## Formalization of GAs

The mathematicians Michael Vose and Gunar Liepins (1991) developed a formal model based on the following simple GA Start with a random population of binary strings of length l. Calculate the fitness f(x) of each string x in the population. Choose (with replacement) two parents from the current population with probability proportional to each string's relative fitness in the population. Cross over the two parents (at a single randomly chosen point) with probability pc to form two offspring. (If...

## Simple Model of the Baldwin Effect

Genetic assimilation is well known in the evolutionary biology community. Its predecessor, the Baldwin effect, is less well known, though it has recently been picked up by evolutionary computationalists because of an interesting experiment performed by Geoffrey Hinton and Steven Nowlan (1987). Hinton and Nowlan employed a GA in a computer model of the Baldwin effect. Their goal was to demonstrate this effect empirically and to measure its magnitude, using a simplified model. An extremely simple...

## Simple Genetic Algorithm

Given a clearly defined problem to be solved and a bit string representation for candidate solutions, a simple GA works as follows Start with a randomly generated population of n 7-bit chromosomes (candidate solutions to a problem). Calculate the fitness f(x) of each chromosome x in the population. Repeat the following steps until n offspring have been created a. Select a pair of parent chromosomes from the current population, the probability of selection being an increasing function of...

## Examples of Fitness Functions

One common application of GAs is function optimization, where the goal is to find a set of parameter values that maximize, say, a complex multiparameter function. As a simple example, one might want to maximize the real-valued one-dimensional function f(y) y + I sin(32.v) . 0 < y < n (Riolo 1992). Here the candidate solutions are values of y, which can be encoded as bit strings representing real numbers. The fitness calculation translates a given bit string x into a real number y and then...

## Thought Exercises

For the fitness function defined by Equation 4.5, what are the average fitnesses of the schemas (a) 1 ** *, (b) 11 * *, and (c) 1 * 1 * * How many schemas are there in a partition with k defined bits in an 7-bit search space Consider the fitness function f(x number of ones in x, where x is a chromosome of length 4. Suppose the GA has run for three generations, with the following populations Define on-line performance at function evaluation step t as the average fitness of all the individuals...

## Computer Exercises

Implement a genetic programming algorithm and use it to solve the 6-multiplexer problem Koza 1992 . In this problem there are six Boolean-valued terminals, a0, ai, d ,di, d2, d3 , and four functions, AND, OR, NOT, IF . The first three functions are the usual logical operators, taking two, two, and one argument respectively, and the IF function takes three arguments. IF X Y Z evaluates its first argument X. If X is true, the second argument Y is evaluated otherwise the third argument Z is...

## The Two Armed Bandit Problem

The tradeoff between exploration and exploitation can be instructively modeled in a simple scenario the Two-Armed Bandit problem. This problem has been studied extensively in the context of statistical decision theory and adaptive control e.g., see Bellman 1961 . Holland 1975 used it as an as a mathematical model of how a GA allocates samples to schemas. The scenario is as follows. A gambler is given N coins with which to play a slot machine having two arms. A conventional slot machine is...

## Predicting Dynamical Systems

Norman Packard 1990 has developed a form of the GA to address this problem and has applied his method to several data analysis and prediction problems. The general problem can be stated as follows A series of observations from some process e.g., a physical system or a formal dynamical system take the form of a set of pairs, where , are independent variables and y is a dependent variable 1didN . For example, in a weather prediction task, the independent variables might be some set of features of...

## Grammatical Encoding

The method of grammatical encoding can be illustrated by the work of Hiroaki Kitano 1990 , who points out that direct-encoding approachs become increasingly difficult to use as the size of the desired network increases. As the network's size grows, the size of the required chromosome increases quickly, which leads to problems both in performance how high a fitness can be obtained and in efficiency how long it takes to obtain high fitness . In addition, since direct-encoding methods explicitly...

## Limitations of Static Schema Analysis

A number of recent papers have questioned the relevance of schema analysis to the understanding of real GAs e.g., Grefenstette 1993 Mason 1993 Peck and Dhawan 1993 . Here I will focus on Grefenstette's critique of the Static Building Block Hypothesis. The following qualitative formulation of the Schema Theorem and the Building Block Hypothesis should now be familiar to the reader The simple GA increases the number of instances of low-order, short-defininglength, high-observed-fitness schemas...

## Rank Selection

Rank selection is an alternative method whose purpose is also to prevent too-quick convergence. In the version proposed by Baker 1985 , the individuals in the population are ranked according to fitness, and the expected value of each individual depends on its rank rather than on its absolute fitness. There is no need to scale fitnesses in this case, since absolute differences in fitness are obscured. This discarding of absolute fitness information can have advantages using absolute fitness can...

## Nextascent hill climbing NAHC

Call this string current-hilltop. For ifrom 1 to l where lis the length of the string , flip bit i if this results in a fitness increase, keep the new string, otherwise flip bit i back. As soon as a fitness increase is found, set current-hilltop to that increased-fitness string without evaluating any more bit flips of the original string. Go to step 2 with the new current-hilltop, but continue mutating the new string starting immediately after the bit position at...

## Genetic Algorithms And Traditional Search Methods

In the preceding sections I used the word search to describe what GAs do. It is important at this point to contrast this meaning of search with its other meanings in computer science. There are at least three overlapping meanings of search Search for stored data Here the problem is to efficiently retrieve information stored in computer memory. Suppose you have a large database of names and addresses stored in some ordered way. What is the best way to search for the record corresponding to a...

## Evolutionary Reinforcement Learning

A second computational demonstration of the Baldwin effect was given by David Ackley and Michael Littman 1992 . Their primary goal was to incorporate reinforcement learning an unsupervised learning method into an evolutionary framework and to see whether evolution could produce individuals that not only behaved appropriately but also could correctly evaluate the situations they encountered as beneficial or dangerous for future survival. In Ackley and Littman's Evolutionary Reinforcement...

## Fitness Proportionate Selection with Roulette Wheel and Stochastic Universal Sampling

Holland's original GA used fitness-proportionate selection, in which the expected value of an individual i.e., the expected number of times an individual will be selected to reproduce is that individual's fitness divided by the average fitness of the population. The most common method for implementing this is roulette wheel sampling, described in chapter 1 each individual is assigned a slice of a circular roulette wheel, the size of the slice being proportional to the individual's fitness. The...