Fast, Effective GAs for Large, Hard Problems David E. Goldberg Illinois Genetic Algorithms Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801 USA Email: [email_address] ; Web: http://www.illigal.uiuc.edu

GAs Had Their Warhol 15, Right? Evolution timeless, GAs so 90s. First-generation GA results were mixed in practice. Sometimes worked, sometimes not & first impressions stuck. But GAs had legs. In 90s, logical continuation of GA thinking has led to Completion of theory in certain sense, & GAs that solve large, hard problems quickly, reliably, and accurately. Consider design theory and designs that have led to reliable solution of difficult problems. Andy Warhol (1928-1987)

Evolution timeless, GAs so 90s.

First-generation GA results were mixed in practice.

Sometimes worked, sometimes not & first impressions stuck.

But GAs had legs.

In 90s, logical continuation of GA thinking has led to

Completion of theory in certain sense,

& GAs that solve large, hard problems quickly, reliably, and accurately.

Consider design theory and designs that have led to reliable solution of difficult problems.

Roadmap One-minute genetic algorithmist. The unreasonableness of GAs. Airplane & toaster design: A lesson from the Wright Brothers. Goals of GA design. Design decomposition step by step. Competent GA design from the fast messy GA to hBOA. From competence to efficiency: When merely fast is not enough. A billion bits or bust.

One-minute genetic algorithmist.

The unreasonableness of GAs.

Airplane & toaster design: A lesson from the Wright Brothers.

Goals of GA design.

Design decomposition step by step.

Competent GA design from the fast messy GA to hBOA.

From competence to efficiency: When merely fast is not enough.

A billion bits or bust.

One-Minute Genetic Algorithmist What is a GA? Solutions as chromosomes. Means of evaluating fitness to purpose. Create initial population. Apply selection and genetic operators: Survival of the fittest. Mutation Crossover Repeat until good enough. Puzzle: operators by themselves uninteresting.

What is a GA?

Solutions as chromosomes.

Means of evaluating fitness to purpose.

Create initial population.

Apply selection and genetic operators:

Survival of the fittest.

Mutation

Crossover

Repeat until good enough.

Puzzle: operators by themselves uninteresting.

Selection Darwinian survival of the fittest. Give more copies to better guys. Ways to do: roulette wheel tournament truncation Gedanken experiment: Run repeatedly without crossover or mutation. By itself, pick best.

Darwinian survival of the fittest.

Give more copies to better guys.

Ways to do:

roulette wheel

tournament

truncation

Gedanken experiment: Run repeatedly without crossover or mutation.

By itself, pick best.

Crossover Combine bits and pieces of good parents. Speculate on, new, possibly better children. Gedanken experiment: 50, 11111 & 50, 00000. By itself, a random shuffle Example Before X After X 11111 11000 00000 00111

Combine bits and pieces of good parents.

Speculate on, new, possibly better children.

Gedanken experiment: 50, 11111 & 50, 00000.

By itself, a random shuffle

Example

Mutation Mutation is a random alteration of a string. Change a gene, small movement in the neighborhood. Gedanken experiment: 100, 11111. By itself, a random walk. Example Before M After M 11111 11011

Mutation is a random alteration of a string.

Change a gene, small movement in the neighborhood.

Gedanken experiment: 100, 11111.

By itself, a random walk.

Example

The Unreasonableness of GAs How do individually uninteresting operators yield interesting behavior? Others talk about emergence. 1983 innovation intuition: Genetic algorithm power like that of human innovation. Separate Selection + mutation as hillclimbing or kaizen. Selection + recombination  Let’s examine. Different modes or facets of innovation or invention.

How do individually uninteresting operators yield interesting behavior?

Others talk about emergence.

1983 innovation intuition: Genetic algorithm power like that of human innovation.

Separate

Selection + mutation as hillclimbing or kaizen.

Selection + recombination  Let’s examine.

Different modes or facets of innovation or invention.

Selection + Recombination = Innovation Combine notions to form ideas. It takes two to invent anything. The one makes up combinations; the other chooses, recognizes what he wishes and what is important to him in the mass of the things which the former has imparted to him . P. Valéry Paul Valéry (1871-1945)

Combine notions to form ideas.

