The Noisy Euclidean Traveling Salesman Problem and Learning

Transcription

1 The Nisy Euclidean Traveling Salesman Prblem and Learning Miki L. Braun, Jachim M. Buhmann Institute fr Cmputer Science, Dept. III, University f Bnn R6merstraBe 164, Bnn, Germany Abstract We cnsider nisy Euclidean traveling salesman prblems in the plane, which are randm cmbinatrial prblems with underlying structure. Gibbs sampling is used t cmpute average trajectries, which estimate the underlying structure cmmn t all instances. This prcedure requires identifying the exact relatinship between permutatins and turs. In a learning setting, the average trajectry is used as a mdel t cnstruct slutins t new instances sampled frm the same surce. Experimental results shw that the average trajectry can in fact estimate the underlying structure and that verfitting effects ccur if the trajectry adapts t clsely t a single instance. 1 Intrductin The apprach in cmbinatrial ptimizatin is traditinally single-instance and wrst-case-riented. An algrithm is tested against the wrst pssible single instance. In reality, algrithms are ften applied t a large number f related instances, the average-case perfrmance being the measurement f interest. This cnstitutes a cmpletely different prblem: given a set f similar instances, cnstruct slutins which are gd n average. We call this kind f prblem multiple-instances and average-case-riented. Since the instances share sme infrmatin, it might be expected that this prblem is simpler than slving all instances separately, even fr NP-hard prblems. We will study the fllwing example f a multiple-instance average-case prblem, which is built frm the Euclidean travelings salesman prblem (TSP) in the plane. Cnsider a salesman wh makes weekly trips. At the beginning f each week, the salesman has a new set f appintments fr the week, fr which he has t plan the shrtest rund-trip. The lcatin f the appintments will nt be cmpletely randm, because there are certain areas which have a higher prbability f cntaining an appintment, fr example cities r business districts within cities. Instead f slving the planning prblem each week frm scratch, a clever salesman will try t explit the underlying density and have a rugh trip pre-planned, which he will nly adapt frm week t week. An idealizing frmulizatin f this setting is as fllws. Fix the number f appintments n E N. Let Xl,..., Xn E ]R2 and (J E 114. Then, the lcatins f the

2 appintments fr each week are given as samples frm the nrmally distributed randm vectrs (i E {1,...,n}) The randm vectr (Xl,...,Xn ) will be called a scenari, sampled appintment lcatins (sampled) instance. The task cnsists in finding the permutatin 7r E Sn which minimizes 7r I-t d7r(n)1f(l) + L~:ll d1f(i)1f(ih), where dij := IIXi - Xj112' and Sn being the set f all bijective functins n the set {1,..., n}. Typical examples are depicted in figure l(a)- (c). It turns ut that the multiple-instance average-case setting is related t learning thery, especially t the thery f cst-based unsupervised learning. This relatinship becmes clear if ne cnsiders the perfrmance measure f interest. The algrithm takes a set f instances It,...,In as input and utputs a number f slutins Sl,, Sn It is then measured by the average perfrmance (l/n) L~=l C(Sk, h), where C(s, I) dentes the cst f slutin s n instance I. We nw mdify the perfrmance measure as fllws. Given a finite number f instances It,...,In, the algrithm has t cnstruct a slutin s' n a newly sampled instance I'. The perfrmance is then measured by the expected cst E (C (s',i')). This can be interpreted as a learning task. The instances 11,...,In are then the training data, E(C(s', I')) is the analgue f the expected risk r cst, and the set f slutins is identified with the hypthesis class in learning thery. In this paper, the setting presented in the previus paragraph is studied with the further restrictin that nly ne training instance is present. Frm this training instance, an average slutin is cnstructed, represented by a clsed curve in the plane. This average trajectry is suppsed t capture the essential structure f the underlying prbability density, similar t the centrids in K-means clustering. Then, the average trajectry is used as a seed fr a simple heuristic which cnstructs slutins n newly drawn instances. The average trajectries are cmputed by gemetrically averaging turs which are drawn by a Gibbs sampler at finite temperature. This will be discussed in detail in sectins 2 and 3. It turns ut that the temperature acts as a scale r smthing parameter. A few cmments cncerning the selectin f this parameter are given in sectin 6. The technical cntent f ur apprach is reminiscent f the "elastic net" -appraches f Durbin and Willshaw (see [2], [5]), but differs in many pints. It is based n a cmpletely different algrithmic apprach using Gibbs sampling and a general technique fr averaging turs. Our algrithm has plynmial cmplexity per Mnte Carl step and cnvergence is guaranteed by the usual bunds fr Markv Chain Mnte Carl simulatin and Gibbs sampling. Furthermre, the gal is nt t prvide a heuristic fr cmputing the best slutin, but t extract the relevant statistics f the Gibbs distributin at finite temperatures t generate the average trajectry, which will be used t cmpute slutins n future instances. 2 The Metrplis algrithm The Metrplis algrithm is a well-knwn algrithm which simulates a hmgeneus Markv chain whse distributin cnverges t the Gibbs distributin. We assume that the reader is familiar with the cncepts, we give here nly a brief sketch f the relevant results and refer t [6], [3] fr further details. Let M be a finite set and f: M -+ lit The Gibbs distributin at temperature T E Il4 is given by (m E M) (1) 9T(m) := exp( - f(m)/t~. Lm/EM exp( - f(m )/T) (2)

