assignment 2: late period until thurs @ ant colony ...cs540/spr2015/more_progress/gas15.pdf ·...

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CS540 February 17, 2015

n  Assignment 2: late period until thurs @ noon n  Copying code or prose

n  If you use publically available code, add a comment to attribute the source

n  Do not look at other student’s assignments n  Do not take text without adding quotes and do

not use too much quoted material

CS540, Artificial Intelligence @ Adele Howe, Spring 2015

Population Based Search

n  Ant Colony Optimization n  Evolutionary Algorithms

n  Evolutionary Strategies n  Genetic Algorithms n  Genetic Programming


Ant Colony Optimization (ACO) [Dorigo 1992]

n  constructive meta-heuristic that shares limited information between multiple search paths

n  Characteristics: n  multi-agent/solution exploration (population based), n  stochastic, n  memory (individual and colony), n  adaptive to problem, n  implicit solution evaluation via “who gets there first”

n  Inspiration: ants follow pheromone paths of other ants to find food. Over time, the best paths have strongest smell, luring more ants.


Inspiration: Ant Behavior

From http://www.nytimes.com/imagepages/2007/11/12/science/2007113_TRAFFIC_GRAPHIC.html CS540, Artificial Intelligence @ Adele Howe, Spring 2015

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Ant Colony Example 1.  An ant (called "blitz") wanders. 2.  If it discovers a food source, it returns

~directly to the nest, leaving a pheromone trail;

3.  Ants will be inclined to follow, more or less directly, the pheromone trail;

4.  Returning to the colony, these ants will strengthen the route;

5.  If there are two routes to reach the same food source then, in a given amount of time, the shorter one will be traveled by more ants than the long route;

6.  The long route will eventually disappear because pheromones are volatile;

7.  Eventually, all the ants have "chosen" the shortest route.

From http://en.wikipedia.org/wiki/Ant_colony_optimization_algorithms CS540, Artificial Intelligence @ Adele Howe, Spring 2015

Ant Colony Simulation

n  http://code.kevinworkman.com/Ants/AntsApplet.html


Core Pseudo-Code for Each Ant 1.  initialize 2.  until reach solution

1.  read local routing table 2.  compute P(neighbors | routing-table, memory, constraints,

heuristics) 3.  select move based on Ps and constraints 4.  update memory for move (pheromone globally and visited arc

locally) 3.  die


ACO Parameters n  when to terminate n  how many ants n  how to schedule ant activities (ant generation and optionals) n  what to keep in memory locally n  how to select a next state n  how to update memory (e.g., amount as function of solution

quality) n  how to “evaporate” pheromone n  optionals:

n  delayed updating of pheromones on trail? n  allow pheromone trail evaporation? n  permit off-line pheromone updates (daemon actions)?


3

ACO Applications n  Scheduling (JSP, FSP…) n  Vehicle routing n  Quadratic assignment n  Image processing… n  TSP

http://en.wikipedia.org/wiki/Travelling_salesman_problem


CS540 February 19

n  A2 will be graded in next week n  Work on project!


Evolutionary Algorithms Overview n  Evolutionary algorithms search a population

representing different sample points in the search space.

n  Each sample point is represented as a string which can be recombined with other strings to generate new sample points in the space.

n  Algorithms are based on biological analogies with “population genetics" and “simulated evolution".


Why Evolutionary Algorithms? n  No Gradient Information Is Needed. These

algorithms do not search along the contours of the function, but rather by hyperplane sampling in Hamming space.

n  The Resulting Search is Global. Since they do not hill-climb, they avoid local optima and so can be applied to multimodal functions.

n  Potential for Massive Parallelism. Can effectively exploit thousands of processors in parallel.

n  They Can Be Hybridized with conventional optimization methods.


4

How to envision search space

n  Local Search? n  Evolutionary

Computation?