It takes two to invent anything. The one makes up combinations; the other chooses, recognizes what he wishes and what is important to him in the mass of the things which the former has imparted to him . P. Valéry

Airplane & Toaster Design Airport story. Why do the rules change? Legacy of Descartes: Separation of mind and body. Material machines (airplanes, toasters, autos, etc.) vs. conceptual machines (GAs, neural nets, computer programs, algorithms). Design is design is design.

Airport story.

Why do the rules change?

Legacy of Descartes: Separation of mind and body.

Material machines (airplanes, toasters, autos, etc.) vs. conceptual machines (GAs, neural nets, computer programs, algorithms).

Design is design is design.

Two Bicycle Mechanics from Ohio Four years, 1899-1903. Three gliders. Orville and Wilbur Wright created powered flight from scratch. Query: Why were the Wright brothers the first to fly?

Four years, 1899-1903.

Three gliders.

Orville and Wilbur Wright created powered flight from scratch.

Query: Why were the Wright brothers the first to fly?

Hypotheses Wrights flew because they were bicycle mechanics. Wrights flew because it was part of the zeitgeist. Wrights flew because they were bachelors! Maybe the Wrights flew because they were better inventors.

Wrights flew because they were bicycle mechanics.

Wrights flew because it was part of the zeitgeist.

Wrights flew because they were bachelors!

Maybe the Wrights flew because they were better inventors.

December 17, 1903: The Most Famous Moment in Aviation History

The Wright Brothers’ Secret Functional decomposition. Three subproblems: Stability : wing-warping plus elevator in 1899 glider model. 1902 glider had three-axis active control. Lift and Drag : wing shape improved on Lilenthal’s through air tunnel experiments. Propulsion : rotary wing with forward lift is a propeller.

Functional decomposition.

Three subproblems:

Stability : wing-warping plus elevator in 1899 glider model. 1902 glider had three-axis active control.

Lift and Drag : wing shape improved on Lilenthal’s through air tunnel experiments.

Propulsion : rotary wing with forward lift is a propeller.

But Decomposition is Old Hat to Moderns Computer science is about one thing: busting big problems up into lots of little problems. Descartes’s theory of decomposition (1637). Discourse on Method of Rightly Conducting the Reason and Seeking Truth in the Sciences. What else distinguishes Wrights’ method of invention? Method of modeling and integration was different.

Computer science is about one thing: busting big problems up into lots of little problems.

Descartes’s theory of decomposition (1637). Discourse on Method of Rightly Conducting the Reason and Seeking Truth in the Sciences.

What else distinguishes Wrights’ method of invention? Method of modeling and integration was different.

Lessons of the Wright Brothers Effective design decomposition of your problem. Facetwise, economic models of subproblem facets. Bounding empirical study and calibration. Scaling laws (dimensional analysis) particularly important.

Effective design decomposition of your problem.

Facetwise, economic models of subproblem facets.

Bounding empirical study and calibration.

Scaling laws (dimensional analysis) particularly important.

Goals of GA Design Solve hard problems, quickly, accurately, and reliably. Call a GA that achieves these goals competent. Can we design competent GAs?

Solve

hard problems,

quickly,

accurately,

and reliably.

Call a GA that achieves these goals competent.

Can we design competent GAs?

Effective Theory in GA Design Many GAs don’t scale & much GA theory inapplicable. Need design theory that works: Understand building blocks (BBs), notions or subideas. Ensure BB supply. Ensure BB growth. Control BB speed. Ensure good BB decisions. Ensure good BB mixing (exchange). Know BB challengers. Can use theory to design scalable & efficient GAs.

Many GAs don’t scale & much GA theory inapplicable.

Need design theory that works:

Understand building blocks (BBs), notions or subideas.

Ensure BB supply.

Ensure BB growth.

Control BB speed.

Ensure good BB decisions.

Ensure good BB mixing (exchange).

Know BB challengers.

Can use theory to design scalable & efficient GAs.

