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Chaos propagation in genetic algorithms: An optimal transport approach

Giacomo Borghi

TL;DR

The paper treats a genetic algorithm for continuous optimization as an interacting-particle system and derives a Boltzmann-type kinetic equation in the many-particle limit. By exploiting an optimal-transport framework with the Kantorovich–Rubinstein norm, it proves a propagation of chaos result and provides a sharp convergence rate for the empirical GA distribution to the kinetic limit, valid both in the time-discrete and time-continuous settings. The analysis elegantly incorporates crossover as a binary collision within the transport coupling, yielding explicit error bounds that separate sampling noise $\varepsilon_1(N)$ from time discretization error $\tau$. This work bridges evolutionary computation with kinetic theory, offering rigorous convergence guarantees and suggesting directions for extending the approach to rank-based selection and other interaction kernels with practical implications for algorithm design and analysis.

Abstract

Genetic algorithms are high-level heuristic optimization methods which enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions following three mechanisms: selection, mutation, and crossover. In this paper, we look at the algorithm as an interacting particle system and show that it is described by a Boltzmann-type equation in the many particles limit. Specifically, we prove a propagation of chaos result with a novel technique that leverages the optimal transport formulation of the Kantorovich-Rubinstein norm and naturally incorporates the crossover mechanism into the analysis. The convergence rate is sharp with respect to the number of particles.

Chaos propagation in genetic algorithms: An optimal transport approach

TL;DR

The paper treats a genetic algorithm for continuous optimization as an interacting-particle system and derives a Boltzmann-type kinetic equation in the many-particle limit. By exploiting an optimal-transport framework with the Kantorovich–Rubinstein norm, it proves a propagation of chaos result and provides a sharp convergence rate for the empirical GA distribution to the kinetic limit, valid both in the time-discrete and time-continuous settings. The analysis elegantly incorporates crossover as a binary collision within the transport coupling, yielding explicit error bounds that separate sampling noise from time discretization error . This work bridges evolutionary computation with kinetic theory, offering rigorous convergence guarantees and suggesting directions for extending the approach to rank-based selection and other interaction kernels with practical implications for algorithm design and analysis.

Abstract

Genetic algorithms are high-level heuristic optimization methods which enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions following three mechanisms: selection, mutation, and crossover. In this paper, we look at the algorithm as an interacting particle system and show that it is described by a Boltzmann-type equation in the many particles limit. Specifically, we prove a propagation of chaos result with a novel technique that leverages the optimal transport formulation of the Kantorovich-Rubinstein norm and naturally incorporates the crossover mechanism into the analysis. The convergence rate is sharp with respect to the number of particles.
Paper Structure (10 sections, 5 theorems, 49 equations, 1 algorithm)

This paper contains 10 sections, 5 theorems, 49 equations, 1 algorithm.

Table of Contents

  1. Introduction and main result
  2. Main result
  3. Literature review and novelty
  4. Notation and preliminaries
  5. Proof of Theorem \ref{['t:main']}
  6. Two particle systems
  7. Coupling of parents
  8. Limit as $N \to \infty$
  9. Limit as $\tau \to 0$ and final estimate
  10. Outlook

Key Result

Theorem 1.3

Let $f_0$ and $\mathbf{F}$ satisfy Assumptions asm:f0 and asm:F, $\mu^\gamma = \textup{Unif}[0,1]^d$, and $\mu^\xi = \mathcal{N}(0,I_d)$. Let $f\in C([0,T],\mathcal{P}_q(\mathbb{R}^d))$ be a weak measure solution to eq:boltzmann in duality with $C_b(\mathbb{R}^d)$ (see Definition def:solution) with with $C = C(d,q, M_q(f_0)^{1/q}, \sigma), C' = C'(q)$ positive constants.

Theorems & Definitions (10)

  • Theorem 1.3
  • Definition 2.1: Weak measure solution to \ref{['eq:boltzmann']}
  • Lemma 2.2: Concentration rate
  • Lemma 2.3: Selection stability
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: Moments evolution
  • proof
  • proof : Proof of Theorem \ref{['t:main']}