Search Spaces

SLS

Current Solution

Neighbor1

Neighbor2

Neighbor3

Neighbor4

Evolutionary Computation 0 0 1 0 0 1 1 0 0 1

1 0 1 0 0 1 1 1 0 0

1 1 1 1 1 0 0 0 0 0

0 1 1 0 1 0 1 1 0 1

0 1 1 0 1 0 0 0 0 0


Issues: Representation

n  Evolutionary model n  Genotype: represents information stored in

chromosomes n  Phenotype: describes how the individual appears

n  Approaches n  Indirect: standard data structure, genotype is

mapped to phenotype n  Direct or natural: specialized representation using

domain specific search operators



Binary: genotype is encoded as bit strings of length l n  Grey codes: similar values have similar

representations n  Binary numbers: may be some issue of

precision in encoded values


5


Nonbinary (integer, continuous, permutation): larger alphabets, real-valued encodings, more natural Arguments against: n  tends to have larger search space n  there will be fewer explicit hyperplane partitions n  the alphabetic characters will not be as well represented in a finite population.


Representation example

n  TSP n  Genotype? n  Phenotype?


Representation: FSP & JSP n  “use an intermediary, encoded representation of schedules that is

amenable to crossover operations, while employing a decoder that always yields legal solutions to the problem.”

n  “A complete schedule for the job shop was derived from a list of preferences for each work station, linked to times. A preference list had an initial member- a time at which the list went into effect. The rest of the list was made up of some permutation of the contracts available, plus the elements ‘wait’ and ‘idle’.”

Davis, Lawrence. "Job shop scheduling with genetic algorithms." Proceedings of an International Conference on Genetic Algorithms and Their Applications. Vol. 140. 1985.


Issues: Fitness Function

n  Based on the objective function n  Allows comparison of different solutions n  Domain specific to goals of problem n  Single value output: multi-objective must be

combined into single function


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Issues: Fitness Function

n  Modifications n  Must be fast! May need to be executed

hundreds of thousands of times n  Sometimes approximate to achieve speed n  Smoothing: replacing fitness with average of

neighbors (useful for plateaus) n  Penalty functions: to relax feasibility

requirements when infeasible solutions cannot be removed


Fitness Function example

n  TSP n  Objective function? n  Fitness function?


Issues: Initializing Population

Tension between good solutions and diversity (low diversity can lead to quick stagnation or large distance from optimum)

Random: generate n strings uniform randomly, within encoding requirements.

Domain specific: use heuristic method to generate “ok”(greedy) solutions that can be refined.


Initialization example

n  TSP n  Greedy initialization?


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CS540 February 24

n  A2 is partly graded n  Take time on the essays… (each essay set

comprises .34*15%=5% of your final grade!) n  We’re looking for insight and thought. n  Write it as an essay, not direct answers to the

questions. Organize your points. n  Do proofread!

n  A3: Timetabling using evolutionary computation!


PA2 Program

n  Significant issues n  28 submitted n  26 passed checkin n  24 produced some output n  16 ran correctly on the trivial instance n  7 programs ran correctly on an instance other

than the trivial one


Evolution Strategies

n  Search method like local search but can be population based

n  Vector of continuous variables n  Mutates each variable by incrementing value using

randomly generated change with zero mean and set standard distribution… Mutation only to create new solutions

n  Survival of fittest between new and previous vectors


Simplest Evolutionary Strategy (ES)

(1 + 1)-ES n  1 “parent” solution plus n  1 “offspring” solution n  Keep the best


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(1+1)-ES Algorithm

1.  Create initial solution x 2.  while termination criterion is not met do

1.  for i = 1 to n do

2.  If then

3.  Update σ

!xi = xi +σNi (0,1)f !x( ) ≥ f x( )x := !x

N(0,1) indicates a normal distribution with 0 mean and 1 standard deviation.


(1+1)-ES Update

How to update σ? n  Convergence proofs on two simple problems deAine a 1/5 rule

n  If ratio is > 1/5, increase σ, else decrease n  Keep it Aixed, which is better for escaping local optima

# search steps that find a better solutionall or last t steps


(µ+λ)-ES

n  µ solutions in population n  Generate λ new solutions n  Choose the best µ from superset at

each iteration n  Can add recombination before mutation n  Can have different σ per variable


Evolution Strategies (cont.)

