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 ■ Selection Operators ■ Heuristic strategies ■ Parent Selection ■ Stochastic Heuristic Searches ■ Truncation Selection ■ Genetic Algorithm ■ Tournament Selection ■ Estimation of Distribution Algorithms ■ Survivor Selection ■ EDAs Pseudo code ■ Worst Replacement ■ Bayesian Optimization Algorithm ■ Restricted Tournament Replacement ■ BOAs Pseudo code ■ Multi-model Approach ■ Bayesian Networks ■ Multi-structural Problems ■ Bayesian Dirichlet Metric ■ Conclusion & Suggestions ■ Find Best Network ■ References ■ Generate New Solutions 2
Intr ntroduction oduction ■ 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 3
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 omness • 4
Stocha chastic stic Heuristic euristic Sea earche ches ■ Store one solution: Simulated Annealing ■ Population-based: Evolutionary Computation. 5
Ge Gene netic tic Alg lgorithms orithms ■ 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. 6
Es Estim timation tion of f Di Distri stribution bution Alg lgorith orithms 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 7
ED EDAs s Ps Pseudo eudo co code de 8
Bayesian esian Optim ptimizat ization ion Alg lgorithm orithm ■ By Pelikan, Goldberg, & Cantu-paz [1998] 9
BOAs As Ps Pseudo eudo co code de 10
Bayesian esian Netw Networ orks 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. ■ 𝑜−1 𝑞 𝑌 𝑗 𝑌 𝑗 𝑞 𝑌 = 𝑗=0 11
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: 𝒏 ′ 𝒚 𝒋, 𝝆 𝒀 𝒋 + 𝒏 𝒚 𝒋, 𝝆 𝒀 𝒋 𝒐−𝟐 𝒏 ′ 𝝆 𝒀 𝒋 ! ! 𝒒 𝑬, 𝑪 𝝄 = 𝒒 𝑪 𝝄 𝒏 ′ 𝒚 𝒋 , 𝝆 𝒀 𝒋 ! 𝒏 ′ 𝝆 𝒀 𝒋 + 𝒏 𝝆 𝒀 𝒋 ! 𝒋=𝟏 𝝆 𝒀𝒋 𝒚 𝒋 𝑛 𝜌 𝑌 𝑗 = 𝑛 𝑦 𝑗 , 𝜌 𝑌 𝑗 𝑦 𝑗 12
Find Fi nd Best est Netw Networ ork ■ 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 o at most k incoming edges into each node o • The algorithms can start with: An empty network o Best network with one incoming edge for each node o Randomly generated network. o 13
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 . 14
Mu Multi lti-model model Appr pproach oach ■ 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]. 22
Implement plementation ation 23
Results esults Average verage Average verage model- Average verage Problem lem Size #Models dels RTR? RT Final al Po Populat ulation ion size algorit orithm hm building lding time fitne ness calls ls time 30 1 0 1312 0.028 0.019 6822 30 3 0 1812 0.015 0.021 16416 30 3 1 734 0.028 0.022 15995 18000 2000 16000 1800 ls ness Calls 1600 14000 ulation Size 1400 12000 1200 erage Fitne 10000 Populatio 1000 8000 800 Average 6000 600 4000 400 2000 200 0 0 Traditional Multi-Models Multi-Models with Traditional Multi-Models Multi-Models with RTR RTR Differen fferent t Approac roaches hes Differen fferent t Approac roaches hes 24
Implement plementation ation 𝑔 𝑗 𝑂 𝑛𝑝𝑒𝑓𝑚 𝑗 = 𝑁𝑝𝑒𝑓𝑚𝑡 𝑘=1 𝑔 𝑘 𝑓 𝑔 𝑗 𝑂 𝑛𝑝𝑒𝑓𝑚 𝑗 = 𝑁𝑝𝑒𝑓𝑚𝑡 𝑓 𝑔 𝑘 𝑘=1 25
Results esults Average verage Average verage Final al Po Populat ulation ion Average verage Problem lem Size #Models dels RTR? RT SoftMa Max? algorit orithm hm model el- size fitne ness calls ls time building lding time 30 1 0 0 1312 0.028 0.019 6822 30 3 0 1 15250 0.126 0.193 143689 30 3 1 1 578 0.039 0.039 26580 160000 18000 16000 140000 ls ness Calls 14000 120000 ulation Size ge Fitness 12000 100000 10000 Populatio 80000 8000 60000 erage 6000 Avera 40000 4000 20000 2000 0 0 Traditional Multi-Models with Multi-Models with Traditional Multi-Models with Multi-Models with Softmax Softmax and RTR Softmax Softmax and RTR Differen fferent t Approac roaches hes Differen fferent t Approac roaches hes 26
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Results esults Average verage Averag age Problem lem Paren ent model el- Average verage #Models dels RTR? SoftMax Max? Final Popu pulati tion on size algo gorit ithm hm Size Displac lacem emen ent? t? building lding fitne ness calls ls time time 30 1 0 0 0 1312 0.028 0.019 6822 30 3 0 0 1 1562 0.038 0.009 7731 30 3 1 0 1 921 0.071 0.012 7204 9000 1800 8000 1600 ness Calls ls 7000 1400 ulation Size 6000 1200 erage Fitne 5000 1000 Populatio 4000 800 Average 3000 600 2000 400 1000 200 0 0 Traditional Multi-Models with Multi-Models with Traditional Multi-Models with Multi-Models with Parent Displacement Parent Displacement Parent Displacement Parent Displacement and RTR and RTR Differen fferent t Approac roaches hes Differen fferent t Approac roaches hes 28
Implement plementation ation 29
Results esults Final al Average verage Problem lem Paren ent Pop. Po Average verage model- #Models odels RTR? RT SoftMax Max? Population Po ulation algorit orithm hm Average verage fitnes ess calls ls Size Displac lacem emen ent? t? Type building lding time size time 30 1 0 0 0 ---- 1312 0.028 0.019 6822 30 3 0 0 1 1562 0.038 0.009 7731 Single 30 3 1 0 1 921 0.071 0.012 7204 Single 30 3 0 0 1 3125 0.058 0.015 14871 Multi 16000 3500 14000 3000 ls ness Calls 12000 2500 ulation Size 10000 2000 erage Fitne 8000 Populatio 1500 Average 6000 1000 4000 2000 500 0 0 Traditional Single Pop. Single Pop. and Multi Pop. Traditional SinglePop. SinglePop. and Multi Pop. RTR RTR Differen fferent t Approac roaches hes Differen fferent t Approac roaches hes 30
Mu Multi lti-st structur ructural al Pr Problems blems ■ Deceptive Behavior! 𝑁𝑇𝑄1 𝑦 = 𝛽 + 𝑔 𝑢𝑠𝑏𝑞 𝑛,𝑙 𝑦 2 , … , 𝑦 𝑜 𝑦 1 = 1 𝑜 − 1 − 𝑔 𝑃𝑜𝑓𝑁𝑏𝑦 𝑦 2 , … , 𝑦 𝑜 𝑦 1 = 0 𝑁𝑇𝑄2 𝑦 = 𝑛𝑏𝑦 𝛽 + 𝑔 𝑢𝑠𝑏𝑞 𝑛,𝑙 𝑦 , 𝑜 − 1 − 𝑔 𝑃𝑜𝑓𝑁𝑏𝑦 𝑦 31
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