Transcript Biologically Inspired Computing

Biologically Inspired Computing:
Introduction to Evolutionary
Algorithms
This is lecture four of
`Biologically Inspired Computing’
Contents:
Local Search
Encodings
Typical Landscapes
Unimodal
Multimodal
Plateau
Deceptive
As we home in on the good areas, we can identify broad types of
Landscape feature.
Most landscapes of interest are predominantly multimodal.
Despite being locally smooth, they are globally rugged
Beyond Hillclimbing
HC clearly has problems with typical landscapes:
There are two broad ways to improve HC, from the
algorithm viewpoint:
1. Allow downhill moves – a family of methods
called Local Search does this in various ways.
2. Have a population – so that different regions can
be explored inherently in parallel – I.e. we keep
`poor’ solutions around and give them a chance
to `develop’.
Local Search
Initialise:
Generate a random solution c; evaluate its
fitness, f(s) = b; call c the current solution,
and call b the best so far.
Repeat until termination conditon reached:
1. Search the neighbourhood of c, and choose one, m
Evaluate fitness of m, call that x.
2. According to some policy, maybe replace c with x, and
update c and b as appropriate.
E.g. Monte Carlo search: 1. same as hillclimbing; 2. If x is better,
accept it as new current solution;if x is worse, accept it with some
probabilty (e.g. 0.1).
E.g. tabu search: 1. evaluate all immediate neighbours of c
2. choose the best from (1) to be the next current solution, unless it is
`tabu’ (recently visited), in which choose the next best, etc.
Population-Based Search
• Local search is fine, but tends to get stuck in local
optima, less so than HC, but it still gets stuck.
• In PBS, we no longer have a single `current
solution’, we now have a population of them. This
leads directly to the two main algorithmic
differences between PBS and LS
– Which of the set of current solutions do we mutate? We
need a selection method
– With more than one solution available, we needn’t just
mutate, we can [mate, recombine, crossover, etc …]
two or more current solutions.
• So this is an alternative route towards motivating
our nature-inspired EAs – and also starts to
explain why they turn out to be so good.
Basics of Encodings
Encoding / Representation
Maybe the main issue in (applying) EC
Note that:
• Given an optimisation problem to solve, we need
to find a way of encoding candidate solutions
• There can be many very different encodings for
the same problem
• Each way affects the shape of the landscape and
the choice of best strategy for climbing that
landscape.
E.g. encoding a timetable I
9:00
mon
tue
E4,
E5
E2
wed
thur
E3,
E7
11:00 E8
4, 5, 13, 1, 1, 7, 13, 2
2:00
E6
4:00 E1
Exam2 in 5th slot
Exam1 in 4th slot
Etc …
• Generate any string of 8 numbers between 1 and 16,
and we have a timetable!
• Fitness may be <clashes> + <consecs> + etc …
• Figure out an encoding, and a fitness function, and
you can try to evolve solutions.
Mutating a Timetable with Encoding 1
9:00
mon
tue
E4,
E5
E2
11:00 E8
4, 5, 13, 1, 1, 7, 13, 2
2:00
4:00
E6
E1
Using straightforward single-gene mutation
Choose a random gene
wed
thur
E3,
E7
Mutating a Timetable with Encoding 1
mon
tue
E4,
E5
E2
11:00 E8
E3
2:00
E6
9:00
4, 5, 6 , 1, 1, 7, 13, 2
4:00
E1
Using straightforward single-gene mutation
One mutation changes position of one exam
wed
thur
E7
Alternative ways to do it
This is called a `direct’ encoding. Note that:
• A random timetable is likely to have lots of
clashes.
• The EA is likely (?) to spend most of its
time crawling through clash-ridden areas of
the search space.
• Is there a better way?
E.g. encoding a timetable II
9:00
mon
tue
E4,
E5
E2
11:00 E8
4, 5, 13, 1, 1, 7, 13, 2
2:00
4:00
E6
E1
Etc …
Use the 13th clash-free slot for exam3
Use the 5th clash-free slot for exam2
Use the 4th clash-free slot for exam1
wed
thur
E3,
E7
E.g. encoding a timetable III
9:00
H1, H3, H2, H2, H1, H1 …
mon
tue
E4,
E5
E2
wed
thur
E3,
E7
11:00 E8
2:00
4:00
E6
E1
Etc …
Use heuristic H2 to schedule exam3 (reduce future conflicts)
Use heuristic H3 to schedule exam2 (best fit based on consecs)
Use heuristic H1 to schedule exam1 (first fit)
One of the very first applications. Determine the internal
shape of a two-phase jet nozzle that can achieve the
maximum possible thrust under given starting conditions
Ingo Rechenberg was the very first, with pipe-bend design
This is slightly later work in the same lab, by Schwefel
Starting point
EA (ES) running
Result
A recurring theme: design freedom  entirely new and better designs
based on principles we don’t yet understand.