Self adaptive mutation and rotation n  Log-normal distribution for mutation n  Adaptive through strategy (σ) and rotation angle (α)

parameters added to chromosome

€

x1,x2,σ1,σ 2,α1,2

Simple Mutations Correlated Mutation via Rotation Figures courtesy of D. Whitley


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Simple Genetic Algorithm (Alg 7 in Rothlauf)

1.  Create initial population P with N solutions 2.  for i = 1 to N do

1.  Calculate 3.  while (termination criterion is not met) and (population has not yet converged) do

1.  Create M with N individuals selected from P 2.  ind = 1 3.  repeat

1.  if random(0,1) ≤ pc then recombine and and place the offspring in P’ 2. Else copy and to P’ 3.  ind = ind + 2

4.  until ind > N 5.  for i = 1 to N do

1.  for j = 1 to l do 1.  if random(0,1) ≤ pm then mutate( ) where

6.  Calculate where 7.  P = P’

xi i ∈ 1,...,N{ }( )

f xi( )

xind ∈M xind+1 ∈Mxind ∈M xind+1 ∈M

x ji xi ∈ "P

f xi( ) xi ∈ "P


Genetic Algorithm Process

String1 String2 String3 String4 ….. …..


OffspringA OffspringB OffspringC OffspringD

….. …..

Selection (Duplication)

Recombination (Crossover)

Current Generation t

Intermediate Generation t

Next Generation t+1


CS540 February 26

n  A2 results: look at comments in Checkin and Canvas n  A3:

n  Check back later today for submitting Part A n  Essays: Q2: Start with the pseudocode for the core

algorithm and explain how you set each parameter/design decision in it.

n  We will be using Checkin n  Strategy for success: read instructions from A2 carefully. After you

tar your code, move it to a fresh directory, un tar it and try running the script on a set of files with different names than before.

n  Run on many test cases!

CS540, Artificial Intelligence @ Adele Howe, Spring 2015 CS540, Artificial Intelligence @ Adele Howe, Spring 2015

A2 Results

P1:Slots Program P1:SC Program P2:Slots Program P2:SC Program 4 A 34.5 A 14 A 54.34 J

4 B 34.5 B 14 C 63.4 E

4 C 34.5 F 14 D 72.45 B 4 D 34.5 H 14 H 91.17 I

4 E 34.5 K 14 M 93.46 C 4 F 34.5 L 15 B 115.4 A 4 G 34.5 N 15 E 126.68 D 4 H 34.5 Q 15 F 130.95 F

4 I 35.5 C 15 I 186.54 K

4 J 35.5 D 15 J 273.61 Q

4 K 35.5 R 16 K 308.53 S

4 L 35.75 E 23 S 321.81 L

4 M 35.75 I 98 Q

4 N 37.25 V

4 O 46.25 W

4 P 50.75 U

5 Q 56.75 X

5 R 59.75 O

5 S 60 J

5 T 129 S

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Simple Genetic Algorithm Selection








xi i ∈ 1,...,N{ }( )

f xi( )


x ji xi ∈ "P

f xi( ) xi ∈ "P


Proportionate Selection

n  Population is evaluated according to a fitness function.

n  Parents are selected for reproduction by ranking according to their relative fitness

ff

i

fi

f jj=1

N

∑


Proportionate Selection Process Stochastic sampling with replacement Map individuals to space on a roulette wheel

CS540, Artificial Intelligence @ Adele Howe, Spring 2015 From clipartbest.com

More fit individuals are allocated proportionally more space. Spin wheel repeatedly until desired population size is achieved

Population Example, Stochastic Sampling w/Replacement String Fit Space copies

001000000 2.0 .095

101010101 1.9 .186

111110011 1.8 .271

010001100 1.7 .352

111100000 1.6 .429

101000110 1.5 .5

011001110 1.4 .567

001111000 1.3 .629

000110100 1.2 .686

100100011 1.1 .738

010000111 1.0 .786

String Fit Space copies

011001111 0.9 .829

000100110 0.8 .867

110001101 0.7 .9

110000111 0.6 .929

100100100 0.5 .952

011011011 0.4 .971

000111000 0.3 .986

001100100 0.2 .995

100111010 0.1 1.0

010010011 0.0 --

Random #s: .93, .65, .02, .51, .20, .93, .20, .37, .79, .28, .13, .70, .80, .51, .76, .45, .61, .07, .76, .86, .29 CS540, Artificial Intelligence @ Adele Howe, Spring 2015

