OneMax in Black-Box Models with Several Restrictions

Abstract

Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the blackbox complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind.We analyze in particular the combined memory-restricted ranking-based blackbox complexity of OneMax for different memory sizes. While its isolated memoryrestricted as well as its ranking-based black-box complexity for bit strings of length n is only of order n/ log n, the combined model does not allow for algorithms being faster than linear in n, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory-and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models.Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in o(n log n) iterations.

The $$(1+1)$$ ( 1 + 1 ) Elitist Black-Box Complexity of LeadingOnes

“…This is in contrast to the situation for the OneMax function, where elitist selection does not substantially harm the running time [DL15b]. Given the much smaller complexity of Lo in many other models, this sheds some light on the cost of elitism.…”

“…Indeed, when the fitness of an individual is k for some k < n, then all but the k + 1 bits determining the fitness of the search point can be used for storing information about previous samples, the number of iterations elapsed, or any other information gathered during the optimization process. We recall that such strategies of constantly storing information about previous samples are at the heart of many upper bounds in black-box complexity, cf., for example, the proofs of the O(n/ log n) bound for the (1+1) memoryrestricted [DW14a], the ranking-based [DW14b] or the (1+1) elitist Monte Carlo [DL15b] black-box complexity of OneMax. We therefore need to show that despite the fact that changing the n − (k + 1) irrelevant bits has no influence on the fitness, the algorithm cannot make effective use of this storage space.…”

“…In [DL15a] examples are presented for which the elitist black-box complexity is much larger than in any of the previously existing black-box models, while in [DL15b] it is shown that the complexity of OneMax is of linear order only even for the most restrictive (1+1) setting. Here in this work we regard another classic problem, a generalized version of the LeadingOnes function.…”

Illustrating the trade-off between time, quality, and success probability in heuristic search

“…Apart from being useful in applications where a high confidence in the success of an algorithm needs to be guaranteed, we believe that the fixed-probability measures will give additional insights into the performance of search heuristics. We note that also in the theoretical running time analysis the success probability is a measure that has recently drawn some attention, for example in terms of the p-Monte Carlo runtime notion defined in [4]. The p-Monte Carlo runtime of an algorithm A is the running time needed to find an optimal solution with probability at least 1 − p. This corresponds to T (A, f , φ = opt, 1 − p) in our notation from Definition 2.1.…”

Bandit-based Random Mutation Hill-Climbing

“…Doerr et al proved that the black-box complexity with memory restriction one is at most 2n [33]. More lower and upper bounds of the complexity of OneMax in the different models are analysed [34], [35] and then summarised in Table 1 of [35]. In their elitist model, only the best-so-far solution can be kept in the memory.…”