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Basics of Hillclimbing in Biologically Inspired Computing

This lecture introduces the concepts of hillclimbing and local search in biologically inspired computing. It explores the limitations and design strategies that can lead to more effective algorithms. A simplified version of the hillclimbing algorithm is discussed using the Travelling Salesperson Problem as an example.

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Basics of Hillclimbing in Biologically Inspired Computing

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  1. Biologically Inspired Computing: Evolutionary Algorithms: Hillclimbing, Landscapes, Neighbourhoods This is lecture 4 of `Biologically Inspired Computing’ Contents: hillclimbing, local search, landscapes

  2. Back to Basics With your thirst for seeing example EAs temporarily quenched, the story now skips to simpler optimization algorithms. 1. Hillclimbing 2. Local Search These have much in common with EAs, but with no population – they just use a single solution and keep trying to improve it. HC and LS can be very effective algorithms when appropriately engineered. But by looking at them, we will discover certain limitations, and this leads us directly to algorithm design strategies that look like Eas. That is, we can ‘arrive’ at EAs from an algorithm design route, not just a ‘bio-inspiration’ route.

  3. The Travelling Salesperson Problem(also seehttp://francisshanahan.com/tsa/tsaGAworkers.htm ) An example (hard) problem, for illustration The Travelling Salesperson Problem Find the shortest tour through the cities. The one below is length: 33 B C E D A

  4. Simplest possible EA: Hillclimbing 0. Initialise: Generate a random solution c; evaluate its fitness, f(c). Call c the current solution. 1. Mutate a copy of the current solution – call the mutant m Evaluate fitness of m, f(m). 2. If f(m) is no worsethan f(c), then replace c with m, otherwise do nothing (effectively discarding m). 3. If a termination condition has been reached, stop. Otherwise, go to 1. Note. No population (well, population of 1). This is a very simple version of an EA, although it has been around for much longer.

  5. Why “Hillclimbing”? Suppose that solutions are lined up along the x axis, and that mutation always gives you a nearby solutions. Fitness is on the y axis; this is a landscape 9 6 10 7 5, 8 4 3 1 2 • Initial solution; 2. rejected mutant; 3. new current solution, 4. New current solution; 5. new current solution; 6. new current soln 7. Rejected mutant; 8. rejected mutant; 9. new current solution, 10. Rejected mutant, …

  6. Example: HC on the TSP We can encode a candidate solution to the TSP as a permutation Here is our initial random solution ACEDB with fitness 32 Current solution B C E D A

  7. Here is our initial random solution ACEDB with fitness 32. We are going to mutate it – first make Mutant a copy of Current. Example: HC on the TSP We can encode a candidate solution to the TSP as a permutation Current solution Copy of current… B B C C E D E A D A

  8. HC on the TSP We randomly mutate it (swap randomly chosen adjacent nodes) from ACEDB to ACDEB which has fitness 33 -- so current stays the same Because we reject this mutant. Current solution Mutant B B C C E D E A D A

  9. HC on the TSP We now try another mutation of Current (swap randomly chosen adjacent nodes) from ACEDB to CAEDB. Fitness is 38, so reject that too. Current solution Mutant B B C C E D E A D A

  10. HC on the TSP Our next mutant of Current is from ACEDB to AECDB. Fitness 33, reject this too. Current solution Mutant B B C C E D E A D A

  11. HC on the TSP Our next mutant of Current is from ACEDB to ACDEB. Fitness 33, reject this too. Current solution Mutant B B C C E D E A D A

  12. HC on the TSP Our next mutant of Current is from ACEDB to ACEBD. Fitness is 32. Equal to Current, so this becomes the new Current. Current solution Mutant B B C C E D E A D A

  13. B C E D A HC on the TSP ACEBD is our Current solution, with fitness 32 Current solution

  14. B C E D A HC on the TSP ACEBD is our Current solution, with fitness 32. We mutate it to DCEBA (note that first and last are adjacent nodes); fitness is 28. So this becomes our new current solution. Current solution Mutant B C E D A

