Center-of-Mass Interpretation of Evolution Strategies

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 $\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$.

We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schr{ö}dinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ''survival of the flattest'' phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

While contemporary Evolution Strategies handle integer optimization problems effectively, their adaptation mechanism is grounded in $\ell_2$-based Gaussian models, which are not native to the integer lattice. In contrast, the $\ell_1$-norm provides the natural measure of displacement on $\mathbb{Z}^n$, with the double geometric distribution as its canonical mutation operator. In this work, we derive a fully $\ell_1$-native step-size adaptation mechanism from first principles and propose an Integer Natural Evolution Strategy. We show that the DG distribution belongs to the exponential family, and that its sufficient statistic $|z|$ yields a natural-gradient signal for dispersion adaptation. By accumulating this signal via an evolution path, we obtain a fading-memory online estimator of the natural gradient, following Ollivier (2018). This establishes that DG-based step-size adaptation arises directly from the statistical structure of the mutation distribution, rather than as a discrete analog of continuous ES mechanisms. Empirical results on integer quadratic benchmarks show that INES learns meaningful coordinate-wise step-sizes and is competitive with integer-handling CMA-ES baselines. Its advantages are most visible in high-dimensional Ellipsoidal problems and in robust convergence at larger dimensions.