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X∈P Problem Space Model, available resources. F Feature Extraction f (x) ∈ F = Rm Feature Space S( f (x), w) Model size, number of resources, etc. p(a, x) p ∈ Rn Performance Measure Space w ∈ Rn Criteria Space User preferences. a∈A Algorithm Space Speed, accuracy, memory consumption, etc. 3: ASP entities and exemplary correspondents from modeling and simulation. allows to compare their virtues and shortcomings on a rather abstract yet precise level. 2 Analytical Algorithm Selection A fundamental approach to assess problem hardness and thereby analytically compare the algorithms to solve them is provided by (computational) complexity theory.

X100 , say ||p(a1 , xi )|| = 10 versus ||p(a2 , xi )|| = 1 for i ∈ [1, 100]. Furthermore, let a2 perform only slightly better than a1 when applied to x101 , . . , x200 , say ||p(a1 , xi )|| = 10 and ||p(a2 , xi )|| = 11 for i ∈ [101, 200]. Now assume that solving the BSMP led to a selection mapping S that chooses a1 for x1 , . . , x70 , and a2 otherwise. S is average-effective since (see def. 15. This means that S is performing worse than a constant selection mapping that does not adapt its decision by considering any problem features.

23), this time on the grounds of a performance tuple set Φ: max φ1 ,φ2 ∈Φ∧( f φ1 ,aφ1 )=( f φ2 ,aφ2 ) ||pφ1 − pφ2 || Moreover, approximation theory can be applied to the given data in Φ: Is it possible to construct a good approximation function F(x, p1 , . . , pn ) → A (see eq. 11, p. 31)? , the deviation from the best possible algorithm selection? How to find the best parameters pi , and which approximation forms are suitable? 3 Algorithm Selection as Learning 37 foundation of two more practical disciplines [49]: statistical learning [122] and machine learning [333].

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