Lower Bounds from Fitness Levels Made Easy
This provides a more accessible and effective method for proving lower bounds in evolutionary computation, addressing a known bottleneck for researchers in the field, though it is incremental as it builds on existing fitness levels techniques.
The paper tackles the problem of proving lower bounds for evolutionary algorithm runtimes by introducing two new variants of the fitness levels method, one for upper and one for lower bounds, which rely on level leaving and visiting probabilities. It applies this method to reprove known results, such as the precise runtime on LeadingOnes and a lower bound on OneMax tight apart from an O(n) term, and proves a tighter lower bound for jump functions showing only O(2^{-n}) probability to avoid jumping over low fitness valleys.
One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters $γ_{i,j}$, $0 \le i < j \le n$. In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the (1+1) EA on \textsc{LeadingOnes}. (ii) A lower bound for the run time of the (1+1) EA on \textsc{OneMax}, tight apart from an $O(n)$ term. (iii) A lower bound for the run time of the (1+1) EA on long $k$-paths. We also prove a tighter lower bound for the run time of the (1+1) EA on jump functions by showing that, regardless of the jump size, only with probability $O(2^{-n})$ the algorithm can avoid to jump over the valley of low fitness.