CMA-ES - Theoretical Foundations - Interpretation As Coordinate System Transformation

Interpretation As Coordinate System Transformation

Using a non-identity covariance matrix for the multivariate normal distribution in evolution strategies is equivalent to a coordinate system transformation of the solution vectors, mainly because the sampling equation

\begin{align} x_i &\sim\ m_k + \sigma_k\times\mathcal{N}(0,C_k) \\ &\sim\ m_k + \sigma_k \times C_k^{1/2}\mathcal{N}(0,I) \end{align}

can be equivalently expressed in an "encoded space" as

\underbrace{C_k^{-1/2}x_i}_{\text{represented in the encode space} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} \sim\ \underbrace{C_k^{-1/2} m_k} {} + \sigma_k \times\mathcal{N}(0,I)

The covariance matrix defines a bijective transformation (encoding) for all solution vectors into a space, where the sampling takes place with identity covariance matrix. Because the update equations in the CMA-ES are invariant under coordinate system transformations (general linear transformations), the CMA-ES can be re-written as an adaptive encoding procedure applied to a simple evolution strategy with identity covariance matrix. This adaptive encoding procedure is not confined to algorithms that sample from a multivariate normal distribution (like evolution strategies), but can in principle be applied to any iterative search method.

Read more about this topic:  CMA-ES, Theoretical Foundations

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