用条件风险框架分析差分进化首次成功概率,揭示其突现式行为模式。
On the Probability of First Success in Differential Evolution: Hazard Identities and Tail Bounds
- 通过条件风险建模生存概率,分解为每代击中目标的条件概率
- 在L-SHADE算法下构造可验证的见证事件,获得依赖采样规则的显式下界
- 实证发现成功集中在短爆发,而非均匀分布,解释了保守边界原因
本文通过条件风险框架研究差分进化(DE)中的首次命中时间。将可测目标集 $A$ 的生存概率表示为各代条件击中概率(风险)$p_t = ext{Prob}(E_t mid ilcal F_{t-1})$ 的乘积,从而在风险有确定下界时导出分布无关的身份关系与显式尾部界。针对采用 current-to-$p$best/1 变异策略的 L-SHADE 算法,我们构造了一个可验证的算法见证事件 $ cal L_t$,在此条件下条件风险可获得仅依赖于采样规则、种群规模和交叉统计量的显式下界。该方法将理论常数与经验事件频率解耦,解释了为何恒定风险上界通常过于保守。结合 CEC2017 基准测试集的 Kaplan--Meier 生存分析,我们识别出三种典型实证行为:(i) 成功高度集中于短爆发;(ii) 尾部近似几何分布,恒定风险模型有效;(iii) 评估期内未观测到命中。结果表明,尽管恒定风险界提供有效的尾部包络,但 L-SHADE 的实际行为由突发式跃迁主导,而非每代均匀的成功概率。
原文摘要 · Abstract (English)
We study first-hitting times in Differential Evolution (DE) through a conditional hazard frame work. Instead of analyzing convergence via Markov-chain transition kernels or drift arguments, we ex press the survival probability of a measurable target set $A$ as a product of conditional first-hit probabilities (hazards) $p_t=\Prob(E_t\mid\mathcal F_{t-1})$. This yields distribution-free identities for survival and explicit tail bounds whenever deterministic lower bounds on the hazard hold on the survival event. For the L-SHADE algorithm with current-to-$p$best/1 mutation, we construct a checkable algorithmic witness event $\mathcal L_t$ under which the conditional hazard admits an explicit lower bound depending only on sampling rules, population size, and crossover statistics. This separates theoretical constants from empirical event frequencies and explains why worst-case constant-hazard bounds are typically conservative. We complement the theory with a Kaplan--Meier survival analysis on the CEC2017 benchmark suite . Across functions and budgets, we identify three distinct empirical regimes: (i) strongly clustered success, where hitting times concentrate in short bursts; (ii) approximately geometric tails, where a constant-hazard model is accurate; and (iii) intractable cases with no observed hits within the evaluation horizon. The results show that while constant-hazard bounds provide valid tail envelopes, the practical behavior of L-SHADE is governed by burst-like transitions rather than homogeneous per-generati on success probabilities.
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