Play the GA Game Give you a population of strings S i . Give a list of associate f i values (bigger is better). Ask you to create a better string. Blind: no equation relating f i and S i .

Give you a population of strings S i .

Give a list of associate f i values (bigger is better).

Ask you to create a better string.

Blind: no equation relating f i and S i .

What Are We Processing? Similarities among strings. Schemata are similarity subsets. Schemata described by similarity templates. Example: *1*** = {strings with 1 second position}. Population contains 2 l - n 2 l schemata .

Similarities among strings.

Schemata are similarity subsets.

Schemata described by similarity templates.

Example: *1*** = {strings with 1 second position}.

Population contains 2 l - n 2 l schemata .

Schema Theorem (Holland 1975) where f - fitness H - schema M - number of schema representatives  - defining length o - schema order P c - probability of crossover P m - probability of mutation l - string length Little schemata grow logistically. A necessary condition for BB growth.

where

f - fitness

H - schema

M - number of schema representatives

 - defining length

o - schema order

P c - probability of crossover

P m - probability of mutation

l - string length

Little schemata grow logistically.

A necessary condition for BB growth.

Practical Schema Theorem for Design Fitness multiplier = s. Overall disruption =  . Goldberg & Sastry, 2001

Fitness multiplier = s.

Overall disruption =  .

Goldberg & Sastry, 2001

Problem Difficulty There are hard problems & easy problems. 3 way decomposition. The core: deception - intra scaling - inter noise - extra

There are hard problems & easy problems.

3 way decomposition.

The core:

deception - intra

scaling - inter

noise - extra

Easy Problems & Hard Problems The OneMax problem: Linear, uniformly scaled Define u, unitation variable # or ones in binary string Needle-in-a-haystack (NIAH) problem: No regularity to infer where good solutions might lie. Nothing does better than enumeration or random search.

The OneMax problem:

Linear, uniformly scaled

Define u, unitation variable

# or ones in binary string

Needle-in-a-haystack (NIAH) problem:

No regularity to infer where good solutions might lie.

Nothing does better than enumeration or random search.

Designing a Harder Problem Low-order estimates mislead GA x * = 111 : f 111 > f i , i  111 . Require complementary schemata better than competitors. Squashed Hamming cube representation:

Low-order estimates mislead GA

x * = 111 : f 111 > f i , i  111 .

Require complementary schemata better than competitors.

Squashed Hamming cube representation:

4-bit Deceptive Trap

Good Decisions: 2- & k -Armed Bandit Competing order-one schemata form a two- armed bandit. Example: 0**** versus 1****. Exponentially increasing trials to the observed best. fff**, a 2 3 = 8-armed bandit. Many bandit problems played in parallel.

Competing order-one schemata form a two- armed bandit. Example: 0**** versus 1****.

Exponentially increasing trials to the observed best.

fff**, a 2 3 = 8-armed bandit.

Many bandit problems played in parallel.

Gambler’s Ruin Population Size Make P bb = Q and solve for n :  = 1 - Q In terms of signal and noise: Compare with populationwise pop-sizing equation:

Make P bb = Q and solve for n :

 = 1 - Q

In terms of signal and noise:

Compare with populationwise pop-sizing equation:

100-bit Onemax

The Complexity Temptation First complexity results for GAs. Calculate the W=nt Function evaluations as product of population size and run duration (single epoch.

First complexity results for GAs.

Calculate the W=nt

Function evaluations as product of population size and run duration (single epoch.

A Sense of Time Truncation selection: make s copies each of top 1/ s th of the population. P(t+1) = s P(t) until P(t) = 1 P(t) = s t P(0) Solve for takeover time t *: time to go from one good guy to all good guys (or all but one). t * = ln n / ln s

Truncation selection: make s copies each of top 1/ s th of the population.

P(t+1) = s P(t) until P(t) = 1

P(t) = s t P(0)

Solve for takeover time t *: time to go from one good guy to all good guys (or all but one).

t * = ln n / ln s

So What? Who cares about selection alone? I want to analyze a “real GA”. How can selection-only analysis help me? Answer: Imagine another characteristic time, the innovation or mixing time.

Who cares about selection alone?

I want to analyze a “real GA”.

