Abstract Goal optimization has long been a topic of great interest in computer science. The literature contains many thousands of papers that discuss methods for the search of optimal solutions to complex problems. In the case of multi-objective optimization, such a search yields iteratively improved approximations to the Pareto frontier, i.e. the set of best solutions contained along a trade-off curve of competing objectives. To approximate the Pareto frontier, one method that is ubiquitous throughout the field of optimization is stochastic search. Stochastic search engines explore solution spaces by randomly mutating candidate guesses to generate new solutions. This mutation policy is employed by the most commonly used tools (e.g. NSGA-II, SPEA2, etc.), with the goal of a) avoiding local optima, and b) expand upon diversity in the set of generated approximations. Such "blind" mutation policies explore many sub-optimal solutions that are discarded when better solutions are found. Hence, this approach has two problems. Firstly, stochastic search can be unnecessarily computationally expensive due to evaluating an overwhelming number of candidates. Secondly, the generated approximations to the Pareto frontier are usually very large, and can be difficult to understand. To solve these two problems, a more-directed, less-stochastic approach than standard search tools is necessary. This thesis presents GALE (Genetic Active Learning). GALE is an active learner that finds approximations to the Pareto frontier by spectrally clustering candidates using a near-linear time recursive descent algorithm that iteratively divides candidates into halves (called leaves at the bottom level). Active learning in GALE selects a minimally most-informative subset of candidates by only evaluating the two-most different candidates during each descending split; hence, GALE only requires at most, $2LogN$ evaluations per generation. The candidates of each leaf are thereafter non-stochastically mutated in the most promising directions along each piece. Those leafs are piece-wise approximations to the Pareto frontier. The experiments of this thesis lead to the following conclusion: a near-linear time recursive binary division of the decision space of candidates in a multi-objective optimization algorithm can find useful directions to mutate instances and find quality solutions much faster than traditional randomization approaches. Specifically, in comparative studies with standard methods (NSGA-II and SPEA2) applied to a variety of models, GALE required orders of magnitude fewer evaluations to find solutions. As a result, GALE can perform dramatically faster than the other methods, especially for realistic models.
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Joseph Krall In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science. College of Engineering and Mineral Resources Faster Evolutionary Multi-Objective Optimization via GALE, the Geometric Active Learner a Ph.D. Final Defense Presentation for the Special Thanks to the NASA Ames Research Center The Lane Department of Computer Science and Electrical Engineering at April 21, 2014 Estimated Duration: 45 minutes
Transcript
Joseph Krall
In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science.
College of Engineering and Mineral Resources
Faster Evolutionary Multi-Objective Optimization via GALE, the Geometric Active Learner
a Ph.D. Final Defense Presentation for the
Special Thanks to the NASA Ames Research Center
The Lane Department of Computer Science and Electrical Engineering
at
April 21, 2014
Estimated Duration: 45 minutes
4/21/2014 Faster Multi-Objective Optimization via GALE
A Thesis Proposal
- “JMOO: Tools for Faster Multi-Objective Optimization”
Comments from Committee
- Lacking Rigor
- Generalizability of Proposal
- Lacking Details / Misunderstandings
- Some Missing Related Works
- Validity Concerns
- Needed More – Not Substantial Enough
Last Time 1. Introduction
November, 2013
SE or CS?
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Final Dissertation - “Faster Multi-Objective Optimization via GALE” Key Changes from Proposal - Focus on Contributions of GALE - Focus on Assessing and Validating GALE - Very rigorous experimental methodology - Addressing Comments from Proposal - Expansive Related Works - Formalizing the Field - MANY more experimental results
This Time
Spring!
…Sort of
April, 2014
1. Introduction
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Search & Optimization of Goals - the art of decision making - e.g. shortest time city navigation - e.g. managing calorie intake for diets Not always trivial - Landing an airplane safely - Maximizing software project profits
MOO = Multi-Objective Optimization - Draft solutions to a problem (red) - Find Pareto Frontiers (green) - Report to a decision maker
This Thesis
Areas on the Pareto frontier
Rejected Solutions
Who do I pick???
1. Introduction
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GALE shown to be a strong rival to NSGA-II & SPEA2
And The Results
Two orders of magnitude fewer evaluations for all
models
Two orders of magnitude faster (seconds) for big
models
Better Solution Quality
SPEA2 much slower
GALE Never worse NSGA-II/SPEA2 Never better
1. Introduction
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Background 2
In this chapter: - Formalities - Definitions
- Related Works
1. Introduction
2. Background
3. MOO Critique
4. GALE
5. Models
6. Experiments
7. Validity
8. Conclusion
10 Slides
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Mathematical Programming: [Dantzig] - The aim is to find solutions that optimize objectives - Transformation functions transform decisions (x) into objectives (y) - Solutions are infeasible if they do not satisfy constraint functions
Formalities 2. Background
objectives
Constraint functions Optimality direction
Transformation functions
a. Defines
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Lab Problems
- Schaffer, Viennet, Tanaka, etc.
