iciency Enhancement of Genetic Algorithms via Building-Block-Wise Fitness Estimation
ra Sastry1, Martin Pelikan2, David E. Goldberg1
is Genetic Algorithms Laboratory, University of Illinois at Urbana-Champaign, 2University of Missouri at St. Louis.
AFOSR F49620-03-1-0129, NSF DMR 03-25939, NSF DMR-0121695, DOE DEFG02-91ER45439, and TRECC-ONR N00014-01-1-0175
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Extended Compact Genetic Algorithm (eCGA) [Harik, 1999] Evaluation Relaxation in eCGA: An Algorithmic Description Scalability and Speed-Up Analysis
ent genetic algorithms—genetic algorithms that solve hard problems quickly,
An estimation distriubtion algorithm (EDA) 1. Initialization: Randomly create individual in the initial population Scalability: # Function evaluations required to
and accurately—have been shown to successfully solve nearly-decomposable
obtain a solution of quality 1-1/m
Builds models of good solutions as linkage groups 2. Evaluation: Compute fitness of all individuals in the population
dly-difficult search problems, requiring only a polynomial (usually subquadratic)
Function-Evaluation ratio:
Replace traditional variation operators of GAs by: 3. Selection: Assign more copies to better individuals
of function evaluations. However, for large-scale problems, the task of computing
o Building a probabilistic model of promising solutions
ubqudratic number of function evaluations can be daunting, especially if the 4. Model building: Optimize the structure and parameters of the probabilistic model to
valuation is a complex simulation, model, or computation.Therefore, when o Sampling the model to generate new candidate solutions. best fit the selected solutions
difficult large-scale problems, we must use one or more efficiency-enhancement Key Idea: Good probability distribution ≡ Linkage learning 5. Fitness-Estimation model building: Estimate the sub-structural fitness values
es such as parallelization, hybridization, time continuation, and evaluation
Key Components: 6. Model sampling: Sample the model to create new candidate solutions (offspring)
n to take tractable competent GAs to practicality.
Model Representation: Marginal product model (MPM) Speed-Up: Ratio of # function evaluations
7. Fitness estimation: Estimate the fitness of an offspring with the internal fitness-
required with evaluation-relaxation scheme to that
estimation model with probability pi
o Partition genes into (mutually) independent groups
ct
without it
o Marginal distribution of gene partitions 8. Evaluation: Compute the fitness of offspring with probability 1-pi
ign an evaluation-relaxation scheme—that provides efficiency enhancement by
Neglect the cost of building and using the
Class-Selection Metric: Minimal description length (MDL) [Rissanen, 1978] 9. Repeat steps 3–8 till one or more convergence criteria is satisfied
g accurate, but expensive fitness evaluations with inexpensive, but less accurate
fitness-estimation model
o Occam’s razor principle: All things being equal, simpler models are better than
stimation models—in a principled manner. The proposed technique builds and
complex ones
babilistic models that automatically and adaptively identify important regularities
o Sum of two components: (1) Model complexity, and (2) Model accuracy
structures of the underlying search problem. The probabilistic model of the
Class-Search Method: Greedy search heuristic
tures is subsequently used as the basis for deriving an endogenous fitness-
o Start with the lowest-complexity model and increase the complexity as long as the
on model.
class-selection metric improves.
orm a scalability and speed-up analysis of the proposed scheme by deriving Only ∼1 – 15% of population requires evaluation
e models for convergence time and population sizing. The scalability analysis
Optimal pi: 0.85 – 0.99
hat for additively separable problems, the proposed method requires fitness Illustration of Probabilistic Model Building in eCGA
Provides maximum speed-up of 1.75 – 53
on for only 1–10% of the individuals, and thereby yielding a speed-up of 2.25–50.
