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This research introduces a novel operator calculus framework to rigorously analyze how modular components in population-based optimizers combine under finite-population constraints. By establishing conditions where leading changes add linearly, the work provides reusable building blocks for proving convergence guarantees in complex, nonconvex evolutionary algorithms. The findings offer practitioners a structured approach to understanding trade-offs between different update rules, such as mutation and selection, within practical evaluation budgets.
Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
arXiv:2606.14289v2 Announce Type: replace-cross 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.
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.
Substantially revised version: finite-population evaluation-complexity guarantees, verified CMA-ES-type, recombinative-ES and CBO instances, a numerical study of operator assemblies, and a practitioner's guide. 8 pages main text plus appendices (52 pages), 10 figures, 13 tables
Optimization and Control (math.OC); Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Numerical Analysis (math.NA); Machine Learning (stat.ML)
arXiv:2606.14289 [math.OC]
(or arXiv:2606.14289v2 [math.OC] for this version)
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