Upper and Lower Bounds on Unrestricted Black-Box Complexity of Jump n,l

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1 Upper ad Lower Bouds o Urestricted Black-Box Complexity of Jump,l Maxim Buzdalov 1, Mikhail Kever 1, ad Bejami Doerr 1 ITMO Uiversity, 49 Kroverkskiy av., Sait-Petersburg, Russia, mbuzdalov@gmail.com, mikhail.kever@gmail.com LIX, École polytechique, 9118 Palaiseau Cedex doerr@lix.polytechique.fr Abstract. We aalyse the urestricted black-box complexity of Jump,l fuctios. For upper bouds, we preset three algorithms for small, medium ad extreme values of l. We preset a matrix lower boud theorem which is capable of givig better lower bouds tha a geeral iformatio theory approach if oe is able to assig differet types to queries ad defie relatioships betwee them. Usig this theorem, we prove lower bouds for Jump separately for odd ad eve values of. For several cases, otably for extreme Jump, the first terms of lower ad upper bouds coicide. 1 Itroductio To uderstad how evolutioary algorithms (ad other black-box optimizers as well) behave whe optimizig certai fuctios, it is possible to costruct upper bouds (by costructig ad studyig various algorithms), as well as lower bouds (by studyig how fast a algorithm ca be i priciple), which complemet each other. Comparig these bouds allows to evaluate how good today s heuristics are ad sometimes to costruct better algorithms by learig from black-box [1]. Black-box complexity studies how may fuctio evaluatios are eeded i expectatio by a optimal black-box algorithm util it queries a optimum for the first time. As radomized search heuristics are black-box optimizers, blackbox complexity of a problem gives a lower boud o the umber of fitess evaluatios of ay search heuristic to solve this problem. I this paper we cosider optimizatio of fuctios mappig bit strigs of fixed legth to itegers the pseudo-boolea fuctios. A famous class of such fuctios is OeMax havig a certai hidde bit strig z of legth, for a bit strig x of legth the fuctio OeMax,z (x) returs the umber of bits coicidig both i x ad z. Jump, aother popular class of fuctios, takes aother parameter l ad zeroes out the values of OeMax for every strig except z that are at the distace of at most l from both z ad its iverse.

2 A more formal defiitio of Jump is as follows: if OeMax,z = Jump,l,z = OeMax,z if l < OeMax,z < l 0 otherwise. Most of times, whe z does ot matter, we write just OeMax ad Jump,l. The special case of l = 1, which is the maximum possible l that does t zero out the middle fitess values, is called extreme Jump. I this paper, we cosider urestricted black-box complexity (which was itroduced i Droste et al [4]) of the Jump,l problem. Aother kid of black-box complexity, the ubiased black-box complexity, was cosidered for Jump i Doerr et al []. The rest of the paper is structured as follows. Sectio is dedicated to the upper bouds o Jump which are prove by givig the correspodig algorithms ad discussig their complexity. I Sectio 3, the matrix lower boud theorem, which is somewhat similar to Theorem from [4] but is able to produce better lower bouds, is described ad prove. Sectio 4 describes lower bouds o Jump which are costructed from the matrix theorem. Sectio 5 cocludes. Upper Bouds for Jump,l Here, the upper bouds for Jump are cosidered. I Sectio.1 several useful helper theorems are refereced or prove. Sectio. is dedicated to smaller l, Sectio.3 is for larger l, ad Sectio.4 cosiders the case of extreme Jump..1 Helper Theorems ( ) Theorem 1. For sufficietly large, for t log log log eve d [; ] it holds that ( ) ( ( d ) ) t d d/ d 3t/4. log ad for a Proof. This is prove i Doerr et al [3] as Statemet 8. Theorem. For sufficietly large, for l < / log ad for x {0, 1} take uiformly at radom the probability for Jump,l (x) to be zero is at most e (log ). Proof. The value of OeMax(x) for a radom x has a biomial distributio with parameters ad p = 1/. From Hoeffdig s iequality [5], for k p, the distributio fuctio for biomial distributio F,p (k) is boud from above (p k) by e. As a cosequece, the probability for Jump,l (x) to be zero is at (/ l) most F,1/ (l) e e ( log ) = e (log ).