Structure MDL value Probabilities
[X0] [X1] [X2] [X3] [X4] [X5] [X6] [X7] [X8] [X9] [X10] [X11] 1.0000 p(X0=0), p(X1=0), …, p(X11=0)
Future work
oud & Motivation [X0] [X1] [X2] [X3] [X4 X5] [X6] [X7] [X8] [X9] [X10] [X11] 0.9933 p(X0),…, p(X3), p(X4X5), p(X6),…, p(X11)
Problems with overlapping building blocks
[X0] [X1] [X2] [X3] [X4 X5 X7] [X6] [X8] [X9] [X10] [X11] 0.9819 p(X0),…, p(X3), p(X4X5X7), p(X6),…, p(X11)
petent Genetic Algorithms
Effect is similar to exogenous noise
[X0] [X1] [X2] [X3] [X4 X5 X6 X7] [X8] [X9] [X10] [X11] 0.9644 p(X0),…, p(X3), p(X4X5X6X7), p(X8),…, p(X11)
Solve hard problems quickly, reliably, and accurately
The speed-up might be lower than the present case
[X0] [X1] [X2] [X3] [X4 X5 X6 X7] [X8 X9] [X10] [X11] 0.9581 p(X0),…, p(X3), p(X4X5X6X7), p(X8X9),…, p(X11)
Facetwise models of convergence time and population sizing
Intractabiliy Tractability
Non-Uniformly scaled problems
[X0] [X1] [X2] [X3] [X4 X5 X6 X7] [X8 X9 X11] [X1o] 0.9454 p(X0),…, p(X3), p(X4X5X6X7), p(X8X9X11), p(X10)
Error in fitness estimation modeled as an additive
Polynomial (usually sub-quadratic) scalability
[X0] [X1] [X2] [X3] [X4 X5 X6 X7] [X8 X9 X1o X11] 0.9273 p(X0),…, p(X3), p(X4X5X6X7), p(X8X9X10X11) Building blocks have different salience
Gaussian noise
o Daunting for complex real-world problems
[X0] [X1 X3] [X2] [X4 X5 X6 X7] [X8 X9 X1o X11] 0.9211 p(X0),p(X1X3),p(X2),p(X4X5X6X7),p(X8X9X10X11) o Sequentiality in solving: Most salient Least salient
o 1000 variables, 10 sec/eval ~120 days!
[X0] [X1 X2 X3] [X4 X5 X6 X7] [X8 X9 X1o X11] 0.9077 p(X0), p(X1 X2 X3), p(X4X5X6X7), p(X8X9X10X11)
Hierarchical problems
ciency Enhancement Techniques [X0 X1 X2 X3] [X4 X5 X6 X7] [X8 X9 X1o X11] 0.8895 p(X0X1 X2 X3), p(X4X5X6X7), p(X8X9X10X11)
Important class of nearly decomposable search problems
Tractability Practicality
Convergence time: Additional design necessary to integrate niching along with fitness-estimation m
Evaluation Relaxation: Replace accurate, but expensive fitness evaluation with
Endogenous Sub-Structural Fitness-Estimation Model Selection-Intensity [Bulmer, 1980] based model Real-World problems
inexpensive, but less accurate evaluation
[Mühlenbein & Schlierkamp-Voosen, 1993;
Probabilistic model provides fitness structure Simple inheritance provides significant speed-up [Smith, Dyke, & Stegmann, 1
Fitness models: Exogenous vs. Endogenous
Thierens & Goldberg 1993; Bäck, 1994; Miller &
Use the sub-structures as the basis for the fitness-estimation model Interactive and human-based evolutionary algorithms
Limited studies on endogenous models Goldberg, 1995 & 1996; Goldberg, 2002; Sastry
Estimate fitness of sub-structures (or linkage groups) Fitness evaluation is very costly
o Fitness inheritance: Some offspring receive fitness from their parents through some & Goldberg, 2002]
Use only evaluated individuals for sub-structural fitness estimation
averag [Smith et al, 1994; Sastry et al 2001; Chen et al 2002] o Infeasible to use large population even if one is warranted
Convergence-Time ratio: Isolate the effect of
o Yields only a modest speed-up of 1.25 when 50% of the offspring individuals receive evaluation-relaxation scheme Use fitness estimation model to screen candidate solutions
Average fitness of all Average fitness of all
inherited fitness and the fitness of the rest are evaluated. Fitness of
= evaluated individuals with – evaluated individuals
substructure instance Summary & Conclusions
the schema instance in the population
Proposed and designed an evaluation relaxation scheme in a principled manne
ive
Probabilistic model of important substructures of the search problem
cipled design of an evaluation-relaxation scheme Neglected cumulative effects of fitness
Estimate the fitness values of different substructure instances
Scalable and provides significant speed-up estimation
Endogenous fitness-estimation model based on sub-structure fitnesses
an endogenous probabilistic fitness model Population size:
Performed scalability analysis of the proposed scheme
Identify and exploit regularities of the search problem Gambler’s ruin model
[Harik, Cantu-Paz, Goldberg, and Miller, 1997] Facetwise models of convergence time and population sizing
Automatically and adaptively identify fitness structure Estimated fitness of an individual is a function of its sub-structural fitness values
Population-Sizing ratio: Requires fitness evaluation of 1 – 15% of the indivduals
form scalability and speed-up analysis
Provides significant speed-up of 1.75 – 53
Sum of indivudal’s sub-structure fitness values
Derive and apply “little models” using dimensional analysis and facetwise theory
o Works well for nearly additively separable problems. Additively separable problems of bounded difficulty
Optimize parameters to provide maximum speed-up
Can use other advanced methods such as neural networks, response surface Drastically alleviates the large population requirement of EDAs
Test against problems of bounded difficulty to ensure scale up to all easier
methods, and design of experiments
problems. Enables solving of real-world problems in practical time