  15. HC on the TSP Our new Current, DCEBA, with fitness 28 Current solution B C E D A

  16. HC on the TSP Our new Current, DCEBA, with fitness 28 . We mutate it, this time getting DCEAB, with fitness 33 – so we reject that and DCEBA is still our Current solution. Mutant Current solution B B C C E E D D A A

  17. And so on …

  18. HC again, on a 1D ‘landscape’ FITNESS x - the ‘genotype’

  19. Initial random point FITNESS x - the ‘genotype’

  20. mutant FITNESS x - the ‘genotype’

  21. Current solution (unchanged) FITNESS x - the ‘genotype’

  22. Next mutant FITNESS x - the ‘genotype’

  23. New current solution FITNESS x - the ‘genotype’

  24. Next mutant FITNESS x - the ‘genotype’

  25. New current solution FITNESS x - the ‘genotype’

  26. ETC ………………………… FITNESS x - the ‘genotype’

  27. How will HC do on this landscape?

  28. Some other landscapes

  29. Landscapes f(s) Recall S, the search space, and f(s), the fitness of a candidate in S members of S lined up along here The structure we get by imposing f(s) on S is called a landscape

  30. Neighbourhoods f(s) Recall S, the search space, and f(s), the fitness of a candidate in S 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Showing the neighbourhoods of two candidate solutions, assuming The mutation operator adds a random number between −1 and 1

  31. Neighbourhoods f(s) Recall S, the search space, and f(s), the fitness of a candidate in S 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Showing the neighbourhood of a candidate solutions, assuming The mutation operator adds a random integer between −2 and 2

  32. Neighbourhoods f(s) Recall S, the search space, and f(s), the fitness of a candidate in S 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Showing the neighbourhood of a candidate solutions, assuming the mutation operator simply changes the solution to a new random Number between 0 and 20

  33. Neighbourhoods f(s) Recall S, the search space, and f(s), the fitness of a candidate in S 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Showing the neighbourhood of a candidate solution, assuming the mutation operator adds a Gaussian (ish) with mean zero.

  34. Neighbourhoods Let s be an individual in S, f(s) is our fitness function, and M is our mutation operator, so that M(s1)  s2, where s2 is a mutant of s1. Given M, we can usually work out the neighbourhood of an individual point s – the neighbourhood of s is the set of all possible mutants of s E.g. Encoding: permutations of k objects (e.g. for k-city TSP) Mutation: swap any adjacent pair of objects. Neighbourhood: Each individual has k neighbours. E.g. neighbours of EABDC are: {AEBDC, EBADC, EADBC, EABCD, CABDE} Encoding: binary strings of length L (e.g. L-item 2-bin-packing) Mutation: choose a bit randomly and flip it. Neighbourhood: Each individual has L neighbours. E.g. neighbours of 00110 are: {10110, 01110, 00010, 00100, 00111}

  35. Landscape Topology Mutation operators lead to slight changes in the solution, which tend to lead to slight changes in fitness. Why are “big mutations” generally a bad idea to have in a search algorithm ??

  36. Typical Landscapes f(s) members of S lined up along here Typically, with large (realistic) problems, the huge majority of the landscape has very poor fitness – there are tiny areas where the decent solutions lurk. So, big random changes are very likely to take us outside the nice areas.

  37. Typical Classes of Landscapes Plateau Unimodal Multimodal 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

  38. Beyond Hillclimbing HC clearly has problems with typical landscapes: There are two broad ways to improve HC, from the algorithm viewpoint: • Allow downhill moves – a family of methods called Local Search does this in various ways. • 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’.

  39. 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: • 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;ifx 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.

  40. 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 selectionmethod • With more than one solution available, we needn’t just mutate, we can [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.

  41. An extra bit about Encodings Direct vs Indirect

  42. 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.

  43. E.g. encoding a timetable I 4, 5, 13, 1, 1, 7, 13, 2 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.

  44. Mutating a Timetable with Encoding 1 4, 5, 13, 1, 1, 7, 13, 2 Using straightforward single-gene mutation Choose a random gene

  45. Mutating a Timetable with Encoding 1 4, 5, 6 , 1, 1, 7, 13, 2 Using straightforward single-gene mutation One mutation changes position of one exam

  46. 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?

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