Pei Jiang, Ying-ping Chen ∗
Lemma 10. The NFL theorem holds over F if and only if F is closed under permutation
4. DLC and subthreshold-seeker
0 ifv = vi; K +1 ifv = vj; K otherwise.
and the permutationπ: π(v) =
vk ifv = vj; tvj ifv = vk; v otherwise.
|fπ(vk) −fπ(vi)| = |f(π(vk)) −f(π(vi))| = |f(vj) −f(vi)| =K+1, so fπ /∈L(G,Y,K), a contradiction.
The condition of completeness implies that the NFL theorem holds over DLC only in the extreme case that the entire search space is in the same neighborhood. While such a situation is theoretically possible, yet somewhat trivial. Taking PDLC as an example, when m>K , the NFL theorem sustains over a PDLC only if there are merely two vertexes in the search space.
4. DLC and subthreshold-seeker
The subthreshold-seeker (STS), introduced by Whitley and Rowe [9] and proved to outperform random search on uniformly sampled polynomials of one variable, is a metaheuristic that employs the threshold as a switch of local search.
In essence, it is a selective local search method as it conducts local search if a given condition is satisfied. In this section, a generalization of seeker is firstly presented, and we will demonstrate that the generalized subthreshold-seeker can outperform random search on DLC.
4.1. Generalized subthreshold-seeker
In Whitley and Rowe’s work, the subthreshold-seeker is an optimization algorithm aiming at functions with a one-dimensional domain, i.e., functions defined on a subsetC⊆R. The subthreshold-seeker will successively select a point from the search space uniformly at random (u.a.r.) until a subthreshold point is encountered. Once encountering a subthreshold point, the subthreshold-seeker will search through the quasi-basin where that subthreshold point resides. In Whitley and Rowe’s definition, a quasi-basin is a set of contiguous points with objective values below the threshold. In other words, the threshold is used to determine whether the subthreshold-seeker enters the local search phase, and the subthreshold-seeker can be viewed as an optimizer with an exhaustively local search operator.
According to this point of view, we generalize the subthreshold-seeker to the extent that it is applicable to any function of which the domain possesses a neighborhood structure as inAlgorithm 1.
Algorithm 1 (Generalized Subthreshold-seeker).
procedure Subthreshold-seeker(X,Y, N :X→2X, f :X→Y) while the stopping criterion is not satisfied do
if Queue is not empty then x←Queue.pop();
else
Select x fromXu.a.r.
end if
if f(x) ≤ θthen Queue.push(N(x)) end if
end while end procedure
Following the NFL framework, the parts of selecting and pushing are both restricted to unvisited points. Such a task can be achieved by a bookkeeping manner. Since the performance of an algorithm is judged by the performance vector, all overheads other than function evaluations will not count under the NFL framework.
The only control parameter of the subthreshold-seeker is the threshold. The elegance of the subthreshold-seeker is that it comprises the two fundamental operations of search heuristics, local search and global restart, and yet still stays in a simple form.
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4.2. Subthreshold-seeker on DLC
Christensen and Oppacher [8] defined the performance measure as the number of submedian points visited by an algorithm, and Whitley and Rowe [9] generalized this notion to any threshold less than or equal to the median. That is, given a predefined stopping time L andα ∈ (0,1/2], the performance measure is the number of points visited in the first L function evaluations with the topα|X|values in the objective space.
This performance measure may seem odd at the first glance, for typically the performance of an optimizer is measured in terms of the time in which the optimum is located. However, even focusing on functions as simple as unimodal functions that are monotone with respect to the distance from the optimum, the time complexity analysis is still a difficult task.
For instance, to the best of our limited knowledge, the time complexity of(1+1)-ES [23] on such functions has not been analyzed until recently [24]. Hence, it seems unlikely to analyze the runtime of an algorithm that is more sophisticated than random search over a broad class of problems. Furthermore, as mentioned in Section2, within the NFL framework, the performance measure can be any function defined on the set containing all performance vectors, and roughly speaking, with more subthreshold points visited, it is more likely to identify a point with a satisfiable objective value. Therefore, Whitley and Rowe’s notion appears in between theoretically analyzable and practically meaningful.
For any function f , we defineβα(f)be the maximum objective value below the performance threshold, i.e., βα(f) :=max
y∈Y
y
−
i=0
|{x∈X|f(x) =i}| ≤α|X|
.
If the set following the ‘‘max’’ notation is empty, thenβα(f)is defined to be−∞. LetΨα,f(v)be a performance measure that maps a performance vectorvto the number of components ofvbelow performance threshold, i.e.,Ψα,f((v1, v2, . . . , vL)) =
|{vi|vi≤βα(f)}|. It is noteworthy that the performance threshold should be distinguished from the algorithmic threshold.
The latter should be regarded as a control parameter of the algorithm and hence is not related to the performance measure.
