When variational regularisation methods are used to solve inverse problems, they suffer from the drawback of having potentially many parameters which the user must specify. A common approach to handle this is to learn these parameters from data. While mathematically appealing, this strategy leads to a bilevel optimisation problem which is difficult to solve computationally. Theoretically, algorithms for bilevel learning rely on access to exact solutions to the lower-level regularisation problem, but this condition is not guaranteed in practice. In this talk, we describe a novel approach using dynamic accuracy derivative-free optimisation for solving bilevel learning problems. This approach still retains convergence guarantees but allows the regularisation problem to be solved inexactly and hence can be implemented in practice. This is joint work with Matthias Ehrhardt (Bath, UK).