A Billion Bits or Bust

A Billion Bits or Bust David E. Goldberg 1 , Kumara Sastry 1,2 , Xavier Llor à 1, 3 1 Illinois Genetic Algorithms Laboratory (IlliGAL) 2 Materials Computation Center (MCC) 3 National Center for Super Computing Applications (NCSA) University of Illinois at Urbana-Champaign Urbana, IL 61801 USA [email_address] , [email_address] , [email_address] http://www- illigal.ge.uiuc.edu Supported by AFOSR FA9550-06-1-0096 and NSF DMR 03-25939. Computational results were obtained using CSE’s Turing cluster.

Billion-Bit Optimization?
Strides w/ genetic algorithm (GA) theory/practice in 1990s.
Solving large, hard problems in principled way.
Moving to practice in important problem domains.
Still GA boobirds claim (1) no theory, (2) too slow, and (3) just voodoo.
How demonstrate results achieved so far in dramatic way? Billion bits or bust whitepaper (10/23/05) .
DEG lunch questions: A million? Sure. A billion? Maybe.
Naïve GA implementation for a billion bits:
~100 terabytes memory for population storage.
~2 72 random number calls.
Naïve approach goes nowhere.

Roadmap
What is a genetic algorithm?
1980s: Trouble in River City.
1990s: IlliGAL solves robustness problems.
2000s: Toward billion-variable optimization
Why this matters in practice: Example, multiscale materials and chemistry modeling
Challenges to using this in industry.
Efficiency enhancement in 4-part harmony.
Supermultiplicative speedups by mining evolved data.

What is a Genetic Algorithm (GA)?
Search procedure based on mechanics of natural selection and genetics .
Representation: GAs operate on codes:
Binary or Gray
Permutation
Integer or real
Program
Fitness function: Objective function, subjective function, ecological co-evolution
Population: Candidate solutions (individuals)
Genetic operators:
Selection: “Survival of the fittest”
Recombination: Combine parental traits to create offspring
Mutation: Modify an offspring slightly

1980s: Trouble in River City
1980s: Promise of robustness not realized.
First generation performance uncertain:
Sometimes effective, sometimes not.
Sometimes fast, sometimes slow.
How can we make GAs scalable? Competence.
How can we make them practical? Efficiency

1990s: Competent and Efficient GAs
90s: IlliGAL solves robustness puzzle.
Competence: Solve hard problems quickly, reliably, and accurately ( Intractable to tractable ).
Efficiency: Develop speedup procedures ( tractability to practicality ).
Principled design : [ Goldberg, 2002 ]
Relax rigor, emphasize scalability/quality.
Use problem decomposition.
Use facetwise models, and patchquilt integration using dimensional analysis.

Competent GAs Then: 1993, GA Kitty Hawk
Early 90s the facetwise theory came together.
Early 90s first competent GAs appeared.
First competent GA in 1993: fast messy GA (fmGA).
Original mGA complexity estimated: O( l 5 )
fmGA subquadratic on deceptive problem.
[Goldberg, Deb, Kargupta, & Harik, 1993]

Competent GAs Now: hBOA
Replace genetics with probabilistic model building: PMBGA or 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
Combines selectionist approach of GAs, statistical methods, and machine learning.
[US utility patent # 7,047,169]

Hierarchical BOA (hBOA) Facetwise model Current population Selection New population Bayesian network

hBOA on Hard Antenna Designs [Yu, Santarelli, & Goldberg, 2006]

Aiming for a Billion
Theory & algorithms in place.
Needed the guts to try.
Focus on key theory, implementation, & efficiency enhancements.
Theory keys :
Problem difficulty.
Parallelism.
Implementation key : compact GA.
Efficiency keys :
Various speedup.
Memory savings.
Results on a billion-variable noisy OneMax.

Theory Key 1: Master-Slave  Linear Speedup
Speed-up:
Max speed-up at
[Cantu-Paz & Goldberg, 1997; Cantú -Paz, 2000 ]
Near linear speed-up until

Theory Key 2: Noise Covers Most Problems
Adversarial problem design [Goldberg, 2002]
Blind noisy OneMax
P Fluctuating Deception Noise Scaling R

Implementation Key: Compact GA
Simplest probabilistic model building GA [Harik, Lobo & Goldberg, 1997; Baluja, 1994; M ü hlenbein & Paa ß , 1996]
Represent population by probability vector
Probability that i th bit is 1
Replace recombination with probabilistic sampling
Selectionist scheme
New population evolution through probability updates
Equivalent to GA with steady-state tournament selection and uniform crossover

