Progress Rate Analysis of Evolution Strategies on the Rastrigin Function: First Results

Area of Science:

• * Evolutionary Computation
• * Optimization Theory
• * Applied Mathematics

Background:

• * The Rastrigin function is a highly multimodal benchmark for global optimization.
• * Evolution Strategies (ES) are stochastic optimization algorithms.
• * Analyzing the progress rate of ES is crucial for understanding their efficiency.

Purpose of the Study:

• * To derive a first-order progress rate for intermediate multi-recombinative Evolution Strategy (μ/μ, λ)-ES.
• * To analyze the performance of this rate on the Rastrigin test function.
• * To investigate the influence of mutation strength and distance to the optimizer on progress.

Main Methods:

• * Linearized model using noisy order statistics.
• * Determination of mutation-induced variance for the Rastrigin function.
• * Iteration of progress using dynamical systems approach and comparison with simulations.

Main Results:

• * A first-order progress rate approximation was derived and validated against simulations.
• * Strengths and limitations of the approximation were identified based on mutation strength and optimizer distance.
• * The property of global convergence was discussed within the derived approximation.

Conclusions:

• * The derived first-order progress rate provides insights into ES performance on multimodal functions.
• * The study highlights the need for improved progress rate models and extensions to higher orders.
• * Future work includes incorporating positional fluctuations for a more comprehensive analysis.