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On the Interaction of Adaptive Population Control with Cumulative Step-Size Adaptation

Amir Omeradzic, Hans-Georg Beyer

Abstract

Three state-of-the-art adaptive population control strategies (PCS) are theoretically and empirically investigated for a multi-recombinative, cumulative step-size adaptation Evolution Strategy $(μ/μ_I, λ)$-CSA-ES. First, scaling properties for the generation number and mutation strength rescaling are derived on the sphere in the limit of large population sizes. Then, the adaptation properties of three standard CSA-variants are studied as a function of the population size and dimensionality, and compared to the predicted scaling results. Thereafter, three PCS are implemented along the CSA-ES and studied on a test bed of sphere, random, and Rastrigin functions. The CSA-adaptation properties significantly influence the performance of the PCS, which is shown in more detail. Given the test bed, well-performing parameter sets (in terms of scaling, efficiency, and success rate) for both the CSA- and PCS-subroutines are identified.

On the Interaction of Adaptive Population Control with Cumulative Step-Size Adaptation

Abstract

Three state-of-the-art adaptive population control strategies (PCS) are theoretically and empirically investigated for a multi-recombinative, cumulative step-size adaptation Evolution Strategy -CSA-ES. First, scaling properties for the generation number and mutation strength rescaling are derived on the sphere in the limit of large population sizes. Then, the adaptation properties of three standard CSA-variants are studied as a function of the population size and dimensionality, and compared to the predicted scaling results. Thereafter, three PCS are implemented along the CSA-ES and studied on a test bed of sphere, random, and Rastrigin functions. The CSA-adaptation properties significantly influence the performance of the PCS, which is shown in more detail. Given the test bed, well-performing parameter sets (in terms of scaling, efficiency, and success rate) for both the CSA- and PCS-subroutines are identified.
Paper Structure (18 sections, 59 equations, 16 figures, 4 tables, 2 algorithms)

This paper contains 18 sections, 59 equations, 16 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. CSA-ES with Adaptive Population Control
  3. The CSA-ES for Large Population Sizes
  4. Scaling of Generation Number and Mutation Strength
  5. Analysis of the CSA-ES
  6. Population Control Schedule
  7. Adaptive Population Control Strategies
  8. Method Comparison
  9. Comparison on the Rastrigin Function
  10. Conclusions and Outlook
  11. Supplementary Material
  12. Mutation Strength Rescaling
  13. PSA-CSA-ES Analysis
  14. Sphere Steady-State of the PSA-CSA-ES
  15. Additional Experiments: APOP
  16. ...and 3 more sections

Figures (16)

  • Figure 1: Progress rate $\varphi^*$ and scale-invariant $\sigma^{*}$ on the sphere.
  • Figure 2: Steady-state ratio $\gamma$ on the sphere function for $\vartheta=1/2$. Measured ratio $\sigma^*_{\mathrm{ss}}/\sigma^*_0$ (solid, with dots) compared to $\gamma$ from \ref{['eq:csa_gamma_b_v2']} (dashed) for the CSA variants \ref{['eq:sqrtN']} (blue), \ref{['eq:linN']} (red), and \ref{['eq:han']} (green).
  • Figure 3: Number of generations $G$, see \ref{['sec:dyn_phi_exp3']}, to reach the target $R^{(g)}/R^{(g_0)}\!=\!10^{-6}$ using a $(1000/1000_I,2000)$-ES (left) and $N=100$ (right). The average over 10 runs is taken. The solid lines show \ref{['eq:sqrtN']} in blue, \ref{['eq:linN']} in red, and \ref{['eq:han']} in green. $G$ from \ref{['sec:dyn_phi_exp3']} at $\gamma=0.9$ is displayed in dash-dotted black. The lower and upper dashed lines show $\gamma=0.8, 0.95$, respectively.
  • Figure 4: Stability of CSA on the sphere using predefined $\mu^{(g)}$-schedule. The gray lines show $\mu^{(g)}$ and the colored lines the $R^{(g)}$-dynamics of \ref{['eq:r1']} (red), \ref{['eq:r2']} (green), and \ref{['eq:r3']} (blue). The tested CSAs are \ref{['eq:sqrtN']}, \ref{['eq:linN']}, and \ref{['eq:han']}, from top to bottom, respectively for (a) and (b).
  • Figure 5: Sphere (left column) and random function (right column) with CSA \ref{['eq:sqrtN']} for $N=100$ using APOP (top, $L=10$), pcCSA (center, $L=10$), and PSA (bottom $\beta=1/10$). The performance is measured with deactivated population control at $\mu=10,100,1000$ (blue, green, and red signals, respectively) The threshold $\mathcal{T}$ is shown in solid black.
  • ...and 11 more figures