Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by an r-factor in lower-order terms of the expected time decreases exponentially with r. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

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使用变量漂移分析的随机搜索启发式命中时间的尾界

漂移分析是随机搜索启发式 (RSH) 运行时分析的最先进技术之一，例如进化算法 (EA)、模拟退火、等等. 绝大多数现有的漂移定理对目标状态的命中时间的期望值产生了界限，例如最优解的集合，而不对这个时间的分布做出额外的陈述。我们通过提供一个通用漂移定理来解决这一缺陷，该定理包括击球时间分布的上尾和下尾的界限。新的尾界用于证明简单 EA 在标准基准问题（包括一般线性函数类）上的运行时间非常精确的锐集中结果。在所有这些问题上，偏离的概率r- 预期时间的低阶项中的因子随r. 该定理在 RSH 理论之外的有用性通过推导随机排列中的循环数的尾界来证明。所有这些结果都处理了一个与位置相关的（可变）漂移，该漂移没有被以前的带有尾界的漂移定理所涵盖。最后，给出了一般漂移定理的用户友好的专业化。