arXiv：cs.LG· Pekka Malo, Lauri Viitasaari, Patrik Nummi, Antti Suominen, Ankur Sinha, Olli Tahvonen·· 5 小时前AI 评分30
面向种群优化的算子演算：模块化收敛与有限种群保证
Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
AI 导读
研究者提出一种"算子演算"框架，将变异、选择、重组等种群更新规则视为可组合算子，在明确的正则性与小步长条件下，各更新带来的主导变化可相加，从而形成可复用的收敛分析模块。该框架区分了找到并保留优质解、降低种群平均目标值、将候选解集中于最优解附近三类目标，并给出有限评估预算保证所需的额外近似条件。在常见非凸问题集上的对照实验显示，同一规则可能恶化平均目标值，却产生更优候选解。
正文
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.
| Comments: | 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 |
| Subjects: | Optimization and Control (math.OC); Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Numerical Analysis (math.NA); Machine Learning (stat.ML) |
| MSC classes: | 90C26 (Primary), 90C59, 35Q84, 60J25, 37N40 (Secondary) |
| ACM classes: | G.1.6 |
| Cite as: | arXiv:2606.14289 [math.OC] |
| (or arXiv:2606.14289v2 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2606.14289 arXiv-issued DOI via DataCite |
Submission history
From: Pekka Malo [view email]
[v1]
Fri, 12 Jun 2026 09:16:58 UTC (261 KB)
[v2]
Fri, 2 Oct 2026 09:21:13 UTC (561 KB)
来源：arXiv：cs.LG · arxiv.org