Parallel Ant Colony for Nonlinear Function Optimization with Graphics Hardware Acceleration

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1 Proceedgs of the 009 IEEE Iteratoal Coferece o Systems Ma ad Cyberetcs Sa Atoo TX USA - October 009 Parallel At Coloy for Nolear Fucto Optmzato wth Graphcs Hardware Accelerato Wehag Zhu Departmet of Idustral Egeerg Lamar Uversty Beaumot Texas USA Wehag.zhu@lamar.edu James Curry Departmet of Idustral Egeerg Lamar Uversty Beaumot Texas USA jcurry@my.lamar.edu Abstract Ths paper presets a massvely parallel At Coloy Optmzato Patter Search (ACO-PS) algorthm wth graphcs hardware accelerato o olear fucto optmzato problems. The objectve of ths study s to determe the effectveess of usg Graphcs Processg Uts (GPU) as a hardware platform for ACO-PS. GPU the commo graphcs hardware foud moder persoal computers ca be used for data-parallel computg a desktop settg. I ths research the classcal ACO s adapted the data-parallel GPU computg platform featurg Sgle Istructo Multple Thread (SIMT). The global optmal search of the ACO s ehaced by the classcal local Patter Search (PS) method. The hybrd ACO- PS method s mplemeted a GPUCPU hardware platform ad compared to a smlar mplemetato a Cetral Processg Ut (CPU) platform. Computatoal results dcate that GPU-accelerated SIMT-ACO-PS method s orders of magtude faster tha the correspodg CPU mplemetato. The ma cotrbuto of ths paper s the parallelzato aalyss ad performace aalyss of the hybrd ACO-PS wth GPU accelerato. Keywords At Coloy Patter Search Graphcs Hardware Accelerato GPU CUDA I. INTRODUCTION At Coloy Optmzato (ACO) s a stochastc populato-based evolutoary meta-heurstc algorthm spred by the behavor of real at coloes. It s a type of Swarm Itellgece whch s based o socal-psychologcal prcple that dvduals commucate ad adjust ther behavor based o the performace of others. The ACO algorthm was frst troduced 99 by Marco Dorgo hs Ph.D. dssertato []. It s spred by the behavor of ats fdg paths from the coloy to food. ACO s computatoally tesve whe t comes to complex problems. Graphcs Processg Ut (GPU) has emerged as a possble desktop parallel computg soluto for large scale scetfc computato wth a desktop PC settg [5]. I ths project we explored ts ablty ad lmtatos wth the ACO algorthm o a set of boud costraed optmzato fuctos. I the local search phase tradtoal Patter Search (PS) method s used [7]. The objectve of ths research s to fd out how ad how much the ACO algorthm ca be potetally accelerated o a GPU platform. The GPU-based SIMT-ACO-PS algorthms are mplemeted wth the Compute Ufed Devce Archtecture (CUDA) TM evromet o a Vda Graphcs Processg Ut (GPU). We have demostrated the potetal of GPU techology Tabu Search o the Quadratc Assgmet Problem a recet paper [9]. We have also desged a GPUaccelerated Partcle Swarm Optmzato ad Patter Search o fucto optmzato problems [0][]. The remader of ths paper s orgazed as follows. Secto presets backgroud formato o the GPU computg At Coloy Optmzato ad Patter Search. Secto 3 provdes a overvew of the SIMT-ACO-PS algorthm. Secto 4 dscusses the parallelzato aalyss ad mplemetato of the SIMT-ACO-PS o a GPU hardware platform. Secto 5 presets computatoal expermet results ad aalyss. Coclusos of our vestgato ad future research tasks are summarzed Secto 6. II. RESEARCH BACKGROUND A. GPU Computg GPU parallel computg follows a dfferet patter amely Sgle Istructo Multple Thread (SIMT). Wth SIMT a GPU executes the same structo set o dfferet data elemets at the same tme. A GPU ca process thousads of threads smultaeously eablg hgh computatoal throughput across large amouts of data. The CUDA techology allows a software developer to program a GPU for geeral purpose computg may possble applcatos [5]. I the CUDA evromet thousads of threads ca ru cocurretly wth a same structo set. Each thread rus a same program called a kerel. A kerel ca employ regsters as fast access memory. The commucato amog threads ca be realzed wth shared memory whch s a type of very fast memory that allows both read ad wrte access. The commucato betwee CPU ad GPU ca be doe through global devce memory costat memory or texture memory o a GPU board. Global devce memory s a relatvely slow memory locato that allows both read ad wrte operatos. Texture memory s relatvely fast memory that s read-oly. We employ texture memory to keep a copy of the at populato the Probablty Vector (wth formato of Pheromoe) ad pre-geerated radom umbers. Costat memory s fast read-oly memory whose sze caot be dyamcally chaged. Texture memory ad costat memory s fast because t s cached for qucker access. The Vda /09/$ IEEE SMC /09/$ IEEE 87

