Abstract
Following up on previous work of Cathabard et al. (in: Proceedings of foundations of genetic algorithms (FOGA’11), ACM, 2011) we analyze variants of the (1 + 1) evolutionary algorithm (EA) for problems with unknown solution length. For their setting, in which the solution length is sampled from a geometric distribution, we provide mutation rates that yield for both benchmark functions OneMax and LeadingOnes an expected optimization time that is of the same order as that of the (1 + 1) EA knowing the solution length. More than this, we show that almost the same run times can be achieved even if no a priori information on the solution length is available. We also regard the situation in which neither the number nor the positions of the bits with an influence on the fitness function are known. Solving an open problem from Cathabard et al. we show that, for arbitrary \(s\in {\mathbb {N}}\), such OneMax and LeadingOnes instances can be solved, simultaneously for all \(n\in {\mathbb {N}}\), in expected time \(O(n (\log (n))^2 \log \log (n) \ldots \log ^{(s-1)}(n) (\log ^{(s)}(n))^{1+\varepsilon })\) and \(O(n^2 \log (n) \log \log (n) \ldots \log ^{(s-1)}(n) (\log ^{(s)}(n))^{1+\varepsilon })\), respectively; that is, in almost the same time as if n and the relevant bit positions were known. For the LeadingOnes case, we prove lower bounds of same asymptotic order of magnitude apart from the \((\log ^{(s)}(n))^{\varepsilon }\) factor. Aiming at closing this arbitrarily small remaining gap, we realize that there is no asymptotically best performance for this problem. For any algorithm solving, for all n, all instances of size n in expected time at most T(n), there is an algorithm doing the same in time \(T'(n)\) with \(T'=o(T)\). For OneMax we show results of similar flavor.
Similar content being viewed by others
Notes
Note here that in the initial segment model, we assume that the set of LeadingOnes instances is not invariant under permutation, that is, for a fixed n there is just the usual LeadingOnes instance measuring the largest segment \([1{\ldots }i] \subseteq [1{\ldots }n]\) that contains only 1-bits. For the unrestricted model, we assume that the LeadingOnes instance is implemented on the set I of relevant bits in an arbitrary ordering of the bit positions. See Sect. 2.1 for precise definitions.
References
Antipov, D., Doerr, B., Fang, J., Hetet, T.: Runtime analysis for the \((\mu +\lambda )\) EA optimizing OneMax. In: Genetic and Evolutionary Computation Conference (GECCO’18). ACM (2018) (to appear)
Ash, J.M.: Neither a worst convergent series nor a best divergent series exists. Coll. Math. J. 28, 296–297 (1997)
Auger, A., Doerr, B.: Theory of Randomized Search Heuristics. World Scientific, Singapore (2011)
Bianchi, L., Dorigo, M., Gambardella, L., Gutjahr, W.: A survey on metaheuristics for stochastic combinatorial optimization. Nat. Comput. 8, 239–287 (2009)
Böttcher, S., Doerr, B., Neumann, F.: Optimal fixed and adaptive mutation rates for the LeadingOnes problem. In: Proceedings of Parallel Problem Solving from Nature (PPSN’10), pp. 1–10. Springer, Berlin (2010)
Cathabard, S., Lehre, P.K., Yao, X.: Non-uniform mutation rates for problems with unknown solution lengths. In: Proceedings of Foundations of Genetic Algorithms (FOGA’11), pp. 173–180. ACM (2011)
Dietzfelbinger, M., Rowe, J.E., Wegener, I., Woelfel, P.: Tight bounds for blind search on the integers and the reals. Comb. Probab. Comput. 19, 711–728 (2010)
Doerr, B.: Better runtime guarantees via stochastic domination. CoRR abs/1801.04487 (2018). http://arxiv.org/abs/1801.04487
Doerr, B.: Probabilistic tools for the analysis of randomized optimization heuristics. CoRR abs/1801.06733 (2018). http://arxiv.org/abs/1801.06733
Doerr, B., Doerr, C.: Optimal static and self-adjusting parameter choices for the \((1+(\lambda,\lambda ))\) genetic algorithm. Algorithmica 80, 1658–1709 (2018)
Doerr, B., Doerr, C., Kötzing, T.: Solving problems with unknown solution length at (almost) no extra cost. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’15), pp. 831–838. ACM (2015)
