A Step Forward in Studying the Compact Genetic Algorithm

The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, we prove that the corresponding ODE converges to local optima and stays there. Consequently, we conclude that the cGA will converge to the local optima of the function to be optimized.

💡 Research Summary

The paper presents a rigorous theoretical study of the compact Genetic Algorithm (cGA), an Estimation‑of‑Distribution Algorithm that requires only O(n) memory by maintaining a probability vector rather than an explicit population. The authors first formalize the cGA’s stochastic dynamics as a discrete‑time Markov chain. In this chain each state is the n‑dimensional vector p = (p₁,…,pₙ), where pᵢ denotes the current probability of a ‘1’ at bit position i. At every iteration two individuals are sampled independently according to p, their fitnesses are evaluated, and the probability vector is updated by adding or subtracting a fixed step Δ = 1/N (N being the virtual population size) to those components where the better individual differs from the worse one. The transition probabilities of the Markov chain are derived directly from this update rule, capturing the exact stochastic behavior of the algorithm.

To obtain analytical insight, the authors consider the limit of large N (or equivalently small Δ) and replace the expected one‑step change of p by a continuous‑time differential equation. By expanding the expectation in a Taylor series and discarding O(Δ²) terms, they arrive at an ordinary differential equation (ODE) of the form

 dpᵢ/dt = E