Outline. CS 6776 Evolutionary Computation. Numerical Optimization. Fitness Function. ,x 2. ) = x 2 1. , x , 5.0 x 1.

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1 Outline CS 6776 Evolutionary Computation January 21, 2014 Problem modeling includes representation design and Fitness Function definition. Fitness function: Unconstrained optimization/modeling Constrained optimization/modeling Multi-objective optimization/modeling Relative fitness: co-evolution Population size and population models Convergence and termination criteria Fitness Function A mathematical function that quantifies how good a solution is. A problem can be modelled as a maximization or a minimization problem. Example: TSP: A tour X = {x i },i =1 n minimize f (X) = n n j =1 d ij d ij = dist(x i, x j ), 0, x i to x i is in the tour otherwise Maximize f (x 1, x 2 ) = x x 2 2, 5.0 x 1,x How many optima? What are they? umerical Optimization 1

Regression Model Example Regression models, such as symbolic regressions, where the output is a real value, the most commonly used fitness function is minimizing error: ( y i y ˆ i ) 2 MSE: mean

2 Regression Model Example Regression models, such as symbolic regressions, where the output is a real value, the most commonly used fitness function is minimizing error: ( y i y ˆ i ) 2 MSE: mean squared error f (x) = Biased more weighted on outliers Weights large errors more heavily than small ones y MAE: mean absolute error i y ˆ i f (x) = Less biased The true value of the outliner point y = 60: Liner model: the predicted value ŷ=80: MSE: ^2=400 MAE: =20 on-liner model: the predicted value ŷ=80: MSE: ^2=100 MAE: =10 Classification Model Classification models, where the output is a discrete label, the most commonly used fitness function is maximizing accuracy: Problematic when the data set is imbalanced. t i f (x) =, t i = 1, y i = ˆ 0, y i ˆ y i y i Constrained Optimization One way to handle solutions that violate the constraints is to assign penalty p in the fitness function: f (x)'= f (x) ± p For maximization problems, p is subtracted from f(x). For minimization problems, p is added to f(x) There are other constraints handling methods, other than sum-penalty. 2

3 Multi-Objective Optimization Problems require satisfying more than one objective: e.g. maximizing profit & minimizing cost. Weighted sum approach: Converts all fitness function for maximization, e.g. convert fitness by multiply -1; m λ = [w 1, w m ],w i 0, w i =1 F(x) = w i f i (x) Maintain a set of Pareto Front solutions: m Fitness is a vector F(x) = [ f 1 (x), f 2 (x), f 3 (x) f m (x)] Select Pareto-optimal solution (discussed in later lecture) Relative Fitness The fitness of an individual is measured in relation to that of other individuals in the same population or in a competing population. Interaction Patterns: Single population Relative (competitive) fitness vs. absolute fitness Multiple populations All vs. previous-best eighborhood interaction Standard Evolutionary Algorithm Individual fitness is based on its absolute performance without interacting with others. Co-evolution Single Population Individuals are evaluated by having them interact with each other, e.g. play a game. Play games with randomly selected individuals from the population 3

4 Co-evolution Multiple Populations In asymmetric games (the strategies for player 1 are different from that of the player 2), each member of Pop. 1 interacts with each member of Pop. 2. Evolutionary Algorithms Workflow To design an effective evolutionary algorithm, one need to consider the problem at hand. Population Size Intuitively, the population size can be viewed as a measure of the degree of parallel search an EA supports. A larger population size provides: better coverage of the search space (diversity) which helps high fitness individuals to be included in the initial population a larger past memory, so that good individuals do not lost so quickly during evolution. Population Size - Continued Depending on the complexity of the problem fitness landscape, different population size is needed to search a solution. Increase population size beyond necessary would take EA longer to find a solution. 4

5 Generational Model on-overlapping Population: gen-0 gen-1 gen-2 gen-n Canonical Genetic Algorithms: Parent selection only; all offspring are kept in the following generation. Evolution Strategies (µ, λ): Random parent selection to generate a large number of offspring; Select fitter offspring to form the new generation. Generational Model - Continued Under stochastic selection, an individual, regardless how fit it is, may only live for one generation, hence has a short-term impact on evolution. During stochastic selection, best solutions might get lost and are never carried over to new generation. These can be fixed by deterministic selection, or elite selection. Steady-state Model Overlapping Population pop replace parents offspring Generation Gap: The proportion of the population that is replaced [Sarma & De Jong, 1995]. Generational model: pop_size/pop_size=1.0 Steady-state model: #_replaced/pop_size Steady-state Model - Continued Offspring and parents compete to survive in the population. Fit individual can live for a long period of time to impact the evolution. A fit offspring can have impact on the evolution immediately after its birth, without waiting until the next generation. Impact evolutionary search? 5

6 EA Termination Criteria EA search process termination criteria: When a specified number of generation is reached. When the known best solution is found. When the population is converged : no further changes in the population may occur. Convergence Practical ways to detect convergence: Measure the degree of homogeneity of the population using spatial dispersion or entropy: When the homogeneity measure approaches 0, the population is converged. Measure the global fitness improvement: When the best fitness does not improve for a certain number of generation (typically 10-20), the population is converged. 6

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