written in terms of the precision, β,as ∇ϕER ¼∂ER ∂β∇ϕβð5Þ Note here that only the first term on the right hand side of Eq. (5) depends on the difficulty Δ, while the second describes how the precision changes with ϕ. Note also that Δitself, as the ‘true’ decision variable, is independent of ϕ. This means that the optimal difficulty for training, that maximizes the change in the parameters, ϕ,at this time point, is the value of the decision variable Δ*that maximizes ∂ER/∂β. Of course, this analysis ignores the effect of changing ϕon the form of the noise—instead assuming that it only changes the scale factor, β, an assumption that likely holds in the relatively simple cases we consider here, although whether it holds in more complex cases will be an important question for future work. In terms of the decision variable, the optimal difficulty changes as a function of precision (Fig. 1c) meaning that the difficulty of training must be adjusted online according to the skill of the agent. Using the monotonic relationship between Δand ER (Fig. 1b) it is possible to express the optimal difficulty in terms of the error rate, ER*(Fig. 1d). Expressed this way, the optimal difficulty is constant as a function of precision, meaning that optimal learning can be achieved by clamping the error rate during training at a fixed value, which, for Gaussian noise is ER ¼1 21 erf 1 ffiffiffi 2 p 0:1587 ð6Þ That is, the optimal error rate for learning is 15.87%, and the optimal accuracy is around 85%. We call this the Eighty Five Percent Rule for optimal learning