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Other Selection Methods

n  Tournament Selection: randomly select two strings, place the best into the new population, repeat until intermediate population is full

n  Ranking: order individuals by rank rather than fitness value


Population Example, Tournament Selection String # copies

001000000 0

101010101 1

111110011 2

010001100 3

111100000 4

101000110 5

011001110 6

001111000 7

000110100 8

100100011 9

010000111 10

String Fit copies

011001111 11

000100110 12

110001101 13

110000111 14

100100100 15

011011011 16

000111000 17

001100100 18

100111010 19

010010011 20

Random pairs: (6 4) (16 7) (13 2) (0 17) (4 14) (19 18) (7 19) (2 2) (11 4) (7 2) (7 20) (18 19) (11 1) (20 0) (10 14) (12 15) (14 3) (20 15) (6 1) (12 13) CS540, Artificial Intelligence @ Adele Howe, Spring 2015

Another Fitness Selection Method

n  Remainder stochastic sampling n  Again map to roulette wheel, but this time add

outer wheel with N evenly spaced pointers. n  Spin once to determine all population members


Population Example, Remainder Stochastic Sampling String Fitness Random copies

001000000 2.0 -- 2

101010101 1.9 0.93 2

111110011 1.8 0.65 2

010001100 1.7 0.02 1

111100000 1.6 0.51 2

101000110 1.5 0.20 1

011001110 1.4 0.93 2

001111000 1.3 0.20 1

000110100 1.2 0.37 1

100100011 1.1 0.79 1

010000111 1.0 -- 1

String Fitness Random copies

011001111 0.9 0.28 1

000100110 0.8 0.13 0

110001101 0.7 0.70 1

110000111 0.6 0.80 1

100100100 0.5 0.51 1

011011011 0.4 0.76 1

000111000 0.3 0.45 0

001100100 0.2 0.61 0

100111010 0.1 0.07 0

010010011 0.0 -- 0

Random #s: .93, .65, .02, .51, .20, .93, .20, .37, .79, .28, .13, .70, .80, .51, .76, .45, .61, .07, .76, .86, .29 CS540, Artificial Intelligence @ Adele Howe, Spring 2015

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CS540 March 2 n  Assignment 3

n  Submitted instances that satisfied requirements are posted – lots of test cases! Some very challenging.

n  Verifier has been made available to help with your debugging ~cs540/verifier/verifier

n  Checkin will be set up tonight. Same instructions and checks as in assignment 2 except for naming as PA3.tar

n  ASAP: make sure your solutions are verified and that checkin works for you


Genetic Algorithm Process



OffspringA OffspringB OffspringC OffspringD

….. …..

Selection (Duplication)

Recombination (Crossover)

Current Generation t

Intermediate Generation t

Next Generation t+1


Simple Genetic Algorithm Recombination




1. if random(0,1) ≤ pc then recombine and and place the offspring in P’ 2. Else copy and to P’ 3.  ind = ind + 2




xi i ∈ 1,...,N{ }( )

f xi( )


x ji xi ∈ "P

f xi( ) xi ∈ "P


Reproduction: Recombination/Crossover

Two parents: binary strings representing an encoding of 5 parameters that are used in some optimization problems. 10010101 11011001 01110100 10100101 10000101 xyxyyyxx xyyyyxyx xyyxyyxy yyxyxxxy yxyxyxyx

Recombination occurs as follows: 10010 \ / 10111011001011101 \ / 001010010110000101 xyxyy / \ yxxxyyyyxyxxyyxy / \ xyyyxyxxxyyxyxyxyx

Producing the following offspring: 10010yxxxyyyyxyxxyyxyy001010010110000101 xyxyy10111011001011101xyyyxyxxxyyxyxyxyx


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Desirable Characteristics of Recombination Operators

n  Respect: any commonalities of the parents are inherited by the offspring

n  Transmission: all components of the offspring must have come from a parent

n  Assortment: all components of the offspring must be compatible/feasible

n  Ergodicity: can reach any combination from all possible starting points


Issues: Combination Operators

1 point crossover: pick single crossover point, split strings at this point, recombine

2 point crossover: pick two crossover points, split strings at these points, recombine (think of ring for string)