3 Theorem 3. Assume that is sufficietly large ad l < / ( ) log. Let z {0, 1}, ad X be a set of t log log log log elemets from {0, 1} chose radomly usig uiform distributio ad mutually idepedetly. The probability that there exists a y {0, 1} such that y z ad Jump,l,z (x) = Jump,l,y (x) for all x X, is at most t/4. Proof. Let s defie A d as a set of poits which differ from z i exactly d positios where 0 d. We say that a poit y {0, 1} agrees with x X if Jump,l,z (x) = Jump,l,y (x). This meas that either Jump,l,z (x) = 0 or OeMax,y (x) = OeMax,z (x). The probability of the former does ot exceed e (log ) by Theorem. The latter holds iff x ad y (as well as x ad z) differ i exactly half of the d bits i which y ad z differ. To sum up, if y A d, the probability for y to agree with a radom x is at most e (log ) for a eve d ad at most e (log ) + ( d d/) d for a odd d. As for large eough it holds that e (log ) ( 1/4 1 ) ( ) d d/ d, the latter is at most 1/4( d d/) d. Let p be the probability that there exists a y {0, 1} \ {z} such that y agrees with all x X. The the followig holds: p = P r y agrees with x = y {0,1} \{z} x X y {0,1} \{z} d=1 y A d x X P r ( x X y agrees with x P r(y agrees with x) ( ) ( ( ) ) t d 1/4 d + ( ) ( e (log )) t d d/ d d eve d odd = ( ) ( ( ) ) t d ( 1/4 d + 1 e (log )). d d/ d eve After applyig Theorem 1, we get that: p + 1 t/4 3t/4 + 1+t e t(log ), ) which is less tha t/4 for sufficietly large.. Upper Boud for Smaller l Theorem 4. If l < / log, the urestricted black-box complexity of Jump,l is at most (1 + o(1)) log, where o(1) is measured relative to.

4 Proof. We use the same algorithm which is used i [3] for provig the lower boud ( for OeMax. ) We select radomly ad idepedetly t queries such that t log log log log ad check if there exists a sigle optimum z which agrees with all these queries (a query q with a aswer a agrees with a optimum z if Jump,l,z (q) = a). The complexity of oe iteratio equals to t ad the probability of ot fidig a optimum is at most t/4 by Theorem 3. Thus the complexity of the algorithm is at most = (1 + o(1)) log..3 Upper Boud for Larger l t 1 t/4 For bigger l, the Jump,l problem ca be solved by reductio to smaller Jump problems for which the algorithm for the previous sectio suffices. Theorem 5. For log l < 1 the urestricted black-box complexity of Jump,l is at most (1 + o(1)) log ( l) where o(1) is measured whe ( l). Proof. Assume that k = l 1 0. We reduce our problem to Jumps, s k 1 where s log s < k. The algorithm is outlied at Fig. 1 First the algorithm fids a maximum eve s such that s log s < k, which would allow applyig Theorem 4 for solvig Jump s, s k 1. After that, the algorithm fids a strig x {0, 1} with exactly correct bits usig radom queries. The probability that Jump,l is equal to for a radom query is ) ( ) which is Θ 1 by Stirlig s formula. This meas that the strig x ( / ca be foud i Θ( ) queries. After fidig the x, the algorithm splits all bit idices ito sets of size s (except for probably oe) i such a way that i each set exactly half of bits coicide with those i the aswer. This is doe i lies 8 15 at Fig. 1, where b i is the i-th such set. B, the set of yet udistributed bits, always cotais idices of which exactly B / idices correspod to correctly guessed bits. To do that, the algorithm geerates radom subsets of size s ad checks each of them if it cotais exactly s correct bits, which is doe by flippig the bits from the chose subset ad checkig whether the fitess fuctio returs. If B = m, the probability of correct selectio is: ( )( )( ) 1 m/ m/ m m/! m/!s!(m s)! p = = s/ s/ s (m s)/! (m s)/!(s/)! ( )( )( s m s m s m s s = s/ (m s)/ ( ) ( m 1 = Θ = Ω s m s s ) 1 = Θ m/ ). m s m m This gives a O( s) boud for oe subset selectio ad a O(/ s) boud o etire process of fidig subsets.

