Availability of Perfect Decomposition in Statistical Linkage Learning for Unitation-based Function Concatenations

TL;DR

This work develops an entropy-based, Chernoff-bounds–driven estimate for the minimal population size $s_{min}$ required to achieve a perfect SLL-based decomposition when solving concatenations of unitation-based symmetric functions. By analyzing the geometry of probability distributions on the entropy simplex and the dependency-structure induced by FIHC-driven DSM construction, the authors provide an explicit criterion linking problem structure, block size, and desired confidence $\alpha$. The theory is validated on bimodal and reverted bimodal deceptive function concatenations, showing that the predicted $s_{min}$ aligns with qualitative difficulty trends and explains why certain SLL-using optimizers struggle on specific problem classes. The results offer a practical hardness measure for SLL and point to extensions for non-symmetric, overlapping, or noisier problems.

Abstract

Statistical Linkage Learning (SLL) is a part of many state-of-the-art optimizers. The purpose of SLL is to discover variable interdependencies. It has been shown that the effectiveness of SLL-using optimizers is highly dependent on the quality of SLL-based problem decomposition. Thus, understanding what kind of problems are hard or easy to decompose by SLL is important for practice. In this work, we analytically estimate the size of a population sufficient for obtaining a perfect decomposition in case of concatenations of certain unitation-based functions. The experimental study confirms the accuracy of the proposed estimate. Finally, using the proposed estimate, we identify those problem types that may be considered hard for SLL-using optimizers.

Figures (7)

• Figure 1: Level sets of entropy function for three values: $\log{2.6}$, $\log{3}$ and $\log{3.8}$, increasing from left to right.
• Figure 2: The interval $\mathcal{I}$ with the center of the simplex $(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4})$ (blue point) and the distribution of dependent pairs $\widetilde{Q}$ (orange point).
• Figure 3: The octahedron contained in $\{P:H(P)\geqslant H(U)\}$ and the plane tangent to the level set $H(P)= H(T)$.
• Figure 4: The triangle $\mathcal{T}$ and the curve $\mathcal{C}$.
• Figure 5: The plots for Examples 1-6.

Theorems & Definitions (23)

• Definition 1
• Remark 2
• Remark 3
• Definition 4
• Lemma 5
• Lemma 6
• proof
• Remark 7
• Lemma 8
• proof