Uniform crossover: randomly pick each element from one of the two parents


Crossover Operators: 1 point

0 1

1

0 0

0

1

0 x

x

y

y x

x

y

y

One end is at beginning/end

of string


Crossover Operators: 2 point

0 1

1

0 0

0

1

0 x

x

y

y x

x

y

y

Both ends can move


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Crossover and Hypercube Paths

1111

0111 1011 1101 1110

0011 0101 0110 1001 1010 1100

0001 0010 0100 1000

0000


Other Combination Operators

HUX: exactly half of the differing bits are swapped Given parents: 100110101111000010110 101100011100001100001 a new individual is: 100100111110000110010


More Combination Operators Reduced Surrogate: Crossover points chosen

within differing bits only 100110101111000010110 101100011100001100001 may become: 100110111100001100001

Domain specific: operations designed to match demands of the domain (e.g., reorder portions for scheduling application)


Why Might Reduced Surrogate Be Important

n  Closely related to HUX n  Key idea: look only at portions of strings

that differ 00011111011010011 00010011010010010 How does probability of new string change with reduced surrogate versus 1-‐point crossover?


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Simple Genetic Algorithm Mutation








xi i ∈ 1,...,N{ }( )

f xi( )


x ji xi ∈ "P

f xi( ) xi ∈ "P


Mutation

n  For each bit in population, mutate with probability

n  should be low, typically < .01 n  can mean randomly select a value for

the bit or flip the bit

mpmp


Issues: Which Strings in New Generation

n  Replace with offspring n  Assumption of canonical GA

n  Best of offspring and parents n  Alternative view which guarantees always

keep best and puts intensive pressure on population for improvement


CS540 March 4 n  Assignment 3

n  Verifier has been made available to help with your debugging ~cs540/verifier/verifier

n  Use Checkin for programs n  Use Canvas for essays – need all 3 essays (a few

students sent Q1 with the problem instance)


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Issues: Termination Criteria

n  Quality of solution: best individual passes some pre-set threshold

n  Time: certain number of generations have been created and tested

n  Diminishing returns: improvement over each generation does not exceed some threshold


Alternatives to the Simple GA Model

n  Genitor n  CHC


Genitor (Whitley et al.)

Differences with canonical genetic algorithm: § Reproduction proceeds one individual at a time. § Worst individual in population is replaced by

new offspring. § Fitness is assigned by rank.

§ Steady State GA


Genitor Algorithm (Whitley)


String n

Parent 1 Parent 2

Child 1 Child 2

String n-1

String1 String2 String3 Child 2 ….. …..

Time t Time t+1

Selection Crossover &

Recombination Insert In

population


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CHC (Eshelman)

Crossover using generation elitist selection

Heterogeneous recombination by incest prevention n  Origin of HUX terminology

Cataclysmic mutation, when population starts to converge


CHC Process Parent1 Parent2 Parent3 Parent4 …..

Parent n

Child1 Child2 Child3 Child4 …..

Child n

Parent b Child f Child h Parent j …..

Parent m

via Random Selection and HUX

Best n strings from parents and children


Genetic Algorithms for Scheduling

n  Represent solution as permutation of tasks n  Requires a schedule builder to convert

solution into schedule and assess objective function. [Indirect search]

n  Syswerda’s permutation crossover was used for scheduling.


GA applied to TSP

n  Applet showing simple GA for TSP http://www.obitko.com/tutorials/genetic-

algorithms/tsp-example.php


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Genetic Programming

n  Combine and/or parameterize code blocks (functions and terminals) to produce program to solve some problem.

n  Follows GA functionality n  Solution is often a parse tree of functions

and terminals. n  Interior nodes are functions (e.g., “+”, “or”),

control structures (e.g., “if”, “while”) or functions with side effects (e.g., “read”, “print”).

n  Terminals are variables and constants. CS540, Artificial Intelligence @ Adele Howe, Spring 2015

GP Steps

1.  Initialize population of programs 2.  Loop

1.  Assess fitness 2.  Construct new generation through selection

and n  Reproduction, n  Mutation or n  Crossover


GP initialization

n  Grow: start with empty tree and iteratively assign nodes to be function or terminal. All nodes at depth kmax are terminals.

n  Full: start with empty tree, randomly add functions up to depth kmax-1, terminals at depth kmax.

