Stagnation Detection with Randomized Local Search

Stagnation Detection in Highly Multimodal Fitness Landscapes

“…Therefore, even for the best setting of the mutation rate, only the expected number of flipping bits equals the gap size while the actual number of flipping bits may be different. This has motivated Rajabi and Witt (2021) to consider the k-bit flip operator flipping a uniform random subset of k bits as known from randomized local search (RLS) (Doerr and Doerr, 2018) and to adjust k via stagnation detection. Compared to the SD-(1+1) EA, this allows a speed-up of ( ne m ) m / n m (up to lower-order terms) on functions with gap size m and a speed-up of up to roughly e = 2.718 .…”

“…on unimodal functions while still being able to search globally. Rajabi and Witt (2021) emphasize that their RLS with self-adjusting k-bit flip operator resembles variable neighborhood search (Hansen and Mladenovic, 2018) but features less determinism by drawing the k bits to be flipped uniformly at random instead of searching the neighborhood in a fixed order. The random behavior still maintains many characteristics of the original RLS, including independent stochastic decisions which ease the runtime analysis.…”

“…However, a drawback of the randomness is that the independent, uniform choice of the set of bits to be flipped leaves a positive probability of missing an improvement within the given time a specific parameter value k is tried. Therefore, the first RLS variant with stagnation detection proposed in Rajabi and Witt (2021) and called SD-RLS there has infinite expected runtime in general, but is efficient with high probability, where the success probability of the algorithm is controlled via the threshold value for the number of steps without fitness improvement that trigger a change of k. We remark here that the problem with infinite runtime is not existent with the independent bit flips as long as each bit is flipped with probability in the open interval (0, 1).…”

“…In this paper, we denote by SD-RLS p the simple SD-RLS just proposed. To guarantee finite expected optimization time, Rajabi and Witt (2021) introduce a second variant that repeatedly returns to lower so-called (mutation) strengths, i. e., number of bits flipped, while the algorithm is still waiting for an improvement. The largest neighborhood size (i. e., number of bits flipped) is denoted as radius r and, in essence, the strength s is decreased in a loop from r to 1 before the radius is increased.…”

“…Interestingly, the additional time spent at exploring smaller strengths in this loop, with the right choice of phase lengths, contributes only a lower-order term to the typical time that SD-RLS p has in the absence of errors. The resulting algorithm (Algorithm 2) is called SD-RLS * in Rajabi and Witt (2021), but in this paper referred to by SD-RLS r , where the label r denotes robust.…”