Compact Genetic Algorithm (cGA)
Random initialization: Set probabilities to 0.5
Model Sampling: Generate two candidate solutions by sampling the probability vector
Evaluation: Evaluate the fitness of two sampled solutions
Selection: Select the best among the sampled solutions
Probabilistic model update: Increase the proportion of winning alleles by 1/n

Parallel cGA Architecture Processor #n p Sample bits 1-  /n p Select best individual Update probabilities Collect partial sampled solutions and combine Parallel fitness evaluation of sampled solutions Broadcast fitness values of sampled solutions Processor #1 Sample bits 1-  /n p Select best individual Update probabilities Processor #2 Sample bits 1-  /n p Select best individual Update probabilities

cGA is Memory Efficient: Θ (  ) vs. Θ (  1.5 )
Orders of magnitude memory savings via efficient GA
Example: ~32 MB per processor on a modest 128 processors for billion-bit optimization
Simple GA:
Compact GA:
Frequencies instead of probabilities (4 bytes)
Parallelization reduces memory per processor by factor of n p

Vectorization Yields Speedup of 4
SIMD instruction set allows vector operations on 128-bit registers
Equivalent to 4 processors per processor
Vectorize costly code segments with AltiVec/SSE2
Generate 4 random numbers at a time
Sample 4 bits at a time
Update 4 probabilities at a time

Other Efficiencies Yield Speedup of 15
Bitwise operations
Limited floating-point operations
Inline functions
Avoid using mod and division operations
Precomputing bit sums and indexing
Parallel, vectorized, and efficient GA:
Memory scales as  (  / n p ); Speedup scales as 60 n p
~32 MB memory, and ~10 4 speedup with 128 processors

GA Population Sizing
Additive Gaussian noise with variance  2 N
Population sizing scales: O( m 0.5 log m ) – O( m log m )
[Harik, et al, 1997] Noise-to-fitness variance ratio Error tolerance Signal-to-Noise ratio # Competing sub-components # Components (# BBs)

GA Convergence Time
Convergence time scales: O( m 0.5 ) – O( m )
GA scales as: O( m log m ) – O( m 2 log m )
[Miller & Goldberg, 1995; Goldberg, 2002; Sastry & Goldberg, 2002] Selection Intensity Problem size ( m · k )

GA Solves Billion-Variable Optimization Problem
Solved 33 million (2 25 ) bit problem to optimality.
Solved 1.1 billion (2 30 ) bit problem with relaxed, but guaranteed convergence
GA scales  (  ¢ log  ¢ (1+  2 N /  2 f ))

Do Problems Like This Matter?
Yes, for three reasons:
Many GAs no more sophisticated than cGA.
Inclusion of noise was important because it covers all difficulty (except deception).
Know how to handle deception and other problems through PMBGAs like hBOA.
Have experience solving tough problems with ordinary genetic and evolutionary algorithms:
Material science.
Chemistry.
Complex versions of these kinds of problems need billion-bit optimization.

Multiscale Nanomaterials Modeling
Accuracy of modeling depends on accurate representation of potential energy surface (PES)
Both values and shape matter
Ab initio methods:
Accurate, but slow (hours-days)
Compute PES from scratch
Faster methods:
Fast (seconds-minutes), accuracy depends on PES accuracy
Need direct/indirect knowledge of PES
Known and unknown potential function/method
Multiscaling quantum chemistry simulations [ Sastry et al, 2006 ]
Multi-timescaling alloy kinetics [ Sastry et al, 2004 ; Sastry et al, 2005 ]

Molecular dynamics (MD): (~10 –9 secs) many realistic processes are inaccessible.
Kinetic Monte Carlo (KMC): (~secs) need all diffusion barriers a priori . ( God or compute )
Efficient Coupling of MD and KMC
Use MD to get some diffusion barriers .
Use KMC to span time .
Use GP to regress all barriers from some barrier info.
Span 10 –15 seconds to seconds ( 15 orders of magnitude )
Multi-timescale Modeling of Alloys
chosen by the AIP editors as focused article of frontier research in Virtual Journal of Nanoscale Science & Technology , 12(9), 2005
Real time Complexity Table Lookup KMC On the fly KMC Symbolically Regressed KMC (sr-KMC)

