The Schema Theorem states that the frequency of short, low-order schemata with above-average fitness increases exponentially through generations. John Holland proposed the theory in the 1970s. It was once regarded as the basis for explaining the effectiveness of genetic algorithms. Nonetheless, this understanding of its consequences has been questioned in several articles where the Schema Theorem is demonstrated as a particular example of the Price equation, with the schema indicator function serving as the macroscopic measurement.

A schema is a template that identifies a collection of strings that share similarities at specific locations. Schemata are a subset of cylinder sets and, as such, constitute a topological space.

Overview

Schema theories can be thought of as large-scale representations of genetic algorithms. It means that they make predictions about the characteristics of a population for the following generation using macroscopic parameters (such as population fitness, schema fitness, the number of individuals in a schema, etc.) measured for the previous generation. These models frequently conceal a GA's enormous degree of flexibility behind their macroscopic quantities. Usually, this results in relatively straightforward equations that are simple to learn and comprehend. Despite many previous schema theorems being approximate or worst-case scenario models, a macroscopic model need not be either.

These characteristics stand in stark contrast to those displayed by microscopic models, such as Vose's model, which tends to give equations with many degrees of freedom despite always being accurate.

Moreover, many people have criticized the usefulness of schemata and the schema theorem (see, for instance. While some objections are legitimate and apply to many schema theories, others are acceptable and not warranted, as discussed. In addition, Schema theorems are criticized for only providing lower bounds on the expected value of the sample size of a specific schema in the following generation. As a result, it is impossible to utilize them to make predictions over numerous generations. There must be some truth to this. Due to these factors, many researchers today think schema theorems are useless trivial tautologies (see, for instance). 

However, this does not imply that we can do nothing to alter the situation or that all schema theories are pointless. On the contrary, recent research has demonstrated that schema theories still need to be fully developed or utilized. For instance, a new schema theorem developed by Stephens and Waelbroeck gives an exact formulation—rather than a lower bound—for the anticipated number of schema instances in the following generation in macroscopic quantities. This result served as the foundation for several additional findings by Stephens and Waelbroeck on the behaviour of a GA.

Conclusion

The schema theorem is true if a genetic algorithm keeps an infinitely large population, but it is not always accurate in (finite) practice. For example, genetic algorithms may converge on schemata with no selective advantage due to sampling error in the initial population. It is especially true in multimodal optimization, where a function can have more than one peak: people may start to favour one peak and ignore the others.

Furthermore, the schema theorem can't explain the power of genetic algorithms because it applies to all problem instances. It needs to find the difference between problems where genetic algorithms do poorly and problems where they do well.