Real-world Problems
- Simulations
- Too complex for math
- Aircraft Safety
- Software Dev. Profit
Kinds of Models
The Schaffer Model
2. Background a. Defines
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Early methods assumed math models
- A bad assumption for real world practicality
They also assume other aspects:
- Concave vs. Convex
- Differentiability
- Linear vs. Non-linear
- Single vs. Multi-objective
- Objective Functions vs. Simulation
Numerical Optimization 2. Background b. Early Methods
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Exterior Search [Dantzig]
- For Linear problems ( [Nelder & Mead 1965] made a non-linear version)
- Embed a simplex with solutions along the vertices
- Traverse along the nodes
- Good average Complexity
- But bad O(N3) worst case
Simplex Search
Nelder, John A.; R. Mead (1965). "A simplex method for function minimization". Computer Journal 7: 308–313.
2. Background b. Early Methods
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Karmarkar’s Algorithm – [Karmarkar 1984]
- Good for big data
- Fast convergence
- Polynomial complexity
- 50x faster than Simplex
- Single-Objective Only
- Requires Concavity
Interior Point Methods
Narendra Karmarkar (1984). "A New Polynomial Time Algorithm for Linear Programming", Combinatorica, Vol 4, nr. 4, p. 373–395.
2. Background b. Early Methods
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Moving onward from Numerical Methods
- Improve a heuristic, not the actual objectives
- Hill Climbing: Accept only improved steps
- Tabu Search: Refuse only recently attempted steps
- Simulated Annealing: Early bad okay, late bad refused
Heuristic-based Searches 2. Background c. Recent Methods
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Particle Swarm Optimization [Kennedy 1995]
- Real life swarms; flocks of birds, etc
- Swarm towards good solutions
- Self best and Pack best
Ant Colony Optimization [Dorigo 1992]
- Ant Colony Path Searches
- Pheromone density = best path
PSO & ACO
Kennedy, J.; Eberhart, R. (1995). "Particle Swarm Optimization". Proceedings of IEEE International Conference on Neural Networks IV. pp. 1942–1948.
M. Dorigo, Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italy, 1992.
2. Background c. Recent Methods
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Standard EA (Evolutionary Algorithm): 1) Build initial population
2) Repeat for max_generations:
a) crossover
b) mutation
c) select
3) Return final population
Evolutionary Algorithms
a+b) Build Offspring: Perturb Population c) Combine Offspring + Population c) Cull the worst solutions to retain Population Size
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NSGA-II [Deb 2002]
- Non-dominated Sorting Genetic Algorithm
- Standard select+crossover+mutation
- Sort by ‘bands’, or domination ‘depth’
- Break ties based on density
- crowding distance
NSGA-II
Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II". IEEE Transactions on Evolutionary Computation 6 (2): 182
2. Background c. Recent Methods
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SPEA2 [Zitzler2002]
- Strength Pareto Evolutionary Algorithm
- Standard select+crossover+mutation
- Sort by ‘strength’: count of solutions someone dominates
- Truncate crowded solutions via nearest neighbor
SPEA2
E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization. Evolutionary Methods for Design Optimization and Control with Applications to Industrial Problems, 95--100, 2001.
2. Background c. Recent Methods
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MOO Critique 3
In this chapter: - Survey - Rigor
1. Introduction
2. Background
3. MOO Critique
4. GALE
5. Models
6. Experiments
7. Validity
8. Conclusion
4 Slides
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Experimental Rigor
- Want to maximize validity
- Because reasons to doubt GALE
- Still does good with few evals?
- Can still run fast?
We looked at literature for advice - Search query targeted these questions:
* J. Demsar, “Statistical comparisons of classifiers over multiple data sets,” ˇ J. Mach. Learn. Res., vol. 7, pp. 1–30, Dec. 2006.
Population size? - 20 ~ 100 is good. - Over 200 is a waste
Number of Repeats? - [Harman 2012]: 30-50 is common. - This Thesis: 20.
* M. Harman et al., Search based software engineering: techniques, taxonomy, tutorial. In Empirical Software Engineering and Verification, Bertrand Meyer and Martin Nordio (Eds.). Springer-Verlag, Berlin, Heidelberg 1-59.
