Errata: Convergence Analysis of Evolutionary Algorithms That Are Based on the
     Paradigm of Information Geometry

Beyer, Hans-Georg

doi:10.1162/evco_x_00281

Four lines below Equation (7), page 682, it must correctly read:

Q = - \frac{1}{2} H_{f} (x) = - \frac{1}{2} \nabla \nabla^{T} f (x)

In Equation (), page 685, the logarithms are missing, it must correctly read:

I_{i j} (θ) = \int \frac{\partial ln p (x | θ)}{\partial θ_{i}} \frac{\partial ln p (x | θ)}{\partial θ_{j}} p (x | θ) d^{N} x = - \int \frac{\partial^{2} ln p (x | θ)}{\partial θ_{i} \partial θ_{j}} p (x | θ) d^{N} x

(27)

On page 687, below Equation (36), the derivations leading to Equation () must be corrected^¹ (the final result in Equation (41) does not change by this correction):

Treating the

C

-related part in (35) using

\frac{\partial C_{a b}}{\partial C_{c d}} = \frac{1}{2} (δ_{a c} δ_{b d} + δ_{a d} δ_{b c})

(here, symmetry of C must be taken into account) yields

\begin{matrix} I_{(α_{1} α_{2}), (β_{1} β_{2})} & = & \frac{1}{2} \sum_{k, l, m, n} C_{k l}^{- 1} \frac{\partial C_{l m}}{\partial C_{α_{1} α_{2}}} C_{m n}^{- 1} \frac{\partial C_{n k}}{\partial C_{β_{1} β_{2}}} \\ = & \frac{1}{8} \sum_{k, l, m, n} C_{k l}^{- 1} (δ_{l α_{1}} δ_{m α_{2}} + δ_{l α_{2}} δ_{m α_{1}}) C_{m n}^{- 1} (δ_{n β_{1}} δ_{k β_{2}} + δ_{n β_{2}} δ_{k β_{1}}) . \end{matrix}

(37)

Thus, one gets for the

C

-related part of

θ

(taking the symmetry of

C^{- 1}

into account)

C : I_{(α_{1} α_{2}), (β_{1} β_{2})} = \frac{1}{4} (C_{α_{1} β_{1}}^{- 1} C_{α_{2} β_{2}}^{- 1} + C_{α_{1} β_{2}}^{- 1} C_{α_{2} β_{1}}^{- 1}) .

(38)

The non-numbered equation below Equation (40), page 687, must be adopted accordingly:

The correctness of

C : I_{(α_{1} α_{2}), (β_{1} β_{2})}^{- 1} = 2 C_{α_{1} β_{2}} C_{β_{1} α_{2}}

(40)

is proven directly by checking

\sum_{β_{1}, β_{2}} I_{(α_{1} α_{2}), (β_{1} β_{2})}^{- 1} I_{(β_{1} β_{2}), (γ_{1} γ_{2})} = \frac{1}{2} (δ_{α_{1} γ_{2}} δ_{α_{2} γ_{1}} + δ_{α_{1} γ_{1}} δ_{α_{2} γ_{2}}) .

On page 694, the derivation of Equation () must be corrected (again without consequences for the result in Equation (84)):

\begin{matrix} \frac{\partial σ_{f}}{\partial C_{m n}} & = & \frac{1}{2} \frac{1}{σ_{f}} \frac{\partial}{\partial C_{m n}} (\sum_{i, j, k, l} (a_{i} - 2 {\bar{x}}_{k} Q_{k i}) C_{i j} (a_{j} - 2 {\bar{x}}_{l} Q_{l j}) + 2 Q_{i j} C_{j k} Q_{k l} C_{l i}) \\ = & \frac{1}{2} \frac{1}{σ_{f}} \sum_{i, j, k, l} ((a_{i} - 2 {\bar{x}}_{k} Q_{k i}) \frac{1}{2} (δ_{i m} δ_{j n} + δ_{i n} δ_{j m}) (a_{j} - 2 {\bar{x}}_{l} Q_{l j}) \\ + Q_{i j} (δ_{j m} δ_{k n} + δ_{j n} δ_{k m}) Q_{k l} C_{l i} + Q_{i j} C_{j k} Q_{k l} (δ_{l m} δ_{i n} + δ_{l n} δ_{i m})) \\ = & \frac{1}{2} \frac{1}{σ_{f}} \sum_{k, l} ((a_{m} - 2 {\bar{x}}_{k} Q_{k m}) (a_{n} - 2 {\bar{x}}_{l} Q_{l n}) + 4 Q_{m k} C_{k l} Q_{l n}) \end{matrix}

(83)

Note

¹

The author is grateful to Zhenhua Li for pointing out this mistake to be corrected here.

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2020

Massachusetts Institute of Technology

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