Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees
Read the original on arXiv Machine Learning →The paper introduces an operator calculus for population-based optimization, treating each update rule—such as mutation, selection, or recombination—as an operator. By establishing explicit regularity and small‑step conditions, the authors show that the leading effects of these operators add, allowing convergence guarantees to be reused as modular building blocks. The framework differentiates between finding and retaining good solutions, reducing the population’s mean objective, and concentrating candidates near an optimizer, and it identifies additional approximation conditions required for finite‑budget guarantees. Applications to distribution adaptation, recombinative evolution, and consensus dynamics are verified, including nonconvex cases, and controlled experiments illustrate how component effects vary with population geometry and performance measures.
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