Whitley and Rowe showed that if f is a uniformly sampled polynomials of one variable, andβα(f)is known in advance, settingθ = βα(f), under certain conditions the subthreshold-seeker outperforms random search on f . In this study, we will show that the subthreshold-seeker withθwithin some range of codomain, rather than a specific value, will outperform random search in the sense that for all functions in the DLC, the expected number of points below the performance threshold visited by the subthreshold-seeker is greater than or equal to that by random search, and there does exist a function such that the inequality is strict.
Theorem 12 (Equal or Better Performance of STS on DLC). LetL(G,Y = {0,1, . . . ,m},K) with m > K be a DLC. For all f ∈ L(G,Y,K)if the algorithmic thresholdθ of a subthreshold-seeker satisfiesθ ≤ βα(f) −K , then E[Ψα,f(Sy(STS,f,L))] ≥ E[Ψα,f(Sy(RS,f,L))]for all L with 1≤L≤ |X|.
Proof. Let f be any function belonging toL(G,Y,K). Suppose S(STS,f,L) = ((Xsi,Ysi))Li=1and S(RS,f,L) = ((Xri,Yri))Li=1. Define the indicator variable Isias Isi = 1 when Ysi ≤ βα(f)and Isi = 0 otherwise, and Iriis defined in a similar way for random search. We can obtain thatΨα,f(Sy(STS,f,L)) = ∑Li=1IsiandΨα,f(Sy(RS,f,L)) = ∑Li=1Iri.
We prove the theorem by induction on L. Let U := |{x ∈ V(G) | f(x) ≤ βα(f)}be the total number of points below the performance threshold. When L = 1, since both strategies select a point u.a.r. fromXin the first move, clearly E[Is1] =U/|X| =E[Ir1]. Suppose E[∑L
i=1Isi] ≥E[∑L
i=1Iri]for 1≤L< |X|. Then, E
L+1
−
i=1
Isi
=E
L
−
i=1
Isi
+E[IsL+1]
=E
L
−
i=1
Isi
+ −
(xi)Li=1∈XL
E
IsL+1|(Xsi)Li=1=(xi)Li=1
Prob{(Xsi)Li=1=(xi)Li=1}. (1)
If Xsiis popped out from the queue, f(Xsi) ≤ θ +K ≤βα(f) −K+K =βα(f), and hence, Isi=1. Otherwise, if Xsiis selected fromXu.a.r., then Prob{Isi=1} =(U−k)/(|X| −i+1), where k is the number of points visited in the first i−1 steps with objective values smaller than or equal toβα(f). Let CLbe the set collecting all(xi)Li=1∈XLsuch that if(Xsi)Li=1=(xi)Li=1, the queue will be nonempty in the (L+1)th move. Therefore,
−
(xi)Li=1∈XL
E[IsL+1|(Xsi)Li=1=(xi)Li=1]Prob{(Xsi)Li=1=(xi)Li=1}
= −
(xi)Li=1∈CL
E[IsL+1|(Xsi)Li=1∈CL]Prob{(Xsi)Li=1=(xi)Li=1}
+ −
(xi)Li=1/∈CL
E[IsL+1|(Xsi)Li=1 /∈CL]Prob{(Xsi)Li=1=(xi)Li=1}
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1620 P. Jiang, Y.-p. Chen / Theoretical Computer Science 412 (2011) 1614–1628
= −
Inequality (3) follows from the induction hypothesis.
Furthermore, next theorem guarantees that for any f ∈L(G,Y,K), if there exists a point above performance threshold and the subthreshold-seeker ever enters the local search phase, the subthreshold-seeker will outperform random search strictly in expectation according to the performance measureΨα,f.
Theorem 13 (Strictly Better Performance of STS on DLC). Let L(G,Y = {0,1, . . . ,m},K) with m > K be a DLC. For all f ∈L(G,Y,K)and for every subthreshold-seeker STS withθ ≤ βα(f) −K satisfy:
(1) ∃v ∈V(G)with f(v) > βα(f), and (2) ∃v ∈V(G)with f(v) ≤ θ,
E[Ψα,f(Sy(STS,f,L))] >E[Ψα,f(Sy(RS,f,L))]for all L∈ [2, |X| −1].
Proof. If there are no such functions inL(G,Y,K), the theorem holds vacuously. Otherwise, let f be any function satisfying the two conditions and define((Xsi,Ysi))Li=1,((Xri,Yri))Li=1, Isi, Iri, U, and CLin the same way as inTheorem 12. We prove
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>E[Is1] + −
x:f(x)≤βα(f)
U−1
|X| −1Prob{Xs1 =x} + −
x:f(x)>βα(f)
U
|X| −1Prob{Xs1=x}
=E[Is1] + −
k∈{0,1}
U−k
|X| −1Prob{Is1 =k}
=E[Ir1] + −
k∈{0,1}
U −k
|X| −1Prob{Ir1=k}
=E[Ir1+Ir2].
