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Feature weighting for data analysis via evolutionary simulation

Aris Daniilidis, Alberto Domínguez Corella, Philipp Wissgott

TL;DR

The paper analyzes a data-analysis problem where feature relevance is learned via a replicator-type dynamical system on the simplex. By normalizing the data into a matrix $\Phi\in[0,1]^{n\times m}$ and defining a combined gain $\Delta_j(\gamma)$ from dominance and balance terms, the weights $\gamma$ evolve to a unique interior equilibrium $\gamma^*$. The equilibrium has a closed-form expression, ensuring non-degenerate limiting weights and enabling Pareto-aware scalarization of multi-objective problems; an illustrative numerical example demonstrates convergence and the role of rare but valuable features. The work positions itself between evolutionary multi-objective optimization and traditional feature-weighting methods, offering analytic tractability, convergence guarantees, and an interpretation of data-driven weights through gene-organism dynamics. Overall, this approach provides a principled, convergence-guaranteed method to learn feature weights from data, with potential applicability across data-driven decision problems and multi-objective analysis.

Abstract

We analyze an algorithm for assigning weights prior to scalarization in discrete multi-objective problems arising from data analysis. The algorithm evolves the weights (the relevance of features) by a replicator-type dynamic on the standard simplex, with update indices computed from a normalized data matrix. We prove that the resulting sequence converges globally to a unique interior equilibrium, yielding non-degenerate limiting weights. The method, originally inspired by evolutionary game theory, differs from standard weighting schemes in that it is analytically tractable with provable convergence.

Feature weighting for data analysis via evolutionary simulation

TL;DR

The paper analyzes a data-analysis problem where feature relevance is learned via a replicator-type dynamical system on the simplex. By normalizing the data into a matrix and defining a combined gain from dominance and balance terms, the weights evolve to a unique interior equilibrium . The equilibrium has a closed-form expression, ensuring non-degenerate limiting weights and enabling Pareto-aware scalarization of multi-objective problems; an illustrative numerical example demonstrates convergence and the role of rare but valuable features. The work positions itself between evolutionary multi-objective optimization and traditional feature-weighting methods, offering analytic tractability, convergence guarantees, and an interpretation of data-driven weights through gene-organism dynamics. Overall, this approach provides a principled, convergence-guaranteed method to learn feature weights from data, with potential applicability across data-driven decision problems and multi-objective analysis.

Abstract

We analyze an algorithm for assigning weights prior to scalarization in discrete multi-objective problems arising from data analysis. The algorithm evolves the weights (the relevance of features) by a replicator-type dynamic on the standard simplex, with update indices computed from a normalized data matrix. We prove that the resulting sequence converges globally to a unique interior equilibrium, yielding non-degenerate limiting weights. The method, originally inspired by evolutionary game theory, differs from standard weighting schemes in that it is analytically tractable with provable convergence.

Paper Structure

This paper contains 17 sections, 3 theorems, 55 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction, data problem, and theoretical results
  2. The data problem
  3. Algorithm and convergence results
  4. Related literature
  5. Proofs
  6. Proof of Proposition \ref{['prop0']}
  7. Proof of Theorem \ref{['thm0']}
  8. Proof of Corollary \ref{['cor0']}
  9. An illustrative numerical experiment
  10. Analogies with genetic algorithms and evolutionary systems
  11. Gene contribution $\Delta_{ij}^{k,\operatorname*{gen}}$
  12. Organisms contribution $\Delta_{ij}^{k,\operatorname*{org}}$
  13. Summary of evolutionary argument
  14. Asymptotic behavior in higher dimensions
  15. A minimal example and its evolutionary interpretation
  16. ...and 2 more sections

Key Result

proposition thmcounterproposition

Let the relative interior of the standard simplex be denoted by Let $(\gamma^k)_{k\in\mathbb N}$ be a sequence generated by Algorithm gAI. If $\gamma^0\in\mathcal{K}^{m-1}$ belongs to $\operatorname{relint}\mathcal{K}^{m-1}$, then any accumulation point of $(\gamma^k)_{k\in\mathbb N}$ also belongs to $\operatorname{relint}\mathcal{K}^{m-1}$.

Figures (3)

  • Figure 1: Illustration of the replicator-type feature weighting algorithm.
  • Figure 2: Evolution of feature weights $\gamma_j^k$ over $k=0,\dots,10$. Dashed lines indicate the analytical solution $\gamma^*$.
  • Figure 3: Evolution of fixed-point weights $\gamma_1^*$ and $\gamma_2^*$ as a function of $\xi$.

Theorems & Definitions (4)

  • proposition thmcounterproposition: Non-degeneracy of limit points of Algorithm \ref{['gAI']}
  • theorem thmcountertheorem: Convergence of Algorithm \ref{['gAI']} to an equilibrium
  • corollary thmcountercorollary: Pareto optimality of the equilibrium weights
  • remark thmcounterremark