The working principles of model-based GAs fall within the PAC framework: A mathematical theory of problem decomposition

TL;DR

This paper establishes a formal, mathematical framework for understanding problem decomposition in model-based genetic algorithms, linking it to PAC learning and providing theoretical guarantees for problem decomposability.

Contribution

It introduces an algorithm-independent definition of linkage, proves decomposability for problems with bounded linkage, and connects problem decomposition to PAC learning theory.

Findings

Problems with bounded linkage degree are decomposable.

Proper linkage learning enables effective problem decomposition.

Global optima of certain problems are PAC learnable and decomposition is polynomial-time decidable.

Abstract

The concepts of linkage, building blocks, and problem decomposition have long existed in the genetic algorithm field and have guided the development of model-based genetic algorithms for decades. However, their definitions are usually vague, making it difficult to develop theoretical support. This paper provides an algorithm-independent definition to describe the concept of linkage. With this definition, the paper proves that any problem with a bounded degree of linkage is decomposable and that proper problem decomposition is possible via linkage learning. The way of decomposition given in this paper also offers a new perspective on nearly decomposable problems with bounded difficulty and building blocks from the theoretical aspect. Finally, this paper relates problem decomposition to probably approximately correct (PAC) learning and proves that the global optima of problems with bounded decomposition difficulty are PAC learnable and the decomposition is decidable in polynomial time under certain conditions.