The inequality follows from C1 ̸= ∅and(U −1)/(|X| −1) < 1, for Condition (1) implies U < |X|. For the induction hypothesis, suppose E[∑L
i=1Isi]>E[∑L
i=1Iri]for L with 2≤L< |X| −1. In the (L+1)th step, from the proof ofTheorem 12, we always have
E
L+1
−
i=1
Isi
≥ U
|X| −L+ |X| −L−1
|X| −L E
L
−
i=1
Isi
> U
|X| −L+ |X| −L−1
|X| −L E
L
−
i=1
Iri
=E
L+1
−
i=1
Iri
. (4)
Since(|X| −L−1)/(|X| −L) > 0 when L < |X| −1, and E[∑L
i=1Isi] > E[∑L
i=1Iri]from the induction hypothesis, Inequality (4) is strict.
The following example illustrates a set of functions on which subthreshold-seekers strictly outperform random search.
Example 14. Given V(G) = {v1, v2, . . . , v2n}and a positive integer K , the function f defined as f(vi) = iK is an instance of the PDLCL(G,Y = {0,1, . . . ,2nK},K). Ifα = 1/2, then all subthreshold-seekers with(n−1)K ≥ θ ≥ K strictly outperform random search on f .
Let d :=max{deg(v) | v ∈V(G)}be the maximum degree of the graph and dis(u, v)be the length of the shortest path from u tov. For any subthreshold-seeker, if we are able to set itsθ within some interval, the following corollary gives a sufficient condition of the existence of functions on which the subthreshold-seeker strictly outperforms random search.
Corollary 15. LetL(G,Y = {0,1, . . . ,m},K)be a DLC with a maximum degree d > 2. Givenα ∈ (0,1/2]and an integer C>1 with CK+1≤m, if
α|V(G)| > d(d−1)C −2 d−2 ,
then there exists a function f ∈ L(G,Y,K)such that E[Ψα,f(Sy(STS,f,L))] > E[Ψα,f(Sy(RS,f,L))]for all L with 2 ≤ L ≤
|X| −1, where STS is a subthreshold-seeker withθ ∈ βα(f) − [K,CK].
Proof. We prove this corollary constructively. Select a vertexv0from V(G)arbitrarily. Consider the function f defined as
f(v) =
0 ifv = v0;
dis(v, v0)K if 1≤dis(v, v0) ≤C; CK +1 otherwise.
Since
|vo| + |{v ∈V(G) |1≤dis(v, v0) ≤C}|
≤1+
d+d(d−1) +d(d−1)2+ · · · +d(d−1)C−1
=1+d
(d−1)C −1 (d−1) −1
= d(d−1)C −2
d−2 < α|V(G)|,
from the definition of βα(f), βα(f) = CK . Furthermore, there must exist v1 ∈ V(G) that f(v1) = CK + 1, for
|vo| + |{v ∈V(G) |1≤dis(v, v0) ≤C}| < |V(G)|. Therefore, we have f(v0) ≤ θ and f(v1) > βα(f). TherebyTheorem 13 can be applied.
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For PDLC, since the maximum degree is upper-bounded by 2, the sufficient condition can be reduced to a simpler form.
Corollary 16. LetL(G,Y= {0,1, . . . ,m},K)be a PDLC. Givenα ∈ (0,1/2]and an integer C>1 with CK+1≤m, if
|V(G)| > (1+2C)α−1,
then there exists a function f ∈ L(G,Y,K)such that E[Ψα,f(Sy(STS,f,L))] > E[Ψα,f(Sy(RS,f,L))]for all L with 2 ≤ L ≤
|X| −1, where STS is a subthreshold-seeker withθ ∈ βα(f) − [K,CK].
Proof. For anyv0∈V(G),|vo| + |{v ∈V(G) |1≤dis(v, v0) ≤C}| ≤1+2C .
CombiningTheorems 12and13, if we manage to setθ ≤ βα(f) −K , the subthreshold-seeker will perform at least as good as random search on a DLC. If the subthreshold-seeker has a chance to conduct local search, it will strictly outperform random search.Corollaries 15and16show that if d and C remain unchanged, we can obtain a DLC satisfying the conditions by increasing|V(G)|, forαas a predefined performance threshold. In other words, with the same neighborhood structure, if the capability of a subthreshold-seeker to sample a decent threshold is unaffected by the increasing domain size, which is generally true and will be discussed later, then the conditions inCorollaries 15and16hold with a sufficiently large domain.
Estimatingθ within some range should be more practical than gauging a specific value such asβα(f). In the next section, we will explore this possibility and empirically confirm the theoretical results obtained in this section by proposing and adopting a sampling-test scheme.