Natural Evolution Strategies follow the natural gradient using the Fisher Information Matrix $\mathbf{F}_\theta$ of a search distribtion $p_\theta$. That is, parameters in natural evolution strategies are updated with:
\begin{equation} \theta \gets \theta + \alpha \mathbf{F}^{-1} \nabla_\theta J \end{equation}
with
\begin{align} \nabla_\theta J =& \frac{1}{n} \sum_{i=1}^n f(x_i) \nabla_\theta p_\theta(x_i)\\ \mathbf{F} = & \frac{1}{n}\nabla_\theta\log p_\theta(x_i) \nabla_\theta\log p_\theta(x_i)^\top \end{align}
where $x_i$ is a drawn sample from $p_\theta(.)$ and $f(x_i)$ is its fitness.
In OpenAI's evolution strategy approach the update looks like this:
\begin{equation} \theta \gets \theta + \alpha \nabla_\theta J \end{equation}
with
\begin{equation} \nabla_\theta J = \frac{1}{n \sigma} \sum_{i=1}^n f(\theta + \sigma \epsilon_i) \epsilon_i \end{equation}
I do not quite understand, why the latter is a natural evolution strategy, i.e., follow the natural gradient, as the authors claim.
