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

Tian-Li Yu, Chi-Hsien Chang, Ying-ping Chen

TL;DR

This work develops an algorithm‑independent mathematical framework for model‑based GAs by formalizing epistasis as a problem‑driven linkage concept via epistatic graphs (EGs). It introduces stationary optima (SO/MSO) and proves the problem decomposition theorem, linking decomposition to EG structure, and the epistasis blanket theorem, showing that small epistatic blocks can force global optima regardless of remaining loci. The authors then establish PAC‑learnability results for nearly decomposable problems with bounded difficulty, using extreme balanced accuracy (EBACC) and iterative partial enumeration (IPE) to bound the required samples and evaluations. They further show that MSOs are PAC‑learnable under reasonable bounds on epistasis order and EG in‑degree, and outline how topological‑order decomposition and PAC guarantees inform practical model building in MBGAs. Overall, the paper provides foundational theory bridging problem structure, decomposition, and learnability to guide principled MBGA design and analysis.

Abstract

The concepts of linkage, building blocks, and problem decomposition have long existed in the genetic algorithm (GA) field and have guided the development of model-based GAs 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 problems with a bounded degree of linkage are 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 PAC learning and proves that the global optima of these problems and the minimum decomposition blocks are PAC learnable under certain conditions.

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

TL;DR

This work develops an algorithm‑independent mathematical framework for model‑based GAs by formalizing epistasis as a problem‑driven linkage concept via epistatic graphs (EGs). It introduces stationary optima (SO/MSO) and proves the problem decomposition theorem, linking decomposition to EG structure, and the epistasis blanket theorem, showing that small epistatic blocks can force global optima regardless of remaining loci. The authors then establish PAC‑learnability results for nearly decomposable problems with bounded difficulty, using extreme balanced accuracy (EBACC) and iterative partial enumeration (IPE) to bound the required samples and evaluations. They further show that MSOs are PAC‑learnable under reasonable bounds on epistasis order and EG in‑degree, and outline how topological‑order decomposition and PAC guarantees inform practical model building in MBGAs. Overall, the paper provides foundational theory bridging problem structure, decomposition, and learnability to guide principled MBGA design and analysis.

Abstract

The concepts of linkage, building blocks, and problem decomposition have long existed in the genetic algorithm (GA) field and have guided the development of model-based GAs 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 problems with a bounded degree of linkage are 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 PAC learning and proves that the global optima of these problems and the minimum decomposition blocks are PAC learnable under certain conditions.
Paper Structure (18 sections, 34 theorems, 11 equations, 5 figures, 4 tables, 3 algorithms)

This paper contains 18 sections, 34 theorems, 11 equations, 5 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Epistasis and Its Properties
  3. Exemplified Test Problems
  4. Basic Operations
  5. Epistasis
  6. Problem Decomposition and Epistatic Graph
  7. Stationary Optima
  8. Problems That Can Be Solved via Proper Decomposition in Polynomial Time with Known EGs
  9. Acyclic EG
  10. EG with boundedly sized SCCs
  11. Global Optima of Nearly Decomposable Problems with Bounded Difficulty Are PAC Learnable
  12. Extreme Balanced Accuracy and PAC
  13. The Global Optimum Is PAC Learnable
  14. PAC-learnable MSOs Provide A Way to Achieve Model Building
  15. Decomposition Follows A Topological Order
  16. ...and 3 more sections

Key Result

Proposition 1

$\forall S \subseteq V, \forall v\in V$, we have $\Psi_{\{(S, g)\}}[v] = \{ g[v]\}$.

Figures (5)

  • Figure 1: Probability of observing a specific weak epistasis in different initial populations and in different generations. The left is at generation 0 (initial population); and the right is with a population of size 500.
  • Figure 2: Epistasis graphs of the exemplified test problems.
  • Figure 3: Graph condensation resulted in the component graph, which is a DAG.
  • Figure 4: In this problem, $a\rightarrow b$ and $a\rightarrow c$. $\mathcal{IN}^*(a)=\{a\}$, $\mathcal{IN}^*(b)=\{a,b\}$, and $\mathcal{IN}^*(c)=\{a,c\}$.
  • Figure 5: Ideal decomposition of LeadingTraps. When the decomposition difficulty and the order of epistasis are bounded by a constant, this happens with a high probability.

Theorems & Definitions (75)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Definition 4
  • Definition 5
  • Proposition 4
  • ...and 65 more