3 The Metrplis algrithm wrks as fllws. We start with any element m E M and set Xl +- m. Fr i ~ 2, apply a randm lcal update m':= (Xi). Then set with prbability min {I, exp( -(f(xi) - f(m'))/t)} else This scheme cnverges t the Gibbs distributin if certain cnditins n are met. Furthermre, a L2-law f large numbers hlds: Fr h: M --t ]R, ~ L:~=l h(xk ) --t L:mEM gt(m)h(m) in L2. Fr TSP, M = Sn and is the Lin-Kernighan tw-change [4], which cnsists in chsing tw indexes i, j at randm and reversing the path between the appintments i and j. Nte that the Lin-Kernighan tw-change and its generalizatins fr neighbrhd search are pwerful heuristic in itself. 3 Averaging Turs Our gal is t cmpute the average trajectry, which shuld grasp the underlying structure cmmn t all instances, with respect t the Gibbs measure at nn-zer temperature T. The Metrplis algrithm prduces a sequence f permutatins 7rl, 7r2,... with P{ 7rn =.} --t gt(.) fr n --t 00. Since permutatins cannt be added, we cannt simply cmpute the empirical means f 7rn. Instead, we map permutatins t their crrespnding trajectries. Definitin 1 (trajectry) The trajectry f 7r E Sn given n pints Xl,...,Xn is a mapping r( 7r): {I,..., n} --t ]R2 defined by r( 7r) (i) := X1C(i). The set f all trajectries (fr all sets f n pints) is dented by Tn (this is the set f all mappings T {I,..., n} --t ]R2 ). Additin f trajectries and multiplicatin with scalars can be defined pintwise. Then it is technically pssible t cmpute t L:~=l r(7rk). Unfrtunately, this des nt yield the desired results, since the relatin between permutatins and turs is nt ne-t-ne. Fr example, the permutatin btained by starting the tur at a different city still crrespnds t the same tur. We therefre need t define the additin f trajectries in a way which is independent f the chice f permutatin (and therefre trajectry) t represent the tur. We will study the relatinship between turs and permutatins first in sme detail, since we feel that the cncepts intrduced here might be generally useful fr analyzing cmbinatrial ptimizatin prblems. Definitin 2 (tur and length f a tur) Let G = (V, E) be a cmplete (undirected) graph with V = {I,...,n} and E = (~). A subset tee is called a tur iff It I = n, fr every v E V, there exist exactly tw el, e2 E t such that v E el and v E e2, and (V, t) is cnnected. Given a symmetric matrix (d ij ) f distances, the length f a tur t is defined by C(t) := L:{i,j}Et dij. The tur crrespnding t a permutatin 7r E Sn is given by n-l t(7r) :={ {7r(I), 7r(n)}} U U {{7r(i),7r(i + I)}}. (4) If t(7r) = t fr a permutatin 7r and a tur t, we say that 7r represents t. We call tw permutatins 7r, 7r' equivalent, if they represent the same tur and write 7r,..., 7r'. Let [7r] dente the equivalence class f 7r as usual. Nte that the length f a permutatin is fully determined by its equivalence class. Therefre,,..., describes the intrinsic symmetries f the TSP frmulated as an ptimizatin prblem n Sn, dented by TSP(Sn). We have t define the additin EB f trajectries such that the sum is independent f the representatin. This means that fr tw turs h, t2 such that h is represented i=l (3)

