The Random Gradient hyper-heuristic was recently shown to be able to learn the optimal neighbourhood size when optimizing the LeadingOnes benchmark via the Randomised Local Search (RLS) meta-heuristic. However, for this to happen, a learning period of a certain length $τ$ had to be used, differently from classic hyper-heuristics, which change their behaviour based on the success of only the previous iteration. In this paper, we show how to automatically set this new parameter value, relieving the user from the non-trivial task of controlling this novel algorithm parameter. We prove that the resulting hyper-heuristic selects the optimal neighbourhood size in a $1-o(1)$ fraction of the iterations and, consequently, optimises the LeadingOnes benchmark in the best possible time (apart from lower-order terms) achievable with these neighborhood sizes.