2 GeForce GTX 80 GPU hardware employed ths paper has 30 mult-processors. Each mult-processor has 8 processors. Ths amouts to 40 data-parallel processors (cores) o oe GPU board. For maagemet purpose all threads are orgazed to blocks. Each block ca have a maxmum of 5 threads. The threads the same block ca commucate through fast shared memory; whle betwee the blocks commucato s possble oly wth slower global devce memory. For a detaled descrpto of the CUDA capabltes readers are referred to the CUDA programmg gude [6]. B. Parallel At Coloy Frst troduced 99 by Marco Dorgo [] the At Coloy Optmzato (ACO) algorthm smulates the behavor of ats fdg paths from the coloy to food. I realty ats start wth radom movemets ad upo fdg food they lay dow pheromoe trals as they retur to ther coloy. If some ats fd a path to food they are lkely to follow the tral wth hgher cocetrato of pheromoe. Pheromoe may evaporate over the tme ad thus the pheromoe tral become less attractve. Ths pheromoe evaporato ca help avod the covergece to local optma. After some teratos of the at swarm the ftess of the global best soluto keeps mprovg. Evetually the swarm coverges to the global optmal. There are may varats of the ACO sce t has bee used dfferet applcatos [3]. I ths paper we used ACO to solve boud costraed cotuous fucto optmzato problems. I our desg the parameter boud rage s dvded to equally spaced steps say 000 steps as the legth of the Pheromoe vector. The ACO starts wth a batch of radomly geerated ats as the tal solutos. The Pheromoe vector s assged a very small value tally. It s typcally decayed each tme t s update as show () where ρ g s the global pheromoe decay costat. τ τ ρ ). () ( g The local pheromoe decay s ot used. The k elte solutos are used to update the Pheromoe vector wth () where Q s a pheromoe update costat value C k s the cost of the k-th elte soluto ad τ ( j k ) s the Pheromoe elemet mappg wth the j-th varable of the k-th elte soluto. Q τ ( j k ) τ ( j k ) f C k!= 0 C τ max { τ } ( j k ) f k = 0 k C () Correspodg to the Pheromoe vector there s a Probablty Vector p whose elemet p s calculated wth (3): p = α τ α τ. (3) where s the pheromoe sestvty costat. Wth the Probablty Vector p each at costructs a caddate soluto. Durg the soluto costructo process f a radom umber s less tha the explorato costat q 0 the Selecto Idex s chose as the dex of the Probablty Vector wth the maxmum probablty. Otherwse a roulette wheel selecto method as the tradtoal ACO s used to choose the Selecto Idex usg the Probablty Vector. Ths Selecto Idex s the coverted to the varable value of the at soluto. The at solutos are evaluated ad sorted from the best to the worst. A small porto of elte (best) solutos are retaed from oe geerato to the ext geerato wthout chage. The ftess of the global best soluto keeps mprovg durg the ACO evoluto. For complex problems ACO requres sgfcat computato tme. Addg parallelsm to ACO s oe opto for mprovemet. ACO s aturally ameable to parallelsm. A detaled classfcato of parallel ACO ca be foud []. Ths research explores a low level parallel mplemetato where all ats teract wth a sgle populato. I our ACO varat dvdual ats are radomly talzed to start the search. These ats serve as caddate solutos. A at soluto vector s composed of elemets coded real umbers. Each elemet represets a dmeso varable a mult-dmesoal boud costraed optmzato problem. Ths model s specfcally desged for the GPU parallel computg platform. C. Parallel Patter Search Whle ACO algorthms are good at quckly fdg a reasoable soluto they may be slow to coverge to the local optmal soluto from a earby soluto ad they may be stuck a local optmum. To overcome ths problem a local search mprovemet phase such as Patter Search (PS) ca be added to the ACO. The basc Patter Search algorthm s a smple drect search method that does ot requre dervatve or secod dervatve formato. Patter Search s tradtoally employed whe the gradet of the fucto s ot relable whe performg the search [7]. Ths research uses At Coloy Optmzato (ACO) as the global search method ad Patter Search (PS) as the local search method. We defed thousads of ats executed parallel o a maxmum of 40 processors wth a sgle GPU. Usg ACO as the global search tool has the advatage of coverg a wde amout of the search space. Each member of the populato s assged a separate Patter Search. Ths approach does ot requre a modfcato to the basc Patter Search algorthm sce each thread s actg o a separate search. III. OVERVIEW OF THE SIMT-ACO-PS METHOD The proposed Sgle Istructo Multple Thread At Coloy Patter Search (SIMT-ACOPS) method has two compoets: At Coloy Optmzato (ACO) ad Patter Search (PS). The method s talored for a evromet wth T ats (threads) that operate a sgle structo multple thread maer. Each thread geerates a ew soluto based o the Probablty Vector ad performs Patter Search mprovemet to mprove the ew soluto. The ats are sorted based o ther costs to fd the best soluto after each geerato. Let x 873 SMC 009

3 represet the set of soluto parameters for dvdual at f ( x ) s the soluto cost of ths at ad c( x ) s the boud costrat. The At Coloy compoet s llustrated Fg.. A) Italze T ats each of whch has P dmeso varables; B) Evaluate the cost ad costrats of all the ats; C) Sort the ats based o the cost; D) Italze ACO ad PS parameters; Whle (Not meetg termate crtero) E) Evaporate pheromoe wth (); F) Use the best few ats to update the pheromoe vector wth (); G) Update the Probablty Vector wth the pheromoe formato wth (3); H) Geerate the ew at solutos based o the Probablty Vector; I) Evaluate the cost f ( x ) ad costrats c( x ) of the ats; J) Improve all the ats wth Patter Search See Fg. ; K) Sort the ats based o the cost f ( x ) ; L) Collect the geerato result; Ed Whle M) Collect fal result. Fgure. Procedure of the At Coloy Compoet a) Italze PS parameters (search step = o ) ad start from the ACO soluto for the thread; For (terato k = k max ) For (dmeso j =... P) b) If f x e Δ ) f ( x ) ad ( j f x e Δ ) f ( x e Δ ) the x ( j f ( x ej Δ ) f ( x = x e Δ c) Else If ) ad f ( x ej Δ ) f ( x e Δ) the x = x e j Δ. Ed For (each dmeso) d) If o mprovemet throughout all d dmesos = / ; If = tol reset = o ; Ed For (k teratos) e) Collect fal result of ths Patter Search Fgure. Procedure of the Patter Search Compoet After