4 by 'lf1, 'If~ and t2 by 'lf2, 'If~ it hlds that f('lf1) EB f('lf2) ~ f('lfd EB f('if~). The idea will be t nrmalize bth summands befre additin. We will first study the exact representatin symmetry f TSP(Sn)' The TSP(Sn) symmetry grup Algebraically speaking, Sn is a grup with cncatenatin f functins as multiplicatin, s we can characterize the equivalence classes f ~ by studying the set f peratins n a permutatin which map t the same equivalent class. We define a grup actin f Sn n itself by right translatin ('If, 9 E Sn): ". " : Sn x Sn -+ Sn, g. 'If:= 'lfg- 1. (5) Nte that any permutatin in Sn can be mapped t anther by an apprpriate grup actin (namely 'If -+ 'If' by ('If,-l'lf). 'If.), such that the grup actin f Sn n itself suffices t study the equivalence classes f ~. Fr certain 9 E Sn, it hlds that t(g 'If) = t('if). We want t determine the maximal set H t f elements which keeps t invariant. It even hlds that H t is a subgrup f Sn: The identity is trivially in H t. Let g, h be t-invariant, then t((gh-1). 'If) = t(g (h- 1. 'If)) = t(h- 1. 'If) = t(h (h- 1. 'If) = t( 'If). H t will be called the symmetry grup f TSP(Sn) and it fllws that ['If] = H t 'If :={h 'If I he Hd. The shift u and reversal (2 are defined by (i E {I,..., n} ) (.). {i + 1 i < n, u z. 1., z = n (2(i) :=n + 1- i, (6) and set H :=((2, u), the grup generated by u and (2. It hlds that (this result is an easy cnsequence f (2(2 = id{l,...,n}, (2U = u-1(2 and un = id{l,...,n}) The fundamental result is H = {uk IkE {I,..., n}} U {(2uk IkE {l,...,n}}. (7) Therem 1 Let t be the mapping which sends permutatins t turs as defined in (4). Then, H t = H, where H t is the set f all t-invariant permutatins and H is defined in (7). Prf: It is bvius that H ~ H t. Nw, let h- 1 E H t. We are ging t prve that t-invariant permutatins are cmpletely defined by their values n 1 and 2. Let he H t and k:= h(l). Then, h(2) = u(k) r h(2) = u - 1(k), because therwise, h wuld give rise t a link {{'If(h(1),'If(h(2»}} 1. t('if). Fr the same reasn, h(3) must be mapped t u ±2(k). Since h must be bijective, h(3) =I- h(l), s that the sign f the expnent must be the same as fr h(2). In general, h(i) = u±(i- 1l(k). Nw nte that fr i,k E {l,...,n}, ui(k) = uk(i) and therefre, { u k- 1 if h(i) = ui-1 (k) h= (2un-k ifh(i)=u-i+1(k)' D Adding trajectries We can nw define equivalence fr trajectries. First define a grup actin f Sn n Tn analgusly t (5): the actin f h E Ht n "( E Tn is given by h "( := "( 0 h- 1. Furthermre, we say that "( ~ 1}, if H t "( = H t 1}. Our apprach is mtivated gemetrically. We measure distances between trajectries as fllws. Let d: ]R2 x ]R2 -+ Il4 be a metric. Then define h, 1} E Tn) dh,1}):= 2::=1 dh(k),1}(k). (8) Befre adding tw trajectries we will first chse equivalent representatins "(', 1}' which minimize d( "(', 1}'). Because f the results presented s far, searching thrugh

