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Implementation of Multi-model Approach for Estimation of Distribution Algorithms (EDA) to Solve Multi-Structural Problems By Mohammad Reza Karimi Dastjerdi Supervisor: Dr. Amin Nikanjam Summer 2016 1 Cont ntent ents Introduction

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By Mohammad Reza Karimi Dastjerdi Supervisor: Dr. Amin Nikanjam

Implementation of Multi-model Approach for Estimation of Distribution Algorithms (EDA) to Solve Multi-Structural Problems

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Summer 2016

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Cont ntent ents

■ Introduction ■ Heuristic strategies ■ Stochastic Heuristic Searches ■ Genetic Algorithm ■ Estimation of Distribution Algorithms ■ EDAs Pseudo code ■ Bayesian Optimization Algorithm ■ BOAs Pseudo code ■ Bayesian Networks ■ Bayesian Dirichlet Metric ■ Find Best Network ■ Generate New Solutions

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■ Selection Operators ■ Parent Selection ■ Truncation Selection ■ Tournament Selection ■ Survivor Selection ■ Worst Replacement ■ Restricted Tournament Replacement ■ Multi-model Approach ■ Multi-structural Problems ■ Conclusion & Suggestions ■ References

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Intr ntroduction

  • duction

■ Optimization is a mathematical discipline that concerns the finding of the extreme of numbers, functions, or systems. ■ All the search strategy types can be classified as:

  • Complete strategies : All possible solutions of the search space
  • Heuristic strategies : Part of them following a known algorithm

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He Heuristic uristic st strat rategies gies

■ Deterministic

  • Same Conditions → Same Solution
  • Main drawback → Risk of getting stuck in local optimu

imum m value ues ■ Non-deterministic

  • Even in Same Conditions → Different Solutions
  • Escape from these local maxima by means of the randomnes
  • mness

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Stocha chastic stic Heuristic euristic Sea earche ches

■ Store one solution: Simulated Annealing ■ Population-based: Evolutionary Computation.

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Ge Gene netic tic Alg lgorithms

  • rithms

■ One of the examples of evolutionary computation is Genetic Algorithms (GAs). ■ Inspired by the process of natural selection ■ Relied on bio-inspired operators such as mutation, crossover and selection. ■ The behavior of GAs depends on too many parameters like:

  • Operators and probabilities of crossing and mutation
  • Size of the population
  • Rate of generational reproduction
  • Number of generations
  • and so on.

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Es Estim timation tion of f Di Distri stribution bution Alg lgorith

  • rithms

ms

■ All these reasons have motivated the creation of a new type of algorithms classified under the name of Estimation of Distribution Algorithms (EDAs). ■ EDAs are:

  • Population-based
  • Probabilistic modelling of promising solutions
  • Simulation of the induced models to guide their search

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ED EDAs s Ps Pseudo eudo co code de

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Bayesian esian Optim ptimizat ization ion Alg lgorithm

  • rithm

■ By Pelikan, Goldberg, & Cantu-paz [1998]

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BOAs As Ps Pseudo eudo co code de

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Bayesian esian Netw Networ

  • rks

ks

■ A Bayesian network encodes the relationships between the variables contained in the modeled data. ■ Bayesian networks can be used to:

  • Describe the data
  • Generate new instances of the variables with similar properties as those of given data

■ Each node → one variable. ■ 𝑞 𝑌 =

𝑗=0 𝑜−1

𝑞 𝑌𝑗 𝑌𝑗

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Bayesian esian Di Diri rich chle let t Metric Metric

■ By Heckerman et al [1994] ■ Prior knowledge about the problem + Statistical data from a given data set. ■ The BD metric for a network B given a data set D and the background information 𝜊 denoted by 𝑞 𝐸, 𝐶 𝜊 is defined as: 𝒒 𝑬, 𝑪 𝝄 = 𝒒 𝑪 𝝄

𝒋=𝟏 𝒐−𝟐 𝝆𝒀𝒋

𝒏′ 𝝆𝒀𝒋 ! 𝒏′ 𝝆𝒀𝒋 + 𝒏 𝝆𝒀𝒋 !

𝒚𝒋

𝒏′ 𝒚𝒋,𝝆𝒀𝒋 + 𝒏 𝒚𝒋,𝝆𝒀𝒋 ! 𝒏′ 𝒚𝒋, 𝝆𝒀𝒋 ! 𝑛 𝜌𝑌𝑗 =

𝑦𝑗

𝑛 𝑦𝑗, 𝜌𝑌𝑗

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Fi Find nd Best est Netw Networ

  • rk

■ K = 0

  • An empty network is the best one (and the only one possible).

