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On Revealing the Hidden Problem Structure in Real-World and Theoretical Problems Using Walsh Coefficient Influence

M. W. Przewozniczek, F. Chicano, R. Tinós, J. Nalepa, B. Ruszczak, A. M. Wijata

TL;DR

This work addresses the challenge of noisy or fully interdependent problem structure in gray-box optimization by extending Walsh decomposition with a dependency-strength measure called the Weighted dynamic VIG (wdVIG). It introduces Weighted PX (wPX) and a parameter-less Gray-box Optimizer for Problems with High Epistasis (GBO-PHE) to selectively exploit strong variable interactions, even under noise. Through toy and real-world experiments, wdVIG reduces spurious dependencies and reveals underlying decomposable structure, improving optimization performance on noisy instances while maintaining parity with state-of-the-art methods in noiseless cases. The results demonstrate the potential to apply gray-box, Walsh-based masking to problems previously deemed unsuitable, with significant practical implications for feature selection, anomaly detection, and related optimization tasks.

Abstract

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

On Revealing the Hidden Problem Structure in Real-World and Theoretical Problems Using Walsh Coefficient Influence

TL;DR

This work addresses the challenge of noisy or fully interdependent problem structure in gray-box optimization by extending Walsh decomposition with a dependency-strength measure called the Weighted dynamic VIG (wdVIG). It introduces Weighted PX (wPX) and a parameter-less Gray-box Optimizer for Problems with High Epistasis (GBO-PHE) to selectively exploit strong variable interactions, even under noise. Through toy and real-world experiments, wdVIG reduces spurious dependencies and reveals underlying decomposable structure, improving optimization performance on noisy instances while maintaining parity with state-of-the-art methods in noiseless cases. The results demonstrate the potential to apply gray-box, Walsh-based masking to problems previously deemed unsuitable, with significant practical implications for feature selection, anomaly detection, and related optimization tasks.

Abstract

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

Paper Structure

This paper contains 17 sections, 1 equation, 3 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Gray-box optimization and Walsh decomposition
  4. Linkage trees and SLL-using optimizers
  5. Dependency types
  6. PX with Weighted Dynamic VIG
  7. Intuitions and motivations
  8. Weighted VIG propositions
  9. Weighted PX
  10. Gray-box Optimizer for Problems with High Epistasis
  11. Revealing Hidden Problem Structure
  12. Noise modelling and filtering
  13. Denoising real-world instances
  14. Main Experiments
  15. Experiment setup and considered optimizers
  16. ...and 2 more sections

Figures (3)

  • Figure 1: The influence of noise volume on the VIG epistasis for various dependency checks (onemax problem). Lines with markers refer to the median. The bottom and upper lines of the same color refer to maximal and minimal values.
  • Figure 2: A landscape cross-section for chosen problem instances (solution opposite to global optimum $\rightarrow$ global minimum $\rightarrow$ global maximum; differing genes are modified in a random order). See Fig. S-1, supplementary material, for more results.
  • Figure 3: Scalablity for chosen instances and increasing noise (X axis: number of random Walsh coefficients per variable).