How can selection-only analysis help me?

Answer: Imagine another characteristic time, the innovation or mixing time.

The Innovation Time, t i Innovation time is the average time to create an individual better than one so far. Under crossover imagine pi, the probability of recomb event creating better individual. Innovation probability in Goldberg, Deb & Thierens (1993) and Thierens & Goldberg (1993).

Innovation time is the average time to create an individual better than one so far.

Under crossover imagine pi, the probability of recomb event creating better individual.

Innovation probability in Goldberg, Deb & Thierens (1993) and Thierens & Goldberg (1993).

Schematic of the Race

Golf Clubs Have Sweet Spots So do GAs. Easy problems, big sweet spots. Monkey can set GA parameters. Hard problems, vanishing sweet spots. [Goldberg, Deb, & Theirens, 1993]

So do GAs.

Easy problems, big sweet spots.

Monkey can set GA parameters.

Hard problems, vanishing sweet spots.

Dr. Jekyll & Mr. Hyde in Practice GA literature full of evidence for this problem. Evolution of the “typical practitioner.” First application goes swimmingly. More complex application needs TLC. Big Kahuna needs compute time = length of universe. Why are we fiddling with codings and operators? Aren’t GAs robust? No. First-generation GAs are not.

GA literature full of evidence for this problem.

Evolution of the “typical practitioner.”

First application goes swimmingly.

More complex application needs TLC.

Big Kahuna needs compute time = length of universe.

Why are we fiddling with codings and operators?

Aren’t GAs robust?

No. First-generation GAs are not.

Simple GAs Are Mixing Limited With growing difficulty, “sweet spot” vanishes. Or populations must grow exponentially.

With growing difficulty, “sweet spot” vanishes.

Or populations must grow exponentially.

The Key: Not the Schema Theorem Much theory focuses on Holland’s schema theorem. Schema theorem a piece of cake. Make sure GA fires on all seven cylinders of the design decomposition. Surprise: Mixing is the key. To mix well, must get building blocks right. Effective GAs identify structure of problem. Data mine early samples for structure of the landscape.

Much theory focuses on Holland’s schema theorem.

Schema theorem a piece of cake.

Make sure GA fires on all seven cylinders of the design decomposition.

Surprise: Mixing is the key.

To mix well, must get building blocks right.

Effective GAs identify structure of problem.

GA Kitty Hawk: 1993 1993 & the fast messy GA. Moveable bits, cutting and splicing, building-block filtering mechanism. Original mGA complexity estimated: O( l 5 ) Compares favorably to hillclimbing, too (Muhlenbein 1992). [Goldberg, Deb, Kargupta, & Harik, 1993]

1993 & the fast messy GA.

Moveable bits, cutting and splicing, building-block filtering mechanism.

Original mGA complexity estimated: O( l 5 )

Compares favorably to hillclimbing, too (Muhlenbein 1992).

Look Ma, No Genetics: hBOA Replace genetics with probabilistic model building  PMBGA or estimation of distribution algorithm: EDA 3 main elements: Decomposition (structural learning): Learn what to mix and what to keep intact. Representation of BBs (chunking): Means of representing alternative solutions. Diversification of BBs (niching): Preserve alternative chunks of solutions. Test on adversarially designed functions so works on yours.

Replace genetics with probabilistic model building  PMBGA or estimation of distribution algorithm: EDA

3 main elements:

Decomposition (structural learning): Learn what to mix and what to keep intact.

Representation of BBs (chunking): Means of representing alternative solutions.

Diversification of BBs (niching): Preserve alternative chunks of solutions.

Test on adversarially designed functions so works on yours.

Schematic of BOA Structure Current population Selection New population Bayesian network

Results on Spin Glasses Pelikan et al. (2002) 64 100 144 196 256 324 400 10 3 Problem Size Number of Evaluations hBOA O(n 3.51 )

From Competence to Efficiency Motivation: Even competent GAs require O ( I 2 ) time. 1000*1000 = a million function evaluations. In real problems, this can be a problem. How can we systematically achieve speedups.

Motivation: Even competent GAs require O ( I 2 ) time.