■ K = 1

  • special case of the so-called maximal branching problem

■ K > 1

  • NP-complete
  • A simple greedy search
  • Only operations are allowed that:
  • keep the network acyclic
  • at most k incoming edges into each node
  • The algorithms can start with:
  • An empty network
  • Best network with one incoming edge for each node
  • Randomly generated network.

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Ge Generat nerate Ne New Solutions lutions

■ Time complexity of generating the value for each variable: 𝑃 𝑙 . ■ Time complexity of generating an instance of all variables: 𝑃 𝑜𝑙 .

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Mu Multi lti-model model Appr pproach

  • ach

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■ Many real-world problems present several global optima. ■ Most EDAs typically bypass the issue of global multi-modality. ■ It is often preferable or even necessary to obtain as many global optima as possible. ■ Used in ECGA by Chuang and Hsu [2010].

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Implement plementation ation

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Results esults

Average verage fitne ness calls ls Average verage model- building lding time Average verage algorit

  • rithm

hm time Final al Po Populat ulation ion size RT RTR? #Models dels Problem lem Size 6822 0.019 0.028 1312 1 30 16416 0.021 0.015 1812 3 30 15995 0.022 0.028 734 1 3 30 24 200 400 600 800 1000 1200 1400 1600 1800 2000 Traditional Multi-Models Multi-Models with RTR Populatio ulation Size Differen fferent t Approac roaches hes 2000 4000 6000 8000 10000 12000 14000 16000 18000 Traditional Multi-Models Multi-Models with RTR Average erage Fitne ness Calls ls Differen fferent t Approac roaches hes

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Implement plementation ation

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𝑂𝑛𝑝𝑒𝑓𝑚𝑗 = 𝑔

𝑗

𝑘=1

𝑁𝑝𝑒𝑓𝑚𝑡

𝑔

𝑘

𝑂𝑛𝑝𝑒𝑓𝑚𝑗 = 𝑓𝑔𝑗 𝑘=1

𝑁𝑝𝑒𝑓𝑚𝑡 𝑓𝑔𝑘

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Results esults

Average verage fitne ness calls ls Average verage model el- building lding time Average verage algorit

  • rithm

hm time Final al Po Populat ulation ion size SoftMa Max? RT RTR? #Models dels Problem lem Size 6822 0.019 0.028 1312 1 30 143689 0.193 0.126 15250 1 3 30 26580 0.039 0.039 578 1 1 3 30 26 2000 4000 6000 8000 10000 12000 14000 16000 18000 Traditional Multi-Models with Softmax Multi-Models with Softmax and RTR Populatio ulation Size Differen fferent t Approac roaches hes 20000 40000 60000 80000 100000 120000 140000 160000 Traditional Multi-Models with Softmax Multi-Models with Softmax and RTR Avera erage ge Fitness ness Calls ls Differen fferent t Approac roaches hes

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Implement plementation ation

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Results esults

Average verage fitne ness calls ls Average verage model el- building lding time

Averag age algo gorit ithm hm time

Final Popu pulati tion

  • n size

Paren ent Displac lacem emen ent? t? SoftMax Max? RTR? #Models dels Problem lem Size 6822 0.019 0.028 1312 1 30 7731 0.009 0.038 1562 1 3 30 7204 0.012 0.071 921 1 1 3 30 28 200 400 600 800 1000 1200 1400 1600 1800 Traditional Multi-Models with Parent Displacement Multi-Models with Parent Displacement and RTR Populatio ulation Size Differen fferent t Approac roaches hes 1000 2000 3000 4000 5000 6000 7000 8000 9000 Traditional Multi-Models with Parent Displacement Multi-Models with Parent Displacement and RTR Average erage Fitne ness Calls ls Differen fferent t Approac roaches hes

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Implement plementation ation

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Results esults

Average verage fitnes ess calls ls Average verage model- building lding time Average verage algorit

  • rithm

hm time Final al Po Population ulation size Po Pop. Type Paren ent Displac lacem emen ent? t? SoftMax Max? RT RTR? #Models

  • dels

Problem lem Size 6822 0.019 0.028 1312

  • 1

30 7731 0.009 0.038 1562 Single 1 3 30 7204 0.012 0.071 921 Single 1 1 3 30 14871 0.015 0.058 3125 Multi 1 3 30 30 500 1000 1500 2000 2500 3000 3500 Traditional SinglePop.

