Lower Bounds for Comparison Based Evolution Strategies Using VC‐dimension and Sign Patterns

We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non‐trivial results in the discrete case. This is achieved by considering the VC‐dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.