5 1: fuctio LargeJump(, l, f Jump,l ) : k l 1 3: s max{w w log w < k; w is eve} 4: τ /s 5: repeat 6: x Uiform() 7: util f(x) = 8: B [1; ] 9: for i [1; τ) do 10: repeat 11: b i ChooseSubsetRadomlyUiformly(B, s) 1: util f(flipbits(x, b i)) = 13: B B \ b i 14: ed for 15: b τ B 16: ω 0 x 17: for i [1; τ] do 18: α i SmallJumpProjectio(b i, x) 19: ω i SetAtPositios(b i, ω i 1, α i) 0: ed for 1: retur ω τ : ed fuctio Fig. 1. Algorithm for Jump,l with log l < 1 Next, the algorithm optimizes separately bits from each of the subsets b i usig the algorithm for small Jump from Theorem 4 (lies 17 0 at Fig. 1). If every query for a subproblem o bits from b i is forwarded to the mai fuctio f with all bits ot from b i take from x, the resultig subproblem becomes exactly a Jump bi, b i problem with the followig correctios: k 1 bi from all ozero aswers, a value of eeds to be subtracted; at the optimum of the subproblem, zero will be retured. The latter correctio, however, does ot chage the algorithm very much, because the algorithm from Theorem 4 does t actually query the optimum poit. The lie 19 from Fig. 1 collects the partial aswers oe by oe: it sets the bits of a i at the correspodig positios from b i to the previous partial aswer ω i 1 ad returs the updated value. The complexity of the algorithm ca be expressed as (here = qs + r, (0 < r s)): O( ( ) s ) + O + q( + o s (1)) s log s + ( + o r r(1)) log r = ( + o s(1)). log s However, due to choice of s, it holds that log s = (+o k (1)) log k, which fially results i (1+o l(1)) log ( l).

6 .4 Upper Boud for Extreme Jump The algorithm from Theorem 5 caot be applied to the case of extreme Jump, because 1 k = 0. I this case we have to use aother algorithm, which will be give i the proof of the followig theorem. Theorem 6. The urestricted black-box complexity of the extreme Jump is at most + Θ( ). Proof. As described i previous theorems, oe ca fid a poit x, such that f(x) =, i Θ( ) queries. After that, if oe flips two bits, the value of f remais the same iff oe of these bits was correct ad the other was ot. Oce x is foud, the algorithm tests f(x 10 i 1 10 i 1 ) for all i [; ], ad if it equals, the value of bi is set to zero, otherwise to oe. This results i 1 queries. After that, if the first bit is correct, the 0b... b is the aswer, otherwise its iverse is the aswer. Oe has to make a sigle query to f(0b... b ) to fid which oe is true. The complexity of this algorithm is + Θ( ). 3 The Matrix Lower Boud Theorem I this sectio we preset a ew theorem which is similar to Theorem from [4] except that the odes correspodig to queries are required to be split i several types. Theorem 7. Let S be the search space of a optimizatio problem, ad for each s S there exists a istace such that s is a uique optimum. Let each query has oe of T types, such that for ay query q of the i-th type the followig holds: there is exactly oe aswer to the query q which meas that q is a optimum; there are at most A i,j aswers such that the ext query after such aswer belogs to the j-th type. Defie B i,j, 1 i, j T +, to be a matrix such that: B i,j = A j,i for 1 i, j T (ote the traspositio); B T +1,j = 1 for 1 j T + 1; B T +,j = 1 for 1 j T + ; B i,j = 0 otherwise. Let the first ever query i the optimizatio process be of type 1. Defie V (d) = B d (1, 0,..., 0) T be a vector, C(d) = V (d) T +1, S(d) = V (d) T +. The the followig statemets are true: 1. C(d) is the maximum total umber of possible queries with depth i [1; d], where depth of a root is equal to oe.. The lower boud o average depth of N odes is d + 1 S(d) N where d is a iteger such that C(d) N C(d + 1). 3. The urestricted black-box complexity of the cosidered optimizatio problem is ot less tha the lower boud o average depth of S odes.