 E calculated: » 3% (256) configurations
Low-energy events: <0.1% prediction error
Overall events: <1% prediction error
GP-Enhanced Kinetics Modeling
Total 2 nd n.n. Active configurations: 8192
Dramatic scaling over MD (10 9 at 300 K)
10 2 decrease in CPU time for calculating barriers
10 3 -10 6 less CPU time than on-they-fly KMC

Chemistry: GA-Enhanced Reaction Dynamics
Accurate excited-state surfaces with semiempirical methods
Permits dynamics of larger systems: proteins, nanotubes, etc.,
Energy & shape of the PES matter
Uknown PES functional form: Multi-timescaling alloy kinetics [ Sastry et al, 2004 ; Sastry et al, 2005 ]
Accurate but slow (hours-days) Can calculate excited states Fast (secs.-mins.), accuracy depends on parameters. Calculate integrals from fit parameters. Ab Initio Quantum Chemistry Methods Semiempirical Methods Tune Semiempirical Parameters [ Best paper and Silver “Humies” award . GECCO (ACM SIG conference) ]

MOGA Finds Physical and Accurate PES
MOGA results have significantly lower errors than current results.
Globally accurate PES yielding physical reaction dynamics
Each point is a set of 11 parameters
10 2 -10 5 increase in simulation time
10-10 3 times faster than current methods
Produces transferable SE parameters
Interpretable semiempirical methods

Do You Make a Million/Billion Decisions?
Materials and chemistry just examples: Increased complexity increases appetite for large optimization.
Modern design increasingly complex.
The Os all have many decisions to make:
Nano
Bio
Info
Generally systems increasingly complex:
~10 5 in a modern automobile
~10 7 parts in commercial jetliner.
Will be driven toward routine million/billion variable problems.
We get the warhead and then hold the world ransom for... 1 MILLION dollars !

Challenges to Routine Billion-Bit Optimization
What if you have large nonlinear solver (PDE, ODE, FEM, KMC, MD, whatever)?
Need efficiency enhancement:
Parallel
Time continuation
Hybridization
Evaluation Relaxation
Take 100-node cluster, and 26% speed increase of other effects:
Combined effect is multiplicative: 100*1.26*1.26*1.26 ≈ 200 .
Good, but not good enough.

Data-Mined Evaluation Relaxation as Key
Steps:
Collect evolutionary stream of data : Solution sets and function values.
Build structural model of key modules & relationships (e.g., hBOA).
Build fitness surrogate using structural model.
Substitute surrogate evaluations in place of expensive evaluations.
Need sample small fraction of expensive evaluations.
Sounds too good to be true: something for nothing?
Not really, analog to human cognition.

Supermultiplicative Speedups
Synergistic integration of probabilistic models and efficiency enhancement techniques
Evaluation relaxation
Learn structural model
Induce surrogate fitness from structural model
Estimate coefficients using standard methods.
Only 1-15% individuals need evaluation
Speed-Up: 30–53
[ Pelikan & Sastry, 2004 ; Sastry, Pelikan & Goldberg, 2004

Summary
What is a genetic algorithm?
1980s: Unrealized promise of robustness.
1990s: Increasing competence (scalability) & efficiency.
2000s: Toward billion-variable optimization.
Why this matters in practice: Multiscale materials and chemistry modeling as two examples.
Challenges to routine billion-bit optimization.
Efficiency enhancement in 4-part harmony.
Supermultiplicative speedups by mining evolved data & fitness surrogates.

Conclusions
Big hardware was a new frontier 20 years ago.
Big optimization is a frontier today:
Take extant cluster computing.
Mix in robustness lessons of the 90s.
Efficiency enhancement of the 2000s.
Integrate into problems.
This technology can create competitive advantage for industries visionary enough to grab hold:
In the O’s (bio, nano, info)
Large-scale complex systems.
Find out more by engaging UIUC and NCSA work, today.

More Information
Billion-bit article in Complexity ( http://www3.interscience.wiley.com/cgi-bin/abstract/114068026/ABSTRACT?CRETRY=1&SRETRY=0 ) and tech report ( ftp://ftp-illigal.ge.uiuc.edu/pub/papers/IlliGALs/2007007.pdf ).
Illinois Genetic Algorithms Laboratory: http://www- illigal.ge.uiuc.edu / .
The Design of Innovation: Lessons from and for Competent Genetic Algorithms (Kluwer, 2002) .
Speakers: [email_address] , [email_address] .
http://www.amazon.com/exec/obidos/tg/detail/-/1402070985/ref=pd_sl_aw_alx-jeb-9-1_book_5065571_2

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A Billion Bits or Bust

by David E. Goldberg on May 22, 2007

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