On the Probability of First Success in Differential Evolution: Hazard Identities and Tail Bounds
Dimitar Nedanovski, Svetoslav Nenov, Dimitar Pilev
TL;DR
The paper develops a hazard-based framework for first-hitting times in Differential Evolution, with a focus on the L-SHADE variant using current-to-\$p\$best/1 mutation. It expresses the survival probability as a product of conditional hazards and introduces a measurable witness $\mathcal{L}_t$ that yields a deterministic lower bound $a_t$ on the one-step hazard when the survival condition holds, separated from an empirical frequency $\gamma_t=\mathbb{P}(\mathcal{L}_t|H_{t-1})$. For Morse functions, the authors derive tighter hazard bounds near local minima and identify a witness-stable regime $G_t$ with a stabilization time $T_{\mathrm{wit}}$, enabling a two-phase analysis that combines almost-sure convergence with gap-free post-$T_{\mathrm{wit}}$ tails. Empirical validation via Kaplan–Meier survival analysis on the CEC2017 suite reveals three regimes—clustered hits, near-geometric tails, and intractable cases—demonstrating that practical DE behavior is governed by burst-like transitions rather than homogeneous per-generation progress, and that constant-hazard envelopes are conservative but valid as tail bounds.
Abstract
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.