7 Proof. Accordig to Yao s miimax priciple [6], the expected rutime of a radomized algorithm o ay iput is ot less tha the average rutime of the best determiistic algorithm over all possible iputs. Thus we costruct a lower boud o complexity of a radomized algorithm by costructig a lower boud o the average performace of ay determiistic algorithm over all possible iputs. A determiistic algorithm ca be represeted as a (rooted) decisio tree with odes correspodig to queries ad arcs goig dowwards correspodig to aswers to these queries. A total lower boud o the average performace of determiistic algorithms, just as i [4], is doe by assigig S differet queries to differet odes of a tree such that their average depth is miimized, ad the by cosiderig all such trees ad takig a miimum over them. It should be oted that, if a (fixed) set of queries is to be assiged to odes of a (fixed) rooted tree such that the average depth of these queries is miimized, a optimal assigmet ca be costructed i a greedy way: each query should be assiged to a free ode with the miimum possible depth. Assume that a optimal assigmet does ot use at least oe ode a with depth d while usig at least oe ode b with depth d > d. The oe ca move a query from the ode b to the ode a, which makes the average depth decrease, so the iitial assigmet is, i fact, ot optimal. Next we show that, i order to miimize the average depth, oe eeds to cosider oly the complete tree, that is, a tree where for ay query of the i-th type, for ay j there are exactly A i,j aswers, each leadig to a query of the j-th type. Ideed, if a optimal assigmet ca be doe for a icomplete tree, it ca be doe for the complete tree as well, because all the odes of ay icomplete tree are preserved i the complete tree. For a complete tree with the costraits determied by the matrix A (as specified i the theorem s statemet) ad with the root vertex of type 1, the umber of( vertices of type i ad depth d (the root has the depth equal to 1) (A T is exactly ) ) d 1 (1, 0,..., 0) T. I the matrix B, the ext-to-last row is i desiged to collect the sum of all umbers of vertices at all previous depths (which is exactly how C(d) is defied), ad the last row, i a similar maer, collects S(d) the sum of C(i) s for all 1 i d. I a more explicit way, S(d) ca be expressed as: d S(d) = C(i) (d + 1 i), i=1 so the expressio C(d) (d+1) S(d) is actually the sum of depths of all vertices up to the depth d: d C(d)(d + 1) S(d) = C(i) i, ad the expressio d + 1 S(d) C(d) is thus exactly the average depth of all such vertices. If we cosider arbitrary iteger N, we ca fid a iteger d such that C(d) N C(d + 1). I this case, the total sum of depths of the first C(d) vertices is i=1

8 C(d) (d + 1) S(d), ad the ext N C(d) vertices have the depth of d + 1. The average depth is thus: C(d) (d + 1) S(d) + (d + 1) (N C(d)) d avg (N) = = d + 1 S(d) N N. It is difficult to use this theorem straightaway, because the lower boud o the average depth of N vertices is ot defied oly i terms of N ad the matrix A, but additioally requires to fid which depth d fulfils C(d) N C(d + 1). However, for several commo usages it is possible to make it more coveiet. Theorem 8. If there is oly oe type of queries i Theorem 7, ad A 1,1 = k such that k, the for the search space S the lower boud o the average depth is at least log k (1 + S (k 1)) 1 k 1. Proof. The value of B d (1, 0, 0) T yields the followig result (itermediate computatios omitted): d k Oe ca see that C(d) = kd 1 k 1 = Cosider a equality N = C(d) = kd 1 k 1 k d k d 1 k 1. k d+1 k d(k 1) (k 1) ad S(d) = kd+1 k d(k 1) (k 1).. It follows that: d(n) = log k (1 + N(k 1)). As for a give N we eed to fid a iteger d such that C(d) N < C(d + 1), we eed to roud it dow: d = d(n). Note that, if d 1 ad k 1, S(d) grows whe d grows, as S(d) > 0. The expressio for a lower boud o the average depth of N queries is at most: d avg (N) = d(n) + 1 S( d(n) ) N log k (1 + N(k 1)) 1 k 1. d(n) + 1 S(d(N)) N Note that the classical result from [4], the log k+1 N 1 lower boud, is actually ot greater tha the give boud. Ideed, for k : log k (1 + N(k 1)) log k+1 N > log k (N(k 1)) log k+1 N = log k N log k+1 N + log k (k 1) > log k N log k+1 N > 0. For the case of k = 1, the lower boud is eve stroger. Theorem 9. If there is oly oe type of queries i Theorem 7, ad A 1,1 = 1, the for the search space S the lower boud o the average depth is at least ( S + 1)/. Proof. I this case oe ca show that C(d) = d ad S(d) = d +d. The average depth for N is N + 1 N +N N = N + 1 (N + 1)/ = (N + 1)/.