3. MOO Critique
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1. Use variety of models – Real World Models: Practicality.
– Standard Models: Reproducibility.
– Constrained and Unconstrained: Generalizability
2. How many Repeats – Pragmatics: Keep repeats low to save on computational cost
– Statistics: Want high repeats for statistical stability
– The middle ground: for n in 20,30,40: no change. So 20 is good.
Half of the papers neglected to report evaluations
Almost no one had a threats to validity section in their paper
3. MOO Critique
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GALE 4
1. Introduction
2. Background
3. MOO Critique
4. GALE
5. Models
6. Experiments
7. Validity
8. Conclusion
In this chapter: - Spectral Learning - Active Learning
5 Slides
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GALE: Geometric Active Learning (Evolution)
- At most O(2Log2N) evaluations per generation
- Exactly Θ(2N) evaluations for NSGA-II, SPEA2
Main Differences in GALE:
- cluster solutions
- evaluate some, not all
- Directed vs random
- More on these later
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Introducing GALE 4. GALE
GALE NSGA-II SPEA2
Asymptotic Notation: Big-O: worst case
Big-Theta: Exact case
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Three key phrases to talk about
1. Active Learning - Minimize cost of evaluation
- Learn more from using less [Settles 2009]
2. Spectral Learning (WHERE) - Reasoning with eigenvectors via covariance matrix
- “Spectral Clustering” – via eigenvectors
- FastMap finds eigenvectors faster than PCA
3. Directed Search - Shove solutions along promising directions
Components to GALE
some, not all
clustered spectrally
Directed mutation
4. GALE
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Algorithm shown here and explained over next several slides - WHERE algorithm - WHERE uses FastMap - Directed Mutation
1. Build initial population, P0. Initialize generation: t = 0. Set Life = 3. 2. Repeat until stopping criteria is met (stop if life == 0):
a. Run WHERE (with pruning) to select Rt = dominant leafs from WHERE. b. Perform Directed Mutation on members of Rt. c. Copy Rt into Pt+1 and generate new random candidates until new population is full. d. Increment generation number t = t + 1. e. Collect stats and evaluate stopping criteria. Decrement life if no improvement to any
objective.
3. Run WHERE (without pruning) to select Rt = dominant leafs from WHERE. 4. Rt contains approximations to the Pareto frontier.
GALE Pseudo-Code GALE
Spectral Learning Active Learning Directed Search
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Spectral clustering is O(n3) [Kumar12]
- Common method: PCA
- The Nystrom Method reduces to near-linear
- Low-rank approx. of covariance matrix
e.g.: FastMap is a Nystrom Algorithm [Platt05] - 1) Pick an arbitrary point, z.
- 2) Let ‘east’ be the furthest point from z.
- 3) Let ‘west’ be the furthest point from ‘east’.
- 4) Project all points onto the line east-west
- 5) east-west is the first principal component
Nystrom Method GALE
east
west
c
b
a x
Active Learning: - Only evaluate East & West!
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WHERE = Spectral Learning in GALE
- Similar to Boley’s PDDP: find first eigenvector and recursively split
- PDDP uses PCA. WHERE uses FastMap.
The WHERE Tool GALE
Initial population
WHERE clusters initial population = Spectral Learning
Only evaluate the best clusters =
Active Learning
Mutate along those clusters = Directed Search
At Most 2Log2(NG) Evaluations (N=Population Size. G=Number of Generations)
Refill the Population
Non-dominated clusters
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Models 5
1. Introduction
2. Background
3. MOO Critique
4. GALE
5. Models
6. Experiments
7. Validity
8. Conclusion
In this chapter: - CDA
- POM3 - Lab Models
4 Slides
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5. Models
Continuous Descent Arrival
- NASA wants to know if CDA is doable
- Standard descents are less efficient than CDA -> more {noise, time, fuel, $$$}
- CDA might unnecessarily strain air traffic control (ATC)
CDA Model a. CDA
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Lots of work - 2 months at NASA Ames Research Center
- CDA not pre-assembled
Inspiration from 2013 Asiana Flight Crash - Pilots had to do unusually more tasks than normal
- Keeping airspeed nominal was a task they ‘forgot’
- Human Factors model a pilot ‘HTM’ = maximum human taskload
Goal of CDA: less forgetting, less time from delays and missed tasks
* based on Work Models that Compute by Pritchett, Kim and Feigh, 2011-2013
Building CDA 5. Models a. CDA
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POM3
- Model of Agile Software Requirements Engineering
