Solving Satisfiability in Fuzzy Logics by Mixing CMA-ES (abstract)

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Solving Satisfiability in Fuzzy Logics by Mixing

CMA-ES

Tim Brys

a

_{Madalina M. Drugan}

a

_{Peter A.N. Bosman}

b

Martine De Cock

c

_{Ann Now´e}

a

_{AI Lab, Vrije Universiteit Brussel, Brussels}

b

_{Centrum Wiskunde}

_{& Informatica (CWI), Amsterdam}

c

_{Dept. of Applied Math., Comp. Sc. and Stat., UGent, Gent}

Abstract

Satisfiability in propositional logic is well researched and many approaches to checking and solving exist. In infinite-valued or fuzzy logics, however, there have only recently been attempts at developing methods for solving satisfiability. In this paper, we analyse the function landscape of different problem classes, focussing our analysis on plateaus. Based on this study, we develop Mixing CMA-ES (M-CMA-ES), an extension to CMA-ES that is well suited to solving problems with many large plateaus. We empirically show the relation between certain function landscape properties and M-CMA-ES performance.

1 SAT

∞

and Landscape Analysis

SAT∞, in contrast to its classical counterpart, involves selecting continuous values, since truth degrees in

fuzzy logics are ∈ [0, 1]. In this section, we analyse some landscape properties of the benchmark problems in Łukasiewicz logic (Ł4 and Ł100) proposed in [1], specifically looking for plateaus, which are expected

to occur given the conjunction operator (max) and objective function definition [1]. We estimate plateau frequency and size by performing random walks through the landscape, counting the number of subsequent steps of the same fitness. Figure 1(a) plots the relation between plateau size and their frequency, while 1(b) plots the relation between plateau size and fitness. Both Łukasiewicz problem classes have similar numbers of plateaus, which decrease with plateau size, i.e. there are many more small plateaus than there are large ones. Interestingly, fitness is highly correlated with plateau size on Ł4problems, while this correlation is

much smaller for the Ł100 problems. For the former, this means that in general, the larger a plateau, the

closer in function value it will be to optimal solutions. We expect this property to ensure that escaping large plateaus needs only be done a few times. On the contrary, for the latter problem class, this means that escaping from a plateau does not guarantee nearness to optimal solutions, and that large plateaus are more distributed with respect to optimal solutions.

2 Mixing CMA-ES

CMA-ES [3] has been consistently among the best-performing optimization algorithms for solving these SAT∞problems, but we can expect that the plateaus identified in the previous section are hindering

CMA-ES performance as it depends on gradient information. In [2], we propose a mechanism for CMA-CMA-ES that can help it deal with plateaus. It is an application of optimal mixing evolutionary algorithms [4] on the multi-population level: we have multiple CMA-ES multi-populations running in parallel, and we recombine their dis-tributions if this leads to improvements. We call this Mixing CMA-ES or M-CMA-ES. This recombination

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101 102 103 100 102 104 106 Plateau Size Number of Plateaus Lukasiewicz T₄ Lukasiewicz T₁₀₀ 101 102 103 0.75 0.8 0.85 0.9 0.95 1 Plateau Size Fitness (a) (b)

Figure 1: (a) The number of plateaus and (b) the fitness of plateaus as a function of their estimated size. Success Evaluations

Ł4 Ł100 Ł4 Ł100

CMA-ES 94.24% 96.72% 297142 266974 10 CMA-ES 94.29% 97.44% 366655 178673 M-CMA-ES 95.06% 98.6% 478158 101670

Table 1: Success percentage after 107evaluations, and average number evaluations in successful runs. All observed differences with the best are significant (Wilcoxon signed-rank test, confidence 95%).

of distributions is executed by exchanging elements of the means of the two populations, and subsequently also recombining their covariance matrix and evolution paths. Table 1 shows the results of an experiment comparing a single CMA-ES population, 10 independent populations, and 10 CMA-ES populations with mixing. We look at the percentage of runs finding a solution, and the mean number of evaluations required to find it. Mixing improves the success rate on both problem classes, but only on the second problem class, with the smaller correlation between plateau size and fitness do we really gain much, as we can more than halve the number of evaluations required to reach a solution. This supports our hypothesis that the distribu-tion of plateaus in these problems has a strong impact on the performance of gradient based optimizers, and that specific mechanisms to handle plateaus can contribute much to performance.

3 Acknowledgement

This work was partially funded by a joint VUB-UGent Research Foundation-Flanders (FWO) project. Also, Tim Brys is funded by a Ph.D grant of the Research Foundation Flanders (FWO).

References

[1] Tim Brys, Yann-Michal De Hauwere, Martine De Cock, and Ann Now´e. Solving satisfiability in fuzzy logics with evolution strategies. In Proceedings of the 31st Annual North American Fuzzy Information Processing Society Meeting, 2012.

[2] Tim Brys, Madalina M. Drugan, Peter A. N. Bosman, Martine De Cock, and Ann Now´e. Solving satisfi-ability in fuzzy logics by mixing cma-es. In Proceedings of the Genetic and Evolutionary Computation Conference, 2013.

[3] N. Hansen. The CMA evolution strategy: a comparing review. In Towards a new evolutionary compu-tation. Advances on estimation of distribution algorithms, pages 75–102. Springer, 2006.

[4] Dirk Thierens and Peter A. N. Bosman. Optimal mixing evolutionary algorithms. In Natalio Krasnogor and Pier Luca Lanzi, editors, GECCO, pages 617–624. ACM, 2011.