| Reading |
| CS 420/594: Read Flake, ch. 22 (Neural Networks and Learning) | |
| CS 594: Read Bar-Yam, sec. 2.3 (Feedforward Networks) |
| Why Does the GA Work? |
| The Schema Theorem |
| Schemata |
| A schema is a description of certain patterns of bits in a genetic string |
| The Fitness of Schemata |
| The schemata are the building blocks of solutions | |
| We would like to know the average fitness of all possible strings belonging to a schema | |
| We cannot, but the strings in a population that belong to a schema give an estimate of the fitness of that schema | |
| Each string in a population is giving information about all the schemata to which it belongs (implicit parallelism) |
| Effect of Selection |
| Exponential Growth |
| We have discovered: m(S, t+1) = m(S, t) × f(S) / fav |
|
| Suppose f(S) = fav (1 + c) | |
| Then m(S, t) = m(S, 0) (1 + c)t | |
| That is, exponential growth in above-average schemata |
| Effect of Crossover |
| Let l = length of genetic strings | |
| Let d(S) = defining length of schema S | |
| Probability {crossover destroys S}: pd = d(S) / (l Ð 1) |
|
| Let pc = probability of crossover | |
| Probability schema survives: |
| Selection & Crossover Together |
| Effect of Mutation |
| Let pm = probability of mutation | |
| So 1 Ð pm = probability an allele survives | |
| Let o(S) = number of fixed positions in S | |
| The probability they all survive
is (1 Ð pm)o(S) |
|
| If pm << 1, (1 Ð pm)o(S) Å 1 Ð o(S) pm |
| Schema Theorem: ÒFundamental Theorem of GAsÓ |
| The Bandit Problem |
| Two-armed bandit: | ||
| random payoffs with (unknown) means m1, m2 and variances s1, s2 | ||
| optimal strategy: allocate exponentially greater number of trials to apparently better lever | ||
| k-armed bandit: similar analysis applies | ||
| Analogous to allocation of population to schemata | ||
| Suggests GA may allocate trials optimally | ||
| GoldbergÕs Analysis of Competent & Efficient GAs |
| Paradox of GAs |
| Individually uninteresting operators: | ||
| selection, recombination, mutation | ||
| Selection + mutation Þ continual improvement | ||
| Selection + recombination Þ innovation | ||
| generation vs.evaluation | ||
| Fundamental intuition of GAs: the three work well together | ||
| Race Between Selection & Innovation: Takeover Time |
| Takeover time t* = average time for most fit to take over population | ||
| Transaction selection: top 1/s replaced by s copies | ||
| s quantifies selective pressure | ||
| Estimate t* Å ln n / ln s | ||
| Innovation Time |
| Innovation time ti = average time to get a better individual through crossover & mutation | |
| Let pi = probability a single crossover produces a better individual | |
| Number of individuals undergoing crossover = pc n | |
| Probability of improvement = pi pc n | |
| Estimate: ti Å 1 / (pc pi n) |
| Steady State Innovation |
| Bad: t* < ti | ||
| because once you have takeover, crossover does no good | ||
| Good: ti < t* | ||
| because each time a better individual is produced, the t* clock resets | ||
| steady state innovation | ||
| Innovation number: | ||
| Feasible Region |
| Other Algorithms Inspired by Genetics and Evolution |
| Evolutionary Programming | ||
| natural representation, no crossover,time-varying continuous mutation | ||
| Evolutionary Strategies | ||
| similar, but with a kind of recombination | ||
| Genetic Programming | ||
| like GA, but program trees instead of strings | ||
| Classifier Systems | ||
| GA + rules + bids/payments | ||
| and many variants & combinationsÉ | ||
| Additional Bibliography |
| Goldberg, D.E. The Design of Innovation: Lessons from and for Competent Genetic Algorithms. Kluwer, 2002. | |
| Milner, R. The Encyclopedia of Evolution. Facts on File, 1990. |