AI 中文总结
本研究提出整数自然进化策略（INES），基于ℓ₁原生机制设计步长适应，经整数二次基准测试，其在高维椭球问题上与CMA-ES基线相当且收敛更稳健。
AI 中文摘要
尽管当前的进化策略能有效处理整数优化问题，但其适应机制基于基于ℓ₂的高斯模型，而高斯模型并非整数格的原生模型。相比之下，ℓ₁范数是ℤⁿ上位移的自然度量，双几何（DG）分布是其典型变异算子。本研究从第一性原理推导了完全基于ℓ₁原生的步长适应机制，提出了整数自然进化策略（INES）。研究表明，DG分布属于指数族，其充分统计量|z|能为分散度适应提供自然梯度信号；通过沿进化路径累积该信号，可得到遵循Ollivier（2018）提出的 fading-memory 在线自然梯度估计器。这表明基于DG的步长适应直接源于变异分布的统计结构，而非连续进化策略机制的离散类似物。在整数二次基准测试中的实验结果显示，INES能学习有意义的坐标方向步长，且与处理整数的CMA-ES基线方法具有竞争力；其优势在高维椭球问题及大维度下的稳健收敛中最为显著。
英文摘要
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 \textsc{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.
