Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
Abstract
Population-based optimizers combine update rules such as mutation, selection, and recombination. When one rule changes, it is often unclear which convergence guarantees survive or how the new combination should be assessed. We develop an operator calculus: an operator is a population-update rule, and the calculus specifies how separately checked effects can be combined. Under explicit regularity and small-step conditions, the leading changes caused by the updates add, yielding reusable building blocks for convergence analysis. The framework distinguishes finding and retaining a good solution, reducing the population's mean objective, and concentrating candidates near an optimizer, and identifies the extra approximation conditions needed for finite evaluation-budget guarantees. Applications include distribution adaptation, recombinative evolution, and consensus dynamics, with verified nonconvex cases. Controlled experiments on a common nonconvex problem collection show how component effects change with population geometry and the performance measure: a rule can worsen the mean objective yet produce better candidates.
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BibTeX
@misc{malo2026operator,
title = {{Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees}},
author = {Pekka Malo and Lauri Viitasaari and Patrik Nummi and Antti Suominen and Ankur Sinha and Olli Tahvonen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.14289},
eprint = {2606.14289},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14289},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
AU - Pekka Malo
AU - Lauri Viitasaari
AU - Patrik Nummi
AU - Antti Suominen
AU - Ankur Sinha
AU - Olli Tahvonen
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2606.14289
UR - https://arxiv.org/abs/2606.14289
AB - Population-based optimizers combine update rules such as mutation, selection, and recombination. When one rule changes, it is often unclear which convergence guarantees survive or how the new combination should be assessed. We develop an operator calculus: an operator is a population-update rule, and the calculus specifies how separately checked effects can be combined. Under explicit regularity and small-step conditions, the leading changes caused by the updates add, yielding reusable building blocks for convergence analysis. The framework distinguishes finding and retaining a good solution, reducing the population's mean objective, and concentrating candidates near an optimizer, and identifies the extra approximation conditions needed for finite evaluation-budget guarantees. Applications include distribution adaptation, recombinative evolution, and consensus dynamics, with verified nonconvex cases. Controlled experiments on a common nonconvex problem collection show how component effects change with population geometry and the performance measure: a rule can worsen the mean objective yet produce better candidates.
ER -