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Perceptron w0 x0 = 1 x1 w1 Activation function Local Field v Input signal Output y x2 w2 Summing function Synaptic weights xm wm Error 16
Activation function 1.Linear function f(x)=ax 2. Step function 3. Ramp function 17
Activation function 4. Logistic function 5. Hyperbolic tangent 6. Gaussian function 18
Least Mean Square learning LMS = Least Mean Square learning Systems, more general than the previous perceptron learning rule. The concept is to minimize the total error, as measured over all training examples, P. O is the raw output, as calculated by We want to minimize the LMS: C-learning rate E W(old) W(new) W 19
BP Network Details Forward Pass: Error is calculated from outputs Used to update output weights Backward Pass: Error at hidden nodes is calculated by back propagating the error at the outputs through the new weights Hidden weights updated 22
BP Neural Networks Signal-flow graph highlighting the details of output neuron k connected to hidden neuron j. 23
Binary Representation selected gene 1 2 3 4 5 6 7 chromosome 0 1 1 0 0 0 1 2 3 7 ratio of cj /wj 1.2 0.33 2.0 selected genes 1 1 1 descending order 7 2 3 1 1 1 7 output: items in knapsack 1 7 2 3 7 2 7 2 1 1 1 1 1 1 1 input: data set W =100 1. Select genes of xj=1 2.Sort items with the value xj = 1 in descending order by the ratio of profit to weight. 3.Select items using the first fit heuristic until the knapsack cannot be filled any further. 34
Order Representation Approach Hinterding, R.:“Mapping, order-independent genes and the knapsack problem”, in Fogel, D., editor, Proc. of the First IEEE Conference on Evolutionary Computation, IEEE Press, Orlando, FL, pp. 13-17, 1994 For n item problem, a chromosome contains n genes and each gene represents an item with an identical integer number. The order of an item in the permutation representation can be viewed as the priority for it to be filled into the knapsack. Fitness function 35
Order Representation Example W=100 chromosome [ 1 6 4 7 3 2 5 ] the feasible solution [ 1 6 4 5 ] 36
Variable-length Representation Hinterding, R.:“Mapping, order-independent genes and the knapsack problem”, in Fogel, D., editor, Proc. of the First IEEE Conference on Evolutionary Computation, IEEE Press, Orlando, FL, pp. 13-17, 1994 The representation belongs to the direct encoding approach. The genes in the chromosome represent a legal knapsack because the total number of items in a legal knapsack is not fixed, the length of a chromosome is variable. The order of items in a chromosome has no meaning. g(x)=30≤W, f(x)=60 7 g(x)=80≤W, f(x)=120 7 2 g(x)=110>W, f(x)=130 gene # : g(x)=80≤W, f(x)=120 chromosome : 1 2 7 2 3 7 2 data set Sample of chromosome W=100 37
1. Injecting a segment insert point parent 1 2 6 5 segment proto-child 2 5 7 6 5 parent 2 4 5 7 3 2. Deleting duplicated genes proto-child 2 5 7 6 3. Culling genes with first fit heuristic offspring 2 5 7 • Injecting crossover input: parent chromosomes output: offspring chromosomes step 1: Choose an insertion point in the first parent, and choose a segment in the second parent. step 2: Inject the segment into the first parent at the insertion point. step 3: Delete duplicate genes outside the segment to get a proto-child. step 4: Cull genes from the proto-child with the first fit heuristic to yield a legal knapsack. step 5: Output offspring chromosomes and stop. data set W=100 38
1. Delete a selected number selected Selected point parent 2 6 5 proto-child 2 5 2. Append the items that were not present in the chromosome proto-child 1 3 4 2 5 7 6 3. Culling genes with first fit heuristic offspring 2 5 7 • Mutation procedure input: parent chromosome output: offspring chromosome step 1: Delete a number of genes at random. step 2: Append the items in random order that were not present in the chromosome. step 3: Cull genes from the proto-child with the first heuristic to yield a legal knapsack. step 4: Output offspring chromosome and stop. data set W=100 39
GP quick overview Developed: USA in the 1990’s Early names: J. Koza Typically applied to: machine learning tasks (prediction, classification…) Attributed features: competes with neural nets and alike needs huge populations (thousands) slow Special: non-linear chromosomes: trees, graphs mutation possible but not necessary (disputed!)
Tree based representation Trees are a universal form, e.g. consider Arithmetic formula Logical formula Program (x true) (( x y ) (z (x y))) i =1; while (i < 20) { i = i +1 }
Ant SystemStep Start Initialization To represent problem with route initialize pheromone State Transition Rule It satisfies the condition? Yes Pheromone Updating Rule No Every ant finds a route End Update the pheromone 47
Ant System While ( termination not satisfied ) create ants Find solutions Transition probability: Pheromone update Daemon activities {optional} Quantity of pheromone Heuristic distance α,β constants 48
Ant System While ( termination not satisfied ) create ants Find solutions Pheromone update Daemon activities {optional} Pheromone laid by each ant that uses edge (i,j) Evaporation rate 49
Ant System • Pheromone update : • Ant-Cycle Model • Ant-Quantity Model • Ant-Density Model • Ant-Cycle model uses the global information while Ant-Quantity and Density Model use the local information 50
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