Too fast unbiased black-box algorithms

Natural Computing Series

2019

A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in general algorithms theory, this running time perspective is most useful when it is complemented by a meaningful complexity theory that studies the limits of algorithmic solutions.In the context of discrete black-box optimization, several black-box complexity models have been developed to analyze the best possible performance that a black-box optimization algorithm can achieve on a given problem. The models differ in the classes of algorithms to which these lower bounds apply. This way, black-box complexity contributes to a better understanding of how certain algorithmic choices (such as the amount of memory used by a heuristic, its selective pressure, or properties of the strategies that it uses to create new solution candidates) influences performance.In this chapter we review the different black-box complexity models that have been proposed in the literature, survey the bounds that have been obtained for these models, and discuss how the interplay of running time analysis and black-box complexity can inspire new algorithmic solutions to well-researched problems in evolutionary computation. We also discuss in this chapter several interesting open questions for future work.

Section: The Unrestricted Black-box Complexity Of Jump Functionsmentioning

confidence: 99%

Complexity Theory for Discrete Black-Box Optimization Heuristics

Natural Computing Series

2019

“…For black-box complexity analyses, the following definition of jump functions was suggested in Doerr et al (2011b). For ∈ [1; n 2 − 1] and z ∈ {0, 1} n , let…”

Section: Jump Functionsmentioning

confidence: 99%

The Unrestricted Black-Box Complexity of Jump Functions

Buzdalov

Evolutionary Computation

Kever

2016

Self Cite

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

“…, n − k − 1} ∪ {n} and Jump k (x) = 0 otherwise. Despite the fact that all common search heuristics need Ω(n k+1 ) fitness evaluations to optimize this function, the unary unbiased black-box complexity of these functions are surprisingly low, see Table 1 for a summary of results presented in [6] ( [9] for k = 1). Interestingly, even for extreme jump functions in which only the fitness value n/2 is visible and all other OneMax values are replaced by zero, polynomial-time unary unbiased black-box algorithms exist.…”

Section: Combining Unbiased and Elitist Black-box Modelsmentioning

confidence: 99%

Elitist Black-Box Models

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

Lengler

2015

Black-box complexity theory provides lower bounds for the runtime of black-box optimizers like evolutionary algorithms and serves as an inspiration for the design of new genetic algorithms. Several black-box models covering different classes of algorithms exist, each highlighting a different aspect of the algorithms under considerations. In this work we add to the existing black-box notions a new elitist black-box model, in which algorithms are required to base all decisions solely on (a fixed number of) the best search points sampled so far. Our model combines features of the ranking-based and the memory-restricted black-box models with elitist selection.We provide several examples for which the elitist blackbox complexity is exponentially larger than that the respective complexities in all previous black-box models, thus showing that the elitist black-box complexity can be much closer to the runtime of typical evolutionary algorithms.We also introduce the concept of p-Monte Carlo black-box complexity, which measures the time it takes to optimize a problem with failure probability at most p. Even for small p, the p-Monte Carlo black-box complexity of a function class F can be smaller by an exponential factor than its typically regarded Las Vegas complexity (which measures the expected time it takes to optimize F).