 1 bits in each word
 Exit if any of the words have all 32 bits set to 1
 Keep the ten words that have the most 1s (discard the rest)
 From each word, generate 9 new words as follows:
 Choose one of the other words
 Take the first half of this word and combine it with the
second half of the other word

12
The example continued
 Half from one, half from the other:
0110 1001 0100 1110 1010 1101 1011 0101
1101 0100 0101 1010 1011 0100 1010 0101
0110 1001 0100 1110 1011 0100 1010 0101
 Or we might choose “genes” (bits) randomly:
0110 1001 0100 1110 1010 1101 1011 0101
1101 0100 0101 1010 1011 0100 1010 0101
0100 0101 0100 1010 1010 1100 1011 0101
 Or we might consider a “gene” to be a larger unit:
0110 1001 0100 1110 1010 1101 1011 0101
1101 0100 0101 1010 1011 0100 1010 0101
1101 1001 0101 1010 1010 1101 1010 0101

13
Comparison of simple examples
 In the simple example (trying to get all 1s):
 The sexual (two-parent, no mutation) approach, if it succeeds,
is likely to succeed much faster
 Because up to half of the bits change each time, not just one bit
 However, with no mutation, it may not succeed at all
 By pure bad luck, maybe none of the first (randomly generated) words
have (say) bit 17 set to 1
 Then there is no way a 1 could ever occur in this position
 Another problem is lack of genetic diversity
 Maybe some of the first generation did have bit 17 set to 1, but
none of them were selected for the second generation
 The best technique in general turns out to be sexual
reproduction with a small probability of mutation

14
Curve fitting with sexual reproduction
 Your formula is y = ax5 + bx4 + cx3 + dx2 +ex + f
 Your “genes” are a, b, c, d, e, and f
 Your “chromosome” is the array [a, b, c, d, e, f]
 What’s the best way to combine two chromosomes into
one?
 You could choose the first half of one and the second half of
the other: [a, b, c, d, e, f]
 You could choose genes randomly: [a, b, c, d, e, f]
 You could compute “gene averages:”
[(a+a)/2, (b+b)/2, (c+c)/2, (d+d)/2, (e+e)/2,(f+f)/2]
 The last may be the best, though it is difficult to know of a good
biological analogy for it

Three main types of rules
 The genetic algorithm uses three main types of
rules at each step to create the next generation
from the current population:
• Selection rules select the individuals, called parents, that
contribute to the population at the next generation.
• Crossover rules combine two parents to form children
for the next generation.
• Mutation rules apply random changes to individual
parents to form children.
15

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Directed evolution
 Notice that, in the previous examples, we formed the
child organisms randomly
 We did not try to choose the “best” genes from each parent
 This is how natural (biological) evolution works
 Biological evolution is not directed—there is no “goal”
 Genetic algorithms use biology as inspiration, not as a set of
rules to be slavishly followed
 For trying to get a word of all 1s, there is an obvious
measure of a “good” gene
 But that’s mostly because it’s a silly example
 It’s much harder to detect a “good gene” in the curve-fitting
problem, harder still in almost any “real use” of a genetic
algorithm

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Probabilistic matching
 In previous examples, we chose the N “best” organisms
as parents for the next generation
 A more common approach is to choose parents
randomly, based on their measure of goodness
 Thus, an organism that is twice as “good” as another is likely
to have twice as many offspring
 This has a couple of advantages:
 The best organisms will contribute the most to the next
generation
 Since every organism has some chance of being a parent, there
is somewhat less loss of genetic diversity

18
Genetic programming
 A string of bits could represent a program
 If you want a program to do something, you might try to
evolve one
 As a concrete example, suppose you want a program to
help you choose stocks in the stock market
 There is a huge amount of data, going back many years
 What data has the most predictive value?
 What’s the best way to combine this data?
 A genetic program is possible in theory, but it might
take millions of years to evolve into something useful
 How can we improve this?

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Concluding remarks
 Genetic algorithms are—
 Fun! They are enjoyable to program and to work with
 This is probably why they are a subject of active research
 Mind-bogglingly slow—you don’t want to use them if you
have any alternatives
 Good for a very few types of problems
 Genetic algorithms can sometimes come up with a solution when you
can see no other way of tackling the problem

Lec 7 genetic algorithms

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Lec 7 genetic algorithms