Download Genetic Algorithms

Genetic Algorithms
7-May-17
Evolution
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Here’s a very 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
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Some of these offspring may survive to produce offspring of their
own—some won’t
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Variation may be due to mutation (random changes)
Variation may be due to sexual reproduction (offspring have some
characteristics from each parent)
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
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Genotypes and phenotypes
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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)
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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
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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
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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:
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Create 100 randomly generated computer words
Repeatedly do the following:
 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
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A more realistic example, part I
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Suppose you have a large number of (x, y) data points
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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
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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:
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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
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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:
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For every actual data point (x, y), (I’m using red to mean “actual data”)
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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):
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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
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A more realistic example, part III
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Your algorithm might be as follows:
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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
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Asexual vs. sexual reproduction
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In the examples so far,
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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,
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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
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The really simple example again
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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:
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Create 100 randomly generated computer words
Repeatedly do the following:
 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
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The example continued
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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
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Comparison of simple examples
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In the simple example (trying to get all 1s):
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The sexual (two-parent, no mutation) approach, if it succeeds,
is likely to succeed much faster
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However, with no mutation, it may not succeed at all
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Because up to half of the bits change each time, not just one bit
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
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Curve fitting with sexual reproduction
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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?
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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]
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This doesn’t happen directly in nature
However, many traits are controlled by multiple genes, so you could
get more or fewer of the relevant genes
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Directed evolution
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Notice that, in the previous examples, we formed the
child organisms randomly
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We did not try to choose the “best” genes from each parent
This is how natural (biological) evolution works
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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
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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
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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
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Thus, an organism that is twice as “good” as another is likely
to have twice as many offspring
This has a couple of advantages:
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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
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Genetic programming
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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
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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
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How can we improve this?
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Shrinking the search space
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There are just too many possible bit patterns!
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An incredible improvement would result if we could
somehow restrict the search space to only valid (even if
nonsensical) programs
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99.9999% of these don’t even represent valid programs
We can do this!
Programs, as you should know by now, can be
represented as trees
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Internal nodes are operators: +, *, if, while, ...
Leaves are values: 2.71818, "AAPL", ...
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Programs as trees
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Given a program represented as a tree, we can mutate it
by changing one of its operators (or one of its values),
or by adding or removing nodes
Given two trees, we can form a new tree by taking parts
of its two parents
The next big problem: How do we evaluate program
trees that are (initially) nothing at all like what we
want?
I realize this is all very vague—I just wanted to give
you the general idea
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Concluding remarks
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Genetic algorithms are—
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Fun! They are enjoyable to program and to work with
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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
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Genetic algorithms can sometimes come up with a solution when you
can see no other way of tackling the problem
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The End
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