5 all equivalent trajectries is cmputatinally tractable. Nte that fr h E H t, it hlds that d( h.,,(, h. rj) = db, rj) as h nly rerders the summands. It fllws that it suffices t change the representatins nly fr ne argument, since d(h,,(, i rj) = db, h- 1 i rj) S the time cmplexity f ne additin reduces t 2n cmputatin f distances which invlve n subtractins each. The nrmalizing actin is defined by b, rj E Tn) n, 1J := argmin d(,,(, n. rj) n EH t (9) Assuming that the nrmalizing actin is unique 1, we can prve Therem 2 Let,,(, rj be tw trajectries, and n, 1J the unique nrmalizing actin as defined in (9). Then, the peratin "( EB rj := "( + n, 1J. rj (10) is representatin invariant. Prf: Let "(I = g.,,(, rjl = h rj fr g, h E Ht. We claim that n,i1j1 The nrmalizing actin is defined by = gn' 1Jh-1. n,i1j1 = argmin db/, n l. rjl) = argmin d(g.,,(, nih rj) = argmin db, g-ln lh rj), n l EHt n l EH t n l EH t (11) by inserting g-l parallelly befre bth arguments in the last step. Since the nrmalizing actin is unique, it fllws that fr the n l realizing the minimum it hlds that g-ln l h = n, 1J and therefre n l = n, I1J1 = gn' 1Jh-1. Nw, cnsider the sum which prves the representatin independence. 0 The sum f mre than tw trajectries can be defined by nrmalizing everything with respect t the first summand, s that empirical sums t EB~=l f(?ri) are nw well-defined. 4 Inferring Slutins n New Instances We transfer a trajectry t a new set f appintments Xl,...,Xn by cmputing the relaxed tur using the fllwing finite-hrizn adaptin technique: First f all, passing times ti fr all appintments are cmputed. We extend the dmain f a trajectry "( frm {I,..., n} t the interval [1, n + 1) by linear interplatin. Then we define ti such that "((ti) is the earliest pint with minimal distance between appintment Xi and the trajectry. The passing times can be calculated easily by simple gemetric cnsideratins. The permutatin which srts (t i)~l is the relaxed slutin f"( t (Xi). In a pst-prcessing step, self-intersectins are remved first. Then, segments f length w are ptimized by exhaustive search. Let?r be the relaxed slutin. The path frm?rei) t?r(i + w + 2) (index additin is mdul n) is replaced by the best alternative thrugh the appintments?r(i + 1),...,?r(i + w + 1). Iterate fr all i E {I,..., n} until there is n further imprvement. Since this prcedure has time cmplexity w!n, it can nly be dne efficiently fr small w. lotherwise, perturb the lcatins f the appintments by infinitesimal changes.

6 5 Experiments Fr experiments, we used the fllwing set-up: We tk the nrm t determine the nrmalizing actin. Typical sample-sizes fr the Markv chain Mnte Carl integratin were 1000 with 100 steps in between t decuple cnsecutive samples. Scenaris were mdeled after eq. (1), where the Xi were chsen t frm simple gemetric shapes. Average trajectries fr different temperatures are pltted in figures l(a)- (c). As the temperature decreases, the average trajectry cnverges t the trajectry f a single lcally ptimal tur. The graphs demnstrate that the temperature T acts as a smthing parameter. T estimate the expected risk f an average trajectry, the pst-prcessed relaxed (PPR) slutins were averaged ver 100 new instances (see figure l(d)-(g)) in rder t estimate the expected csts. The csts f the best slutins are gd apprximatins, within 5% f the average minimum as determined by careful simulated annealing. An interesting effect ccurs: the expected csts have their minimum at nn-zer temperature. The crrespnding trajectries are pltted in figure l(e),(f). They recver the structure f the scenari. In ther wrds, average trajectries cmputed at temperatures which are t lw, start t verfit t nise present nly in the instance fr which they were cmputed. S cmputatin f the glbal ptimum f a nisy cmbinatrial ptimizatin prblem might nt be the right strategy, because the slutins might nt reflect the underlying structure. Averaging ver many subptimal slutins prvides much better statistics. 6 Selectin f the Temperature The questin remains hw t select the ptimal temperature. This prblem is essentially the same as determining the crrect mdel cmplexity in learning thery, and therefre n fully satisfying answer is readily available. The prblem is nevertheless suited fr the applicatin f the heuristic prvided by the empirical risk apprximatin (ERA) framewrk [1], which will be briefly sketched here. The main idea f ERA is t carse-grain the set f hyptheses M by treating hyptheses as equivalent which are nly slightly different. Hyptheses whse 1 mutual distance (defined in a similar fashin as (8)) is smaller than the parameter "( E Il4 are cnsidered statistically equivalent. Selecting a subset f slutins such that l -spheres f radius "( cver M results in the carse-grained hypthesis set M,. VC-type large deviatin bunds depending n the size f the carse-grained hypthesis class can nw be derived: ( n(c - "()2 ) p{ C2 (m "! ) - min C2 (m) > 2c} :::; 21M"! 1 sup exp - ( )' mem mem., am + c c - "( am depending n the distributin. The bund weighs tw cmpeting effects. On the ne hand, increasing "( intrduces a systematic bias in the estimatin. On the ther hand, decreasing "( increases the cardinality f the hypthesis class. Given a cnfidence J > 0, the prbability f being wrse than c > 0 n a secnd instance and "( are linked. S an ptimal carsening "( can be determined. ERA then advcates t either sample frm the,,(-sphere arund the empirical minimizer r average ver these slutins. Nw it is well knwn, that the Gibbs sampler is cncentrated n slutins whse csts are belw a certain threshld. Therefre, the ERA is suited fr ur apprach. In the relating equatin the lg cardinality f the apprximatin set ccurs, which is usually interpreted as micr cannical entrpy. This relates back t statistical physics, the starting pint f ur whle apprach. Nw interpreting "( as energy, we can cmpute the stp temperature frm the ptimal T Using the well-knwn (13)

