A simplified run time analysis of the univariate marginal distribution algorithm on LeadingOnes
Introduction
Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that maintain a probabilistic model of the search space and refine it iteratively. In each iteration, the current model of an EDA is used to create some samples which, in turn, are used to adjust the model such that better solutions are more likely to be created in the following iteration. Thus, the model evolves over time into one that creates very good solutions. EDAs have been applied to real-world problems with great success [1].
Within the last few years, the theoretical analysis of EDAs has gained increasing interest (see, for example, the survey by Krejca and Witt [2]). One of the first papers in this period was by Dang and Lehre [3], who proved run time guarantees for the univariate marginal distribution algorithm (UMDA, [4]) when optimizing the two classical benchmark functions OneMax and LeadingOnes. While their run time bound for OneMax has been improved since then independently by Lehre and Nguyen [5] and Witt [6], [7], the run time bound of is the best known result so far on LeadingOnes. Here, n is the problem dimension and λ is the offspring population size of the UMDA, which is required to be for the result to hold. In an extension of their result, Dang et al. [8] show the same run time bound but slightly improve a restriction on the population size by a factor of 13.
In this work, we improve in Theorem 5 the second term of this bound from to when , where μ is the size of the subpopulation selected for the model update, and where C is a sufficiently large constant. In the regime of , the UMDA shows the generally desirable behavior of low genetic drift, that is, the sampling frequencies stay in the middle range of, say, until a sufficiently strong fitness signal moves them into the right direction.
Using the same proof method as for the upper bound, we prove a lower bound (see Theorem 6) that matches the upper bound for polynomial λ. This improves the previously best known lower bounds by Lehre and Nguyen [9] for the regime of . For the regime of , the bound by Lehre and Nguyen remains the best known lower bound. Additionally, Lehre and Nguyen prove a lower bound of for and , for an appropriately chosen constant C, which remains untouched by our result.
From our lower bound and the upper bound by Dang and Lehre [3], we observe that the UMDA generally performs better on LeadingOnes in the regime of strong genetic drift, as the regime results in a run time bound of , whereas the regime results in a bound of . Interestingly, for the regime of , it remains open whether a run time of is possible. We believe that this is a particularity of the LeadingOnes problem and that, in general, genetic drift in EDAs leads to a performance loss, since it may take long to move a frequency from the wrong boundary value back into the middle range. This has been rigorously shown by Lengler et al. [10] for the OneMax problem, and by Lehre and Nguyen [11] and Doerr and Krejca [12] for the DeceptiveLeadingBlocks problem.
Equally interesting as the improved run time guarantee is our elementary proof method. While it was truly surprising that Dang and Lehre [3] could use the level-based population method to analyze an EDA (which does not have a population that is transferred from one iteration to the next), this method is a highly advanced tool restricted to algorithms whose population follows a distribution entirely characterized by the previous population. In contrast to this, our proofs only use elementary arguments common in the analysis of evolutionary algorithms and are applicable for deriving upper and lower bounds. We are thus optimistic that our arguments can more easily be applied to other EDAs as well.
We note that both of our bounds do not require the fraction to be constant, which is a common requirement of many other analyses of the UMDA [5], [7], [8], [13] (although this is not always explicitly stated in the result). In particular, our bounds show that the gain from reducing the selection rate (which often requires a costly increase of λ) is very small, namely, only logarithmic in .
Another advantage of our approach is that it gives run time guarantees that hold with high probability, whereas the level-based method, relying on drift arguments, can only give bounds on expected run times. Consequently, the result of Dang and Lehre [3] also concerns the expectation only. We believe that a result that holds with high probability is often more relevant, as has also been argued by Doerr [14]. A high-probability bound provides can be interpreted as a deterministic upper with a (very) low failure probability, which is more closely connected to classical run time analysis. In contrast, an upper bound on the expected run time only guarantees that the algorithm terminates almost surely, and it provides (without further deeper analysis) only rather weak tail bounds via Markov's inequality.
Section snippets
Preliminaries
We are concerned with the run time analysis of algorithms optimizing pseudo-Boolean functions, that is, functions , where denotes the dimension of the problem. Given a pseudo-Boolean function f and a bit string x, we refer to f as a fitness function, to x as an individual, and to as the fitness of x.
For an , we define . From now on, if not stated otherwise, the variable n always denotes a natural number. For a vector x of length n, we denote its component at
Upper bound
In the following, we present our simple and intuitive run time analysis for the upper bound of the UMDA optimizing LeadingOnes, which gives the following theorem.
Theorem 5 Let be a constant, and let . Consider the UMDA optimizing LeadingOnes with and . Further, let . Then the UMDA samples the optimum after at most fitness function evaluations with a probability of at least .
As discussed in the introduction, we only want to consider
Lower bound
Our main insight, which gave our sharper upper bound with a proof simpler than in previous works, was that the UMDA, when optimizing LeadingOnes in the regime of low genetic drift, makes a steady progress in each iteration: It sets the frequencies to the maximum value in a left-to-right fashion, keeping the other frequencies close to the middle value of . The increase of the number of frequencies at the maximum value, with a simple Chernoff bound argument, could be shown to be
Conclusion
We improved the best known upper bound for the run time of the UMDA on LeadingOnes for the case of from to . This result improves the previous best result both by removing an unnecessary factor and, not discussed in previous works, by gaining a factor and thus showing an advantage of using a low selection rate . We obtained these results via a different proof method that avoids the technically deeper level-based method. Our arguments can
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences. This publication is based upon work from COST Action CA15140, supported by COST.
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