Doerr, B., Doerr, C., Kötzing, T.: The right mutation strength for multi-valued decision variables. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’16), pp. 1115–1122. ACM (2016)
Doerr, B., Doerr, C., Kötzing, T.: Unknown solution length problems with no asymptotically optimal run time. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’17), pp. 1367–1374. ACM (2017)
Doerr, B., Fouz, M., Witt, C.: Quasirandom evolutionary algorithms. In: Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference (GECCO’10), pp. 1457–1464. ACM (2010)
Doerr, B., Fouz, M., Witt, C.: Sharp bounds by probability-generating functions and variable drift. In: Proceedings of the 13th Annual Genetic and Evolutionary Computation Conference (GECCO’11), pp. 2083–2090. ACM (2011)
Doerr, B., Jansen, T., Witt, C., Zarges, C.: A method to derive fixed budget results from expected optimisation times. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’13), pp. 1581–1588. ACM (2013)
Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64, 673–697 (2012)
Doerr, B., Künnemann, M.: Optimizing linear functions with the (1+\(\lambda \)) evolutionary algorithm–different asymptotic runtimes for different instances. Theor. Comput. Sci. 561, 3–23 (2015)
Doerr, B., Le, H.P., Makhmara, R., Nguyen, T.D.: Fast genetic algorithms. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’17), pp. 777–784. ACM (2017)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)
Hardy, G.H.: Orders of Infinity. Cambridge University Press, Cambridge (1910)
Hwang, H., Panholzer, A., Rolin, N., Tsai, T., Chen, W.: Probabilistic analysis of the (1+1)-evolutionary algorithm. Evol. Comput. 26, 299–345 (2018)
Jansen, T.: Analyzing Evolutionary Algorithms—The Computer Science Perspective. Springer, Berlin (2013)
Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evol. Comput. 13, 413–440 (2005)
Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 9, 303–317 (2005)
Ladret, V.: Asymptotic hitting time for a simple evolutionary model of protein folding. J. Appl. Probab. 42, 39–51 (2005)
Lehre, P.K., Yao, X.: Runtime analysis of the (1 + 1) EA on computing unique input output sequences. Inf. Sci. 259, 510–531 (2014)
Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization—Algorithms and Their Computational Complexity. Springer, Berlin (2010)
Sudholt, D.: A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evol. Comput. 17, 418–435 (2013)
Witt, C.: Runtime analysis of the (\(\mu \) + 1) EA on simple pseudo-Boolean functions. Evol. Comput. 14, 65–86 (2006)
Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear functions. Comb. Probab. Comput. 22, 294–318 (2013)
Acknowledgements
Parts of this work have been done while Timo Kötzing was visiting the École Polytechnique. This work was supported in part by the German Research Foundation under Grant FR 2988 (TOSU), by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and from the Gaspard Monge Program for Optimization and Operations Research (PGMO) of the Jacques Hadamard Mathematical Foundation (FMJH).
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Summable Sequences
In this section we recall definitions and results regarding summable sequences.
A sequence \(\vec {p}=(p_i)_{i \in {\mathbb {N}}}\) is summable if the sequence \((S_n)_{n \in {\mathbb {N}}}\) of partial sums of absolute values \(S_n:=\sum _{i=1}^n{|p_i|}\) converges.
The run time guarantees in Theorems 15, 16, 19, and 20 all decrease with increasing sequence \(\vec {p}\). In the hope of finding best possible parameters for the \((1 + 1)~\text{ EA }_{\vec {p}}\) and the \((1 + 1)~\text{ EA }_{Q}\) it is therefore natural to ask for a largest summable sequence p. The following statement shows that such a sequence does not exist. More precisely, for every summable sequence \(\vec {p}\), there is a summable sequence which decreases more slowly than \(\vec {p}\). Such constructions are well known in the mathematics literature, cf. [2] for an easily-accessible example proving Theorem 41.