9 4 Lower Bouds for Jump,l First, let s apply Theorem 8 immediately to the Jump problem. Theorem 10. For ay ad l < /, the urestricted black-box complexity of Jump,l is at least log l (1 + ( l 1)) 1 l 1. Proof. I Jump,l, the search space has a size of. There are l+1 possible aswers to a query, but oe of them termiates the search process immediately, so k = l. The result follows straightaway from Theorem 8. Theorem 11. The urestricted black-box complexity of extreme Jump for eve is at least 1. Proof. Follows from Theorem 10 by assumig l =. The preseted bouds are already a improvemet over the curretly kow bouds (say, for extreme Jump ad eve, as follows from [4]). However, log 3 for odd Theorem 10 reports log 3 ( ) 1/, which is still quite far away from the best kow algorithms. Fortuately, the Jump problem possesses a particular property, which ca be used to refie the lower bouds usig Theorem 7 with two types of queries. Theorem 1. For Jump,l, defie a aswer to the query to be o-trivial if it is either 0 or. After receivig the first o-trivial aswer for every subsequet query it is possible to determie a priori the parity of ay o-trivial aswer. Proof. Cosider the optimum ad a query. We itroduce the followig values: q 00 : umber of positios with zeros i both the optimum ad the query; q 01 : umber of positios with zeros i the optimum ad oes i the query; q 10 : umber of positios with oes i the optimum ad zeros i the query; q 11 : umber of positios with oes i both the optimum ad the query. The umber of zeros i the optimum modulo, which is q 00 q 01, is fixed. The umber of oes i the query modulo is q 01 q 11, ad the aswer to the query modulo is q 00 q 11. The followig equality holds: (q 01 q 11 ) (q 00 q 11 ) = q 00 q 01, which meas that the parity of the o-trivial aswer is uiquely determied by the parity of the umber of oes i the query. As a result, if a algorithm receives the first o-trivial aswer, all subsequet queries will provably have fewer possible aswers. Usig Theorem 1, we ca defie two types of queries to use with Theorem 7, amely, the queries happeed before ad after a o-trivial aswer.

10 Theorem 13. The urestricted black-box complexity of Jump,l for odd is at least: ( log l+1 ( l 1) + 1 ) l 1. Proof. For odd there are l + 1 = k + possible aswers: oe aswer equal to 0, oe aswer equal to ad k pairs of o-trivial aswers. For the Theorem 7, the first type of queries has k + 1 o-termiatig aswers, ad the secod type of queries, which occurs after oe of k o-trivial aswers is received from a query of the first type, has oly k + 1 o-termiatig aswers. The value of B d (1, 0, 0, 0) T is thus: B d 0 0 = k k d = ( (k + 1) d 1 ) (k+1) d dk k 0 (k+1) d+1 (dk+) +dk +4k k A problem of defiig d i terms of N is more difficult this time: as C(d) = (k+1) d dk k, the equality N = C(d) caot be easily solved i terms of d. Istead, we itroduce a fuctio d(n) such that the followig equality holds: N = (k + 1)d(N) d(n)k. k We fid the lower boud o the average depth d avg (N), keepig i mid that S(d) grows as d grows ad that d(n) 1 for N 1: d avg (N) = d(n) + 1 S( d(n) ) N = d(n) + 1 = d(n) + 1 d(n) + 1 k + 1 k d(n) + 1 S(d(N)) N (k+1) 1+d(N) (d(n)k+) +d(n)k +4k k (k+1) d(n) d(n)k k. (k+1) 1+d(N) d(n)k d(n)k k d(n)k (d(n) 1) k (k+1) 1+d(N) d(n)k d(n)k k k+1 = d(n) 1 k. We ca also obtai a good lower boud o d(n) by throwig out the ( d(n)k part i the defiitio of d(n) above, which leads to d(n) > log Nk k+1 + 1). Together, d avg (N) ( log Nk k+1 + 1) 1 k. For Jump,l, it holds that N = ad k + = l + 1, which costitutes: ( log l+1 ( l 1) + 1 ) l 1. Theorem 14. The urestricted black-box complexity of extreme Jump for odd is at least.

