We introduce a new simple model to study the fitness progress of Evolution Strategies (ES) in generic problems. In this model, we bypass the underlying fitness landscape and assume that the mutation of any individual produces an offspring whose fitness relative to the parent is given by an invariant distribution $Z$, such as a mean-shifted Gaussian. This serves as a prototypical model for the optimisation landscape when an evolution algorithm operates far from the global optimum. This simple model can be used to approximate the optimisation process for problems where it is intractable to model the exact fitness function, including tasks such as hyperparameter tuning in machine learning models. We rigorously analyse the expected growth rate $\mathcal{R}_μ$ of the continuous steady-state $(μ+1)$-ES in this model. Unlike comma-selection strategies, the steady-state $(μ+1)$-ES maintains overlapping generations, introducing complex mathematical dependencies among surviving parents that make it harder to analyse. We give a general technique to analyse the the $(μ+ 1)$-ES by constructing modified processes whose growth rates provably sandwich that of the original process. These modified processes are then easier to analyse but still close enough to the true process to give a tight bound on the expected growth rate. When $Z = \mathcal{N}(-δ, 1)$ and $μ\le e^δ$, we show that $\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$.

翻译：我们引入了一种新的简单模型，用于研究进化策略（ES）在一般问题中的适应度进度。在该模型中，我们绕过潜在的适应度景观，假设任何个体的变异产生的子代，其相对于父代的适应度由一个不变分布$Z$给出（例如均值偏移高斯分布）。当进化算法在远离全局最优解的区域运行时，该模型可作为优化景观的原型模型。对于难以建模精确适应度函数的问题（包括机器学习模型中的超参数调优等任务），此简单模型可用于近似优化过程。我们严格分析了连续稳态$(μ+1)$-ES在该模型中的预期增长率$\mathcal{R}_μ$。与逗号选择策略不同，稳态$(μ+1)$-ES保留了重叠世代，引入了存活父代之间复杂的数学依赖关系，使其分析更加困难。我们给出了一种通用技术，通过构造修改过程来分析$(μ+1)$-ES，这些过程的增长率被严格证明夹逼原始过程的增长率。这些修改过程更易于分析，同时与真实过程足够接近，从而给出预期增长率的紧界。当$Z = \mathcal{N}(-δ, 1)$且$μ\le e^δ$时，我们证明$\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$。