1000*1000 = a million function evaluations.

In real problems, this can be a problem.

How can we systematically achieve speedups.

IlliGAL Decomposition of Efficiency 1. Space: parallelization 2. Time: continuation 3. Fitness: Evaluation relaxation 4. Specialization: Hybridization

1. Space: parallelization

2. Time: continuation

3. Fitness: Evaluation relaxation

4. Specialization: Hybridization

Computation time: Communications time: Less computations, more communications Master-Slave Parallel GAs

Computation time:

Communications time:

Less computations, more communications

Account for Time (and Quality) Use perspective of the master Minimize time:

Use perspective of the master

Minimize time:

Master-Slave Example Dummy function T f = 4 ms Communications time T c = 19ms Pop size: 120, length = 80 Cantu-Paz, E. and Goldberg, D. E.(1999). On the scalability of parallel genetic algorithms, Evolutionary Computation, 7 (4), 429-449.)

Dummy function T f = 4 ms

Communications time T c = 19ms

Pop size: 120, length = 80

Speedups and Efficiency

My Dr. Evil Moment Lunchtime question: do real large problems draw attention to theoretical & design findings? Dr. Evil’s mistake: Wondered if GAs could go to 10 6 vars. Decided to go for a billion. Use simple underlying problem (OneMax) with Gaussian noise (0.1 variance of deterministic problem) Don’t try this at home!!! We get the warhead and then hold the world ransom for... 1 MILLION DOLLARS !

Lunchtime question: do real large problems draw attention to theoretical & design findings?

Dr. Evil’s mistake: Wondered if GAs could go to 10 6 vars.

Decided to go for a billion.

Use simple underlying problem (OneMax) with Gaussian noise (0.1 variance of deterministic problem)

Don’t try this at home!!!

Road to Billion Paved with Speedups Naïve implementation: 100 terabytes & 2 72 random number calls. cGA  Memory O(ℓ) v. O(ℓ 1.5 ). Parallelization  speedup n p . Vectorize four bits at a time  speedup 4. Other doodads (bitwise ops, limit flops, inline fcns, precomputed evals)  speedup 15. Gens & pop size scale as expected.

Naïve implementation: 100 terabytes & 2 72 random number calls.

cGA  Memory O(ℓ) v. O(ℓ 1.5 ).

Parallelization  speedup n p .

Vectorize four bits at a time  speedup 4.

Other doodads (bitwise ops, limit flops, inline fcns, precomputed evals)  speedup 15.

Gens & pop size scale as expected.

A Billion Bits or Bust Simple hillclimber solves 1.6(10 4 ) or (2 14 ). Souped-up cGA solves 33 million (2 25 ) to full convergence. Solves 1.1 billion (2 30 ) with relaxed convergence. Growth rate the same  Solvable to convergence.

Simple hillclimber solves 1.6(10 4 ) or (2 14 ).

Souped-up cGA solves 33 million (2 25 ) to full convergence.

Solves 1.1 billion (2 30 ) with relaxed convergence.

Growth rate the same  Solvable to convergence.

Design Fast, Effective GAs GA design advanced by taking GA ideas and running with them. Large, difficult problems in grasp. Theory and practice in sync. These direct lessons are crucial. Meta-lessons of this style of thinking as important for complex systems & interdisciplinary work, generally. This style of theory works for all GEC.

GA design advanced by taking GA ideas and running with them.

Large, difficult problems in grasp.

Theory and practice in sync.

These direct lessons are crucial.

Meta-lessons of this style of thinking as important for complex systems & interdisciplinary work, generally.

This style of theory works for all GEC.

More Information Goldberg, D. E. (2002). The design of innovation: Lessons from and for competent genetic algorithms. Boston, MA: Kluwer Academic Publishers. Lab: www.illigal.org iFoundry: www.ifoundry.illinois.edu Email: [email_address]

Goldberg, D. E. (2002). The design of innovation: Lessons from and for competent genetic algorithms. Boston, MA: Kluwer Academic Publishers.

Lab: www.illigal.org

iFoundry: www.ifoundry.illinois.edu

Email: [email_address]