11 Proof. For extreme Jump ad odd, l + 1 = 4. The from Theorem 13 it follows that the lower boud is at least: log ( + 1 ) = log ( ) 1. Theorem 15. The urestricted black-box complexity of Jump,l for eve is at least: ) 1 ( l) log l+ (1 + l 1 l. Proof. For eve there are l + 1 = k + 3 possible aswers (k 0): oe aswer equal to 0, oe aswer equal to, oe aswer equal to / ad k more pairs of o-trivial aswers. For Theorem 7, the first type of queries has k + o-termiatig aswers, ad the secod type of queries ca have either k +1 or k o-termiatig aswers, depedig o the parity of the umber of oes i a query. As we caot predict the parity for all possible algorithms, the maximum umber of queries is limited to k + 1. The matrix B has the followig form: B = k + 1 k We omit the itermediate computatios ad just state that: C(d) = (k + 1)(k + )d dk dk k 1 (k + 1) S(d) = (k + )d (k + 5k + ) d k 3 +dk 3 +d k +6dk +4k +d+d k+7dk+10k+4 (k + 1) 3. Followig the same approach as i the proof of Theorem 13, we defie d(n) such that C(d(N)) = N ad produce the followig lower boud: d avg (N) d(n) 1 k + 1. The lower boud o d(n) ca be achieved from the value of C(d) by throwig out the dk + dk part, which yields: ( d(n) log k+ 1 + (k + ) 1) N k + 1 ad, together: d avg (N) ( log k+ 1 + (k + ) 1) N 1 k + 1 k + 1. Substitutio of N with ad k + with l proves the theorem. Note that Theorem 15 does ot improve the boud for extreme Jump ad eve it remais equal to 1 whe oe sets k = 0 because i this case the umber of possible aswers does ot chage after receivig the first o-trivial aswer.

12 5 Coclusio New black-box algorithms for solvig Jump,l problem are preseted, givig the followig upper bouds: for l < / log : (1+o(1)) log, where o(1) is measured whe ; for / log l < 1: (1+o(1)) log ( l), where o(1) is measured whe l ; for l = 1: + Θ( ). A ew theorem for costructig lower bouds o urestricted black-box complexity of problems is proposed. The uderlyig idea is that ifluece of particular aswers to queries to all subsequet queries ca be formalized by assigig a type to each query ad writig the relatios i a form of a matrix. Several followig steps for costructig the lower bouds are automated ad ca be performed usig tools like Wolfram Alpha. We hope that this theorem ca be used to obtai better lower bouds i other problems. Usig the proposed theorem, the lower bouds for Jump,l are updated: ) for eve : log l+ (1 + 1 ( l) l 1 l l+ 1; log ( for odd : 1 + ( l 1) ) l 1 1. log l+1 1 log l+1 I particular, for extreme Jump the lower bouds become equal to 1 for eve ad for odd. This meas that the quotiets at the first term of lower ad upper bouds coicide. I the case of large, but ot extreme l, these quotiets seem to coicide as well, however, the (1 + o(1)) multiple ca hide as much as ( ) l+1 log ( l)/ log, which makes it hard to see exactly. This work was partially fiacially supported by the Govermet of Russia Federatio, Grat 074-U01. Refereces 1. Doerr, B., Doerr, C., Ebel, F.: Lessos from the black-box: fast crossover-based geetic algorithms. I: Proceedigs of Geetic ad Evolutioary Computatio Coferece. pp (014). Doerr, B., Doerr, C., Kötzig, T.: Ubiased black-box complexities of jump fuctios, 3. Doerr, B., Johase, D., Kötzig, T., Lehre, P.K., Wager, M., Wize, C.: Faster black-box algorithms through higher arity operators. I: Proceedigs of Foudatios of Geetic Algorithms. pp (011) 4. Droste, S., Jase, T., Wegeer, I.: Upper ad lower bouds for radomized search heuristics i black-box optimizatio. Theory of Computig Systems 39(4), (006) 5. Hoeffdig, W.: Probability iequalities for sums of bouded radom variables. Joural of the America Statistical Associatio 58(301), (1963) 6. Yao, A.C.C.: Probabilistic computatios: Toward a uified measure of complexity. I: Foudatios of Computer Sciece, 1977., 18th Aual Symposium o. pp. 7 (1977)