The (μw/ρ, λ)-CMA-ES maintains 𝒩(m, σ²C) and adapts all three each generation. The step-size path pσ drives σ via CSA: |pσ|/χn ≈ 1 is ideal — above 1 → σ grows, below → shrinks. The evolution path pc accumulates clean directional steps for the rank-1 covariance update. The rank-μ update stretches C toward the empirical covariance of selected offspring. Trail shows past mean positions. This explorer minimizes f(x,y) — darker regions on the heatmap are lower (better). Tip: Rosenbrock's narrow curved valley is a classic CMA-ES stress test — σ can stagnate before reaching (1,1); try increasing λ or reducing cσ×.
Fitness Landscape · minimizing f(x,y)
●m (mean)
●best-ever ★
●best sample
●top ρ
⊕optimum
Fitness History
3-D View

Controls

Covariance Structure

C (2×2)
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σ²
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λ₁(C)
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λ₂(C)
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θ₁
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Evolution Paths & Step-Size

pσ
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pc
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|pσ|/χn
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Fitness Function

Display

Strategy Parameters

10
5
5
0.50

Learning Rate Multipliers

1.00×
1.00×
1.00×
1.00×
8

Reference

σGlobal step size. All sample std. devs. scale as σ·√λᵢ(C). Adapted by CSA each generation.
χnExpected norm of an n-dim. standard Gaussian: √n·(1−1/(4n)+1/(21n²)) ≈ √n. Normalizes |pσ| so ratio ≈ 1 means σ is well-calibrated.
|pσ|/χnStep-size control signal. ≈1 = ideal; >1 → σ grows; <1 → σ shrinks.
cond(C)Condition number λ₁/λ₂. 1 = spherical; large → highly elongated distribution.
λ₁, λ₂(C)Eigenvalues of C; effective std. dev. along eigenvector i is σ√λᵢ.
θ₁Angle of dominant eigenvector (°, from +x axis).
m (dist. mean)Fitness evaluated at current distribution mean; not a sample — may lie in a poor region early on.
ρRecombination arity: # parents used in weighted mean and rank-μ update (ρ ≤ μ ≤ λ).
⊕ on mapWhite circle + × marks all known global optima of the current function.
★ on mapBest-ever sample found this run (red filled circle).