n  Ramped-half-and-half: divide population into (kmax-1) parts, half of each is created by grow and other by full, depth of nodes in ith part is varied from 2 to kmax


GP: Assessing Fitness

n  Use program to solve problem n  Quality of solution n  Efficiency of solution

n  May require simulation, e.g., UAV


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GP: Representation

Domain Dependent! n  E.g., Simple Lisp

functions (math, cons, list, append…)

+

^

*

-

a b

3 c a


GP: Mutation

n  Substitute one function (terminal) for another

n  Substitute one subtree for another

+

^

/

-

a b

3 c d

+

^

*

-

a b

3 a sqrt

c

*

d

Animation: http://www.genetic-programming.com/mutation.gif CS540, Artificial Intelligence @ Adele Howe, Spring 2015

GP: Crossover +

^ *

-

a b

3 a sqrt

c *

d

+ ^

* -

a b

3 a abs

b *

d

+ ^ -

3 a sqrt

c *

d

-

a abs

cb *

d Animation: http://www.genetic-programming.com/crossover.gif


GP for Software Repair (Forrest et al. GECCO 2009)

Given n  C source code (represented as abstract syntax trees), n  a negative test case that exercises the fault to be

repaired (leading to weighted paths through the tree), n  positive test cases that encode the required behavior of

the program Use GenProg to create better/fixed code

n  selects a subset (top ranked 50%) of programs based on fitness (sum of passed test cases), and

n  modifies the programs with mutation and crossover.


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The Zune Bug

days is # of days since Jan 1, 1980. When days is the last day of a leap year, the program enters an infinite loop.


void zunebug(int days) { int year = 1980; while (days > 365) { if (isLeapYear(year)){ if (days > 366) { days -= 366; year += 1; } else{ } } else { days -= 365; year += 1; } } printf("current year is %d\n", year); }

Repairing the Zune Bug


void zunebug(int days) { int year = 1980; while (days > 365) { if (isLeapYear(year)){ if (days > 366) { days -= 366; year += 1; } else{ } } else { days -= 365; year += 1; } } printf("current year is %d\n", year); }

void zunebug(int days) { int year = 1980; while (days > 365) { if (isLeapYear(year)){ if (days > 366) { //days -= 366; //GP deletes year += 1; } else{ } days -= 366; //GP inserts } else { days -= 365; year += 1; } }

GenProg Movie


http://dijkstra.cs.virginia.edu/genprog/#videos

Underlying Theory: Hyperplane Sampling


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Another View of Hyperplane Sampling


Population Example for Hyperplane Sampling String Fitness Random copies

001##...## 2.0 -- 2

101##...## 1.9 0.93 2

111##...## 1.8 0.65 2

010##...## 1.7 0.02 1

111##...## 1.6 0.51 2

101##...## 1.5 0.20 1

011##...## 1.4 0.93 2

001##…## 1.3 0.20 1

000##…## 1.2 0.37 1

100##…## 1.1 0.79 1

010##…## 1.0 -- 1

String Fitness Random copies

011##…## 0.9 0.28 1

000##…## 0.8 0.13 0

110##…## 0.7 0.70 1

110##…## 0.6 0.80 1

100##…## 0.5 0.51 1

011##…## 0.4 0.76 1

000##…## 0.3 0.45 0

001##…## 0.2 0.61 0

100##…## 0.1 0.07 0

010##…## 0.0 -- 0


Some Schemata and Fitness Values Schema Mean Count Expect Observe 101**…* 1.70 2 3.4 3

111**…* 1.70 2 3.4 4

1*1**…* 1.70 4 6.8 7

**01*…* 1.38 5 6.9 6

***1*…* 1.30 10 13.0 14

**11*…* 1.22 5 6.1 8

11***…* 1.175 4 4.7 6

001**…* 1.166 3 3.5 3

1****…* 1.089 9 9.8 11

0*1**…* 1.033 6 6.2 7

10***…* 1.020 5 5.1 5

**1**…* 1.010 10 10.1 12

*****…* 1.000 21 21.0 21 CS540, Artificial Intelligence @ Adele Howe, Spring 2015

Hyperplane Deception Since genetic algorithms are driven by hyperplane sampling a misleading problem can be constructed as follows. f(0**) > f(1**) f(*0*) > f(*1*) f(**0) > f(**1) f(00*) > f(01*), f(10*), f(11*) f(0*0) > f(0*1), f(1*0), f(1*1) f(*00) > f(*01), f(*10), f(*11) BUT f(111) > f(000) where f(x) gives the average fitness of all strings in the hyperplane slice represented by x.


