Online algorithm selection (OAS) aims to adapt the optimization process to changes in the fitness landscape and is expected to outperform any single algorithm from a given portfolio. Although this expectation is supported by numerous empirical studies, there are currently no theoretical results proving that OAS can yield asymptotic speedups (apart from some artificial examples for hyper-heuristics). Moreover, theory-based guidelines for when and how to switch between algorithms are largely missing. In this paper, we present the first theoretical example in which switching between two algorithms -- the $(1+λ)$ EA and the $(1+(λ,λ))$ GA -- solves the OneMax problem asymptotically faster than either algorithm used in isolation. We show that an appropriate choice of population sizes for the two algorithms allows the optimum to be reached in $O(n\log\log n)$ expected time, faster than the $Θ(n\sqrt{\frac{\log n \log\log\log n}{\log\log n}})$ runtime of the best of these two algorithms with optimally tuned parameters. We first establish this bound under an idealized switching rule that changes from the $(1+λ)$ to the $(1+(λ,λ))$ GA at the optimal time. We then propose a realistic switching strategy that achieves the same performance. Our analysis combines fixed-start and fixed-target perspectives, illustrating how different algorithms dominate at different stages of the optimization process. This approach offers a promising path toward a deeper theoretical understanding of OAS.

翻译：在线算法选择（Online Algorithm Selection, OAS）旨在使优化过程适应适应度景观的变化，并期望优于给定策略池中的任何单一算法。尽管这一期望得到了大量实证研究的支持，但目前尚无理论结果证明OAS能够实现渐近加速（除了超启发式的一些人工示例）。此外，关于何时以及如何在算法之间进行切换的基于理论的指导原则也基本缺失。本文提出了首个理论实例，其中在两种算法——(1+λ) EA 与 (1+(λ,λ)) GA——之间切换，能够比单独使用任一算法更渐近快速地解决OneMax问题。我们证明，为这两种算法选取合适的种群规模，可在期望时间$O(n\log\log n)$内达到最优解，快于这两者中参数最优的算法运行时间$Θ(n\sqrt{\frac{\log n \log\log\log n}{\log\log n}})$。我们首先在理想切换规则下建立该界，该规则在最优时刻从(1+λ) EA切换至(1+(λ,λ)) GA。随后提出一种能实现相同性能的现实切换策略。我们的分析结合了固定起点与固定目标视角，揭示了不同算法在优化过程的不同阶段如何占据主导地位。该研究为更深入地理解OAS理论提供了有前景的路径。