7 relatin frm statistical physics ~ee:t:~:: = T - 1, we can derive a lwer bund n the ptimal temperature depending n variance estimates f the specific scenari given. 7 Cnclusin In reality, ptimizatin algrithms are ften applied t many similar instances. We pinted ut that this can be interpreted as a learning prblem. The underlying structure f similar instances shuld be extracted and used in rder reduce the cmputatinal cmplexity fr cmputing slutins t related instances. Starting with the nisy Euclidean TSP, the cnstructin f average turs is studied in this paper, which invlves determining the exact relatinship between permutatin and turs, and identifying the intrinsic symmetries f the TSP. We hpe that this technique might prve t be useful fr ther applicatins in the field f averaging ver slutins f cmbinatrial prblems. The average trajectries are able t capture the underlying structure cmmn t all instances. A heuristic fr cnstructing slutins n new instances is prpsed. An empirical study f these prcedures is cnducted with results satisfying ur expectatins. In terms f learning thery, verfitting effects can be bserved. This phenmenn pints at a deep cnnectin between cmbinatrial ptimizatin prblems with nise and learning thery, which might be bidirectinal. On the ne hand, we believe that nisy (in cntrast t randm) cmbinatrial ptimizatin prblems are dminant in reality. Rbust algrithms culd be built by first estimating the undistrted structure and then using this structure as a guideline fr cnstructing slutins fr single instances. On the ther hand, hardness f efficient ptimizatin might be linked t the inability t extract meaningful structure. These cnnectins, which are subject f further studies, link statistical cmplexity t cmputatinal cmplexity. Acknwledgments The authrs wuld like t thank Naftali Tishby, Sctt Kirkpatrick and Michael Clausen fr their helpful cmments and discussins. References [1] J. M. Buhmann and M. Held. Mdel selectin in clustering by unifrm cnvergence bunds. Advances in Neural Infrmatin Prcessing Systems, 12: , [2] R. Durbin and D. Willshaw. An analgue apprach t the travelling salesman prblem using an elastic net methd. Nature, 326: , [3] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi Optimisatin by simulated annealing. Science, 220: , [4] S. Lin and B. Kernighan. An effective heuristic algrithm fr the traveling salesman prblem. Operatins Research, 21: , [5] P.D. Simic. Statistical mechanics as the underlying thery f "elastic" and "neural" ptimizatins. Netwrk, 1:89-103, [6] G. Winkler. Image Analysis, Randm fields and Dynamic Mnte Carl Methds, vlume 27 f Applicatin f Mathematics. Springer, Heidelberg, 1995.

8 -sigma2 = O.03 T.,...,:0.15OO:Xl Lenglt. : i 17.7 &: 17.6 " j 17.5 f 17.4 <"Il temperaturet (d) e.i H -si ma = O.025 T.,...,: Lenglt. : ~ 11.5 ~ &: " j 11.4 ~ ~ temperaturet (f) CD n 5O"",11I>1S20_ _ () _ _ (l.Q a1 ~ g Figure 1: (a) Average trajectries at different temperatures fr n = 100 appintments n a circle with a 2 = (b) Average trajectries at different temperatures, fr multiple Gaussian surces, n = 50 and a 2 = (c) The same fr an instance with structure n tw levels. (d) Average tur length f the pst-prcessed relaxed (PPR) slutins fr the circle instance pltted in (a). The PPR width was w = 5. The average fits t nise in the data if the temperature is t lw, leading t verfitting phenmena. Nte that the average best slutin is :s: (e) The average trajectry with the smallest average length f its PPR slutins in (d). (f) Average tur length as in (d). The average best slutin is :s: (g) Lwest temperature trajectry with small average PPR slutin length in (f).