  • SinglePop. and

RTR Multi Pop. Populatio ulation Size Differen fferent t Approac roaches hes 2000 4000 6000 8000 10000 12000 14000 16000 Traditional Single Pop. Single Pop. and RTR Multi Pop. Average erage Fitne ness Calls ls Differen fferent t Approac roaches hes

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Mu Multi lti-st structur ructural al Pr Problems blems

■ Deceptive Behavior! 𝑁𝑇𝑄1 𝑦 = 𝛽 + 𝑔

𝑢𝑠𝑏𝑞 𝑛,𝑙 𝑦2, … , 𝑦𝑜

𝑦1 = 1 𝑜 − 1 − 𝑔

𝑃𝑜𝑓𝑁𝑏𝑦 𝑦2, … , 𝑦𝑜

𝑦1 = 0 𝑁𝑇𝑄2 𝑦 = 𝑛𝑏𝑦 𝛽 + 𝑔

𝑢𝑠𝑏𝑞 𝑛,𝑙 𝑦 , 𝑜 − 1 − 𝑔 𝑃𝑜𝑓𝑁𝑏𝑦 𝑦

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MS MSP1 P1 Test est Results esults

32 Average verage fitnes ess calls ls Average verage model- building lding time Average verage algorit

  • rithm

hm time Final al Po Populat ulation ion size Paren ent Replac acem emen ent? t? RT RTR? #Models dels Problem lem Size 24897 0.066 0.093 5750

  • 1

31 27840 0.022 0.1 6875 1 3 31 23347 0.022 0.163 4875 1 1 3 31 33907 0.042 0.311 8250 1 5 31 31812 0.019 0.218 6250 1 1 5 31 1000 2000 3000 4000 5000 6000 7000 8000 9000 Traditional Three Models Three Models and RTR Five Models Five Models and RTR Populatio ulation Size Diffe fferen rent t Approac roache hes 5000 10000 15000 20000 25000 30000 35000 40000 Traditional Three Models Three Models and RTR Five Models Five Models and RTR Average erage Fitne ness Calls ls Differen fferent t Approac roaches hes

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MS MSP2 P2 Test est Results esults

33 Average verage fitnes ess calls ls Average verage model el-building uilding time Average verage algori

  • rithm

hm time Final al Po Populat ulation ion size Paren ent Replac acem emen ent? t? RT RTR? #Models els Problem lem Size 596960 2.949 4.231 164000

  • 1

30 238680 0.379 2.327 78000 1 3 30 201750 0.089 0.697 25000 1 1 3 30 256660 0.237 2.797 82000 1 5 30 185925 0.067 0.87 18500 1 1 5 30 20000 40000 60000 80000 100000 120000 140000 160000 180000 Traditional Three Models Three Models and RTR Five Models Five Models and RTR Populatio ulation Size Diffe fferen rent t Approac roache hes 100000 200000 300000 400000 500000 600000 700000 Traditional Three Models Three Models and RTR Five Models Five Models and RTR Average erage Fitne ness Calls ls Different fferent Approa roaches hes

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Conclusions nclusions an and Sug uggest gestions ions

■ The proposed method seems Nice! ■ Method of learning: Sporadic or Incremental ■ Number of Models

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Ref eference erences

■

  • E. Bengoetxea, "Inexact Graph Matching Using Estimation of Distribution Algorithms," Paris, 2002

■

  • M. Pelikan, D. E. Goldberg and E. E. Cantú-paz, "Linkage Problem, Distribution Estimation, and

Bayesian Networks," Evolutionary Computation, pp. 311-340, 2000 ■

  • T. Blickle and L. Thiele, "A comparison of selection schemes used in evolutionary algorithms,"

Evolutionary Computation, vol. 4, no. 4, pp. 361-394, 1996 ■

  • A. E. Eiben and E. J. Smith, Introduction to Evolutionary Computing, Heidelberg: springer, 2003

■

  • C. F. Lima, C. Fernandes and F. G. Lobo, "Investigating restricted tournament replacement in ECGA

for non-stationary environments," in Genetic and evolutionary computation, New York, 2008 ■ C.-Y. Chuang and W.-L. Hsu, "Multivariate multi-model approach for globally multimodal problems," in Genetic and evolutionary computation, New York, 2010. ■

  • A. Nikanjam and H. Karshenas, "Multi-Structure Problems: Difficult Model Learning in Discrete

EDAs," in IEEE Congress on Evolutionary Computation (CEC2016), Vancouver, Canada, 2016 ■

  • M. Pelikan, K. Sastry and D. E. Goldberg, "Sporadic model building for efficiency enhancement of

the hierarchical BOA," Genetic Programming and Evolvable Machines, vol. 9, no. 1, p. 53–84, 2008 ■ Martin Pelikan’s personal website ■ Wikipedia

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Special Thanks to…

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  • Dr. Nikanjam
  • Dr. Naser Sharif

Saeed Ghadiri

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Special Thanks to…

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  • Dr. Taghirad

KN2C Robotic Team

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