$(\mu,\lambda)$-ES and $(\mu+\lambda)$-ES are two well-known Evolution Strategies. One variant of these algorithms is when we set a constant value for $\sigma$ and generate off-springs for $x$ as:

$$ x'=x+\mathcal{N}(0,\sigma) $$

The value of $\sigma$ has a direct effect on convergence of algorithm. In page 91 of (1) the relation between $\sigma$ and convergence speed, $\varphi'_2$, for several $(1,\lambda)$-ES and $(1+\lambda)$-ES strategies is plotted as:

Convergence velocity over $\sigma$

The dotted curve, $\varphi^{'*}_2$, indicates the location of the maxima of the curves. This curve can be approximated with a line (especially for $\sigma>1$). My question is, how could this fact help us and how we can take advantage of it?

The very first idea is we can predict best convergence speed given an optimal $\sigma$, but this does not help very much in practical problems since:

  1. We usually don't know optimal value of $\sigma$, rather we are looking for optimal $\sigma$,
  2. While this line can predict best convergence speed, it doesn't tell us which setup for $(\mu,\lambda)$ or $(\mu+\lambda)$ could reach to that speed (i.e. we don't know which of two ES algorithms and values for $\mu$ and $\lambda$ to use).

So, can this linear relation between $\sigma$ and best convergence speed (indicated by $\varphi^{'*}_2$ curve) help us to have a better use of $(\mu,\lambda)$-ES or $(\mu,\lambda)$-ES algorithms?

In the other hand, what would we lose if this relation weren't linear at the first place?

(1) Thomas Bäck. Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, Dortmund, Germany, 1996


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