Theorem 41
(Folklore) For every summable sequence \(\vec {p}:=(p_n)_{n \in {\mathbb {N}}}\) of positive terms \(p_n>0\) there exists a summable sequence \(\vec {q}:=(q_n)_{n \in {\mathbb {N}}}\) such that \(\vec {q}=\omega (\vec {p})\).
Likewise, for every non-summable sequence \(\vec {q}\) there exists a non-summable sequence \(\vec {p}\) such that \(\vec {p}=o(\vec {q})\).
It is interesting to note that, in both statements of Theorem 41, the sequence \(\vec {q}\) can be chosen in a way that the expected run times of the \((1 + 1)~\text{ EA }_{\vec {p}}\) when fed with sequence \(\vec {p}\) and \(\vec {q}\), respectively, satisfy \({{\mathrm{E}}}[T_{\vec {q}}]=o({{\mathrm{E}}}[T_{\vec {p}}])\) (this is not automatically the case when \(\vec {q}=\omega (\vec {p})\)). But since this result also follows from Theorem 30, we do not prove these statements explicitly.
We have introduced in equation (5) (Sect. 4.2) the sequences \((p_b^{s,\varepsilon }(n))_{n \in {\mathbb {N}}}\). We have also discussed that it was shown in [21, page 48] that for every \(\varepsilon >0\) and every \(s \ge 1\), the sequence \((p_2^{s,\varepsilon }(n))_{n \in {\mathbb {N}}}\) is summable. In the same work, Hardy also mentions that, for all \(s \in {\mathbb {N}}\), the sequence \((p_2^{s,0}(n))_{n \in {\mathbb {N}}}\) is not summable; he attributes this result to De Morgan and Bertrand. From this result we easily get the following statement, which we need in the proof of our lower bounds in Sect. 6.2.3.
Lemma 42
Let \(b>1\) and let \(\vec {p}\) be a summable sequence of positive terms \(0<p_n<1\). Then, for all \(s \in {\mathbb {N}}\), \(1/p_n = \omega (1/p_b^{s,0}(n))\); i.e., \(1/p_n = \omega (n \log _b(n) \log ^{(2)}_b(n)\ldots \log ^{(s)}_b(n))\).
Proof
We show the statement for \(b=2\). The general case follows from observing that, for all \(s \in {\mathbb {N}}\) and all \(b>1\), \(1/p_b^{s,0}(n) = {\varTheta }(1/p_2^{s,0}(n))\).
We thus need to show that for all summable sequences \(\vec {p}\) of positive terms \(0<p_n<1\), all integers \(s \in {\mathbb {N}}\), and all positive constants \(C>0\) there exists an integer \(n_0=n_0(\vec {p},s,C)\) such that, for all \(n \ge n_0\), it holds that \(1/p_n \ge C/p_2^{s,0}(n) = C n \log _2(n) \log ^{(2)}_2(n) \ldots \log _2^{(s)}(n)\).
For the sake of contradiction we assume that this is not the case. Then there exists a summable sequence \(\vec {p}\) of positive terms \(0<p_n<1\), an integer \(s \in {\mathbb {N}}\), and a constant \(C>0\) such that for all \(n_0 \in {\mathbb {N}}\) there exists an integer \(n \ge n_0\) such that \(1/p_n \le C/p_2^{s,0}(n) = C n \log _2(n) \log ^{(2)}_2(n) \ldots \log _2^{(s)}(n)\). This implies that there exists a subsequence \(\vec {q}\) of \(\vec {p}\) such that \(q_n \ge 1/\left( C n \log _2(n) \log ^{(2)}_2(n) \ldots \log _2^{(s)}(n)\right) \). However, the result of De Morgan and Bertrand states that this sequence is not summable. Hence, \(\vec {p}\) cannot be summable. \(\square \)
Furthermore, we have mentioned in Sect. 4.2 that the summability of the sequence \((p_b^{\infty }(n))_{n \in {\mathbb {N}}}\) (cf. equation (7)) crucially depends on the base b of the logarithm. Indeed, using Cauchy’s condensation test, one can show the following.
Theorem 43
The sequence \((p_b^{\infty }(n))_{n \in {\mathbb {N}}}\) is summable if and only if \(b<e:=\exp (1)\). More precisely,
-
1.
