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}


\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}


\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.

  • $\begingroup$ Did you end up coming up with an explanation to your question? $\endgroup$
    – s1624210
    Aug 27, 2022 at 13:13
  • $\begingroup$ It boils down to usage of ${\it natural gradient}$, Amari's definition used KL distance and the first paper you mentioned used Fisher's matrix. It is about accounting for uncertainty in second order updates.In OpenAI's approach using $\sigma$, noise standard-deviation accounts for the uncertainty, so it makes the apprach natural evaluation strategy. $\endgroup$ Aug 28, 2022 at 13:43
  • $\begingroup$ @msuzen I don't think that's correct as stated. Yes, the natural gradient may be about accounting for uncertainty, but it is about accounting for it in a very particular way. Not any constant factor depending on the standard-deviation makes the method a NES. In contrast to the update rule used by OpenAI, using the Fisher information would give you a sigma in the numerator and not in the denominator. $\endgroup$
    – s1624210
    Aug 29, 2022 at 10:58
  • $\begingroup$ @s1624210 Yes, you are right. It looks like there might be some intricacies and deeper insights there. Though, the natural gradient is indeed about accounting for uncertainty. $\endgroup$ Aug 29, 2022 at 16:51


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