A Real Encoding (and: How EAs can
innovate, rather than just optimize)
D1,
D2,
D3,
D4
D5
D6
D1 >= D2 >= D3, D4 <= D5 <= D6
Fixed at six diameters, five sections
E.g. How EAs can innovate,
rather than just optimize
2,
1.8,
1.1,
1.3
1.3
1.5
D1 >= D2 >= D3, D4 <= D5 <= D6
Fixed at six diameters, five sections
E.g. How EAs can innovate,
rather than just optimize
Num sections before smallest
Z1, Z2,
Section diameters
D1, D2, D3 Dsmall…, Dn, Dn+1, …
Num sections after smallest
Middle section constrained to be smallest,
That’s all
Mutations can change diameters, add sections,
and delete sections
Constructive Methods
Problems like timetabling, scheduling, and
other `logistics’ activities are often `solved’
in practice via constructive heuristics, These
are also called greedy heuristics. A
constructive method is a technique that
builds a single solution step by step, trying
to be clever (often) about each step.
Examples
Prim’s algorithm for building the minimal spanning
tree (see an earlier lecture) is an example.
Djikstra’s shortest path algorithm is also an example.
In both of these cases, the optimal solution is
guaranteed to be found, since MST and SP are
easy problems.
But usually we see constructive methods used to
give very fast `OK’ solutions to hard problems.
A constructive method for the TSP
Start with a random current city c; mark c as
visited:
Initialise Tour = {} (empty)
Repeat ncities-1 times:
choose, BTR, the closest unvisited city to c
(call it d)
add the edge cd to Tour
mark d as visited
Let d be the current city
Try it yourself a few times. Can you construct examples
where this will give a very poor result?
A constructive method for exam timetabling
Repeat nexams times:
choose an exam, e, randomly.
let V be the set of valid timeslots for e – I.e.
slots it can go in without introducing a clash.
If V is empty, mark e as unplaced
Else choose random slot t from V, and assign e to t.
Is this how people do timetabling, or is there an even better
way?
A (usually) better constructive method for
exam timetabling
Assign a difficulty score to each exam – e.g. this could
be the number of other exams with which it clashes.
Repeat nexams times:
choose an unscheduled exam e with highest difficulty,BTR.
Find V, the set of slots it can go in without introducing
a clash.
If V is empty, mark e as unplaced
Else for each slot in V, find its usability score – e.g.
the number of unplaced exams that could go in that slot
without introducing a clash
Choose a slot t with minimal usability score.
Assign e to t.
Back to encoding …
We can use constructive methods as encodings in the
following sort of way; this is sometimes called a
`hybrid’ approach.
The EA searches through the space of orderings of
items (e.g. exams to schedule, edges to put in a graph,
etc…).
When evaluating fitness, a constructive method builds a
solution using the ordering provided in the
chromosome, and then evaluates fitness in the normal
way.
Think about these things
How could you design a `smart’ mutation operator
for the direct timetable encoding?
(hint – when you’ve randomly chosen a gene to
mutate, can you do better than give it a random
new slot?)
How could you design a smart mutation operator for
the indirect timetable encoding?
(hint – hard)
Direct vs Indirect Encodings
Direct:
• straightforward genotype (encoding)  phenotype (individual) mapping
• Easy to estimate effects of mutation
• Fast interpretation of chromosome (hence speedier fitness evlaluation)
Indirect/Hybrid:
• Easier to exploit domain knowledge – (e.g. use this in the constructive
heuristic)
• Hence, possible to `encode away’ undesirable features
• Hence, can seriously cut down the size of the search space
• But, slow interpretation
• Neighbourhoods are highly rugged.
Back to Bin-Packing
The bin-packing encoding that you will use in your
assignment is a direct one.
But there are some well-known constructive heuristics for
bin-packing: the following ones are used when the bins
have fixed capaities, and the problem is to pack the items
into the smallest number of bins:
First-fit-random (FFR):
Repeat nitems times:
Choose an item i randomly and place it in the
first bin in which it will fit.
First-fit-descending (FFD):
Order the items from heaviest to lightest (BTR)
For each item in order: place it into the first
bin in which it will fit.
How might you invent an indirect encoding for bin-packing?
An important aside about
constructive methods
Some Constructive Heuristics are deterministic. I.e. they give
the same answer each time.
Some are stochastic – I.e. they may give a different solution
in different runs.
Usually, if we have a deterministic constructive method such
as FFD, we can engineer a stochastic version of it. E.g.
instead of choosing the next-lightest item in each step, we
might choose randomly between the lightest three unplaced
items.
When applying EAs, it is often found that a stochastic
constructive heuristic is very useful for building an initial
population. But care has to be taken with such an approach
– why?
This week’s additional material
Further encodings:
Grouping problems, Rules, Trees.