For all \(b>1\) and all \(s \in {\mathbb {N}}\) it holds that \(\sum _{i=1}^{n}{p^{(s,0)}_b(i)} = {\varTheta }\left( \log ^{(s+1)}(n)\right) \).
-
2.
For all \(1<b<e:=\exp (1)\), \(\sum _{i=1}^{n}{p^\infty _b(i)} = {\varTheta }(1)\).
-
3.
\(\sum _{i=1}^{n}{p^\infty _e(i)} = {\varTheta }(\log ^*(n))\).
-
4.
For all \(b>e\), \(\sum _{i=1}^{n}{p^\infty _b(i)} = \exp ({\varTheta }(\ln ^*(n)))\).
Proof
For this proof, we set
Let \(s \in {\mathbb {N}}\) and \(b>1\). We apply integration by substitution, in the formulation of Theorem 46 below and with \(\varphi = \exp _b\), \(s+1\) times to the function \(x \mapsto p^{(s,0)}_b(x)\). Using \(b^x p^{(s,0)}_b(b^x) = p^{(s-1,0)}_b(x)\) and that the derivative of \(\exp _b\) is \(\exp _b\) times \(\ln (b)\) we obtain, for \(n \ge e(s)\),
Thus, we also get (using \(p^\infty _b\) monotonically decreasing)
Clearly, for \(b<e\), this sum converges for \(n \rightarrow \infty \). Furthermore, for \(b=e\), this partial sum equals \(\ln ^*(n)\), while for \(b > e\) it is \(2^{O(\ln ^*(n))}\) as desired. A corresponding lower bound can be found analogously. \(\square \)
From Theorem 43 we also get the following result.
Corollary 44
Let \(b\ge e\). For every summable sequence \(\vec {p}\) it holds that \(1/p_n = \omega (1/p_b^{\infty }(n)) = \omega (n \log _b(n) \log ^{(2)}_b(n) \ldots )\).
Corollary 44 can be proven in the same way as Lemma 42.
Appendix B: Useful Tools
We use this section to recall two useful theorems.
The following well-known lemma is applied a few times in our proofs.
Lemma 45
Let \(\vec {a}=(a_n)_{n \in {\mathbb {N}}}\) and \(\vec {b}=(b_n)_{n \in {\mathbb {N}}}\) be sequences of positive terms satisfying \(\vec {a}= o(\vec {b})\). Set \(A:=(A_n)_{n \in {\mathbb {N}}}\) with \(A_n:=\sum _{i=1}^n{a_i}\) and \(B:=(B_n)_{n \in {\mathbb {N}}}\) with \(B_n:=\sum _{i=1}^n{b_i}\). Assume that \(B=\omega (1)\). Then \(A=o(B)\).
Proof
We need to show that for all \(\varepsilon >0\) there exists an integer \(n(\varepsilon ) \in {\mathbb {N}}\) such that \(A_n/B_n \le \varepsilon \). Let \(\varepsilon >0\). Since \(\vec {a}= o(\vec {b})\) there exists an integer \(n_{\varepsilon } \in {\mathbb {N}}\) such that \(a_n/b_n < \varepsilon /2\) for all \(n \ge n_{\varepsilon }\). Since \(B=\omega (1)\) there exists \(n_{\varepsilon ,B} \in {\mathbb {N}}\) such that \(\sum _{i=1}^{n_{\varepsilon }-1}{a_i}/B_n \le \varepsilon /2\) for all \(n \ge n_{\varepsilon ,B}\). We therefore obtain for all \(n \ge \max \{n_{\varepsilon }, n_{\varepsilon ,B}\}\)
\(\square \)
When we apply integration by substitution, we refer to the following formulation.
Theorem 46
(Integration by Substitution) Let \(m<n\) and let I be an Interval. Let \(f: I \rightarrow {\mathbb {R}}\) be continuous and \(\varphi : [m,n] \rightarrow I\). Then we have
Rights and permissions
About this article
Cite this article
Doerr, B., Doerr, C. & Kötzing, T. Solving Problems with Unknown Solution Length at Almost No Extra Cost. Algorithmica 81, 703–748 (2019). https://doi.org/10.1007/s00453-018-0477-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-018-0477-7