22

Fitness Landscape Analysis

n  A fitness landscape (X,f,d) of a problem instance consists of a set of solutions X, an objective function f and a distance measure d.

n  We can analyze problems based on characteristics derived from this definition and determine (roughly) problem difficulty.


Locality

n  How well distances correspond to differences in fitness between solutions

n  High locality if nearby solutions have similar fitness

n  Low locality problems are difficult for local search


Fitness-Distance Correlation

n  Metric of locality

n  c is the covariance, <> are means. n  Positive correlation (i.e., better is closer) is easy,

uncorrelated is difficult, negative correlation is misleading.

ρFDC =cfd

σ f( )σ dopt( )where

c fd =1m

fi − f( )i=1

m

∑ di,opt − dopt( )


Big Valley Topology Original observation

from (Boese, Kahng & Muddu 1994)

Figure: benchmark FlowShop scheduling problem from Watson et al. 1999


23

Ruggedness n  Statistical property on relation of objective values to intra-

solution distances n  Random walk correlation between solutions separated by s

steps

xi is the solution at step i n  Correlation length…high means smooth, easy for guided

search

r s( ) =f xi( ) f xi+s( ) − f 2

f 2 − f 2

lcorr = −1

ln r 1( )( )


No Free Lunch…

n  Wolpert and Macready, 1997 “For any algorithm, any elevated

performance over one class of problems is exactly paid for in performance over another class.”

n  Proven for finite optimization problems solved with deterministic “black-box” algorithms


NFL Theorem I For any pair of algorithms a1 and a2: f is an optimization problem, m is number of distinct

points sampled by the algorithm, y is the cost value, d is the best cost value over the sample.

Meaning… average performance for an algorithm over all problems is the same as for any other algorithm, assuming equal sampling size.

€

P(dmy | f ,m,a1) =

f∑ P(dm

y | f ,m,a2)f∑


Implications of NFL

n  Emphasizes need for different algorithms n  Does not preclude superior performance on

some interesting class(es) of problem!

n  However, if no problem knowledge is used, then P(f) is essentially uniform.

€

P(dmy |m,a) = P(

f∑ dm

y | f ,m,a)P( f )


24

Implications (cont.)

n  Structure of problem must be understood and used to inform choice of search.

n  Knowledge of cost function properties is folded into P(f) or

n  “performance of an algorithm is determined by … how aligned is with the problems “

€

P(dmy |m,a) =

v d my ,m,a •

p

€

p

€

p

€

v d my ,m,a


Comparing Performance

n  Assume we can construct a histogram of cost values produced by a run of an algorithm , then use these to gauge performance: n  Over all cost functions:

n  Average of n  for the random algorithm

n  Given f and m, % of algorithms with

€

c

€

P min c ( ) > ε | f ,m,a( )

€

P min c ( ) > ε | f ,m,a( )

€

min c > ε( )


Minimax & Fixed f

n  NFL focuses on average performance. n  One algorithm may be much better than another on a

subset of f (the other is a little better on remaining to even out average).[Minimax]

n  May occur when samples intersect

n  If f is fixed, observing performance until time t does not indicate what will happen later. n  Intelligent choice requires knowledge of both f and a


Phase Transitions

Cheeseman, Kanefsky & Taylor (1991) observed that:

n  The hardest problems in an NP-complete class (e.g., TSP, SAT) were at the boundary between feasibility and infeasibility.

n  Problems that are clearly feasible (or not) are easy to solve (or easy to recognize there’s no solution).

n  “phase transition” depends on identifying a key scale up parameter (e.g., constraints per variable, standard deviation of costs)

0

20

40

60

80

100

120

140

0 5 10 15

Feasible Not


assignment 2: late period until thurs @ ant colony ...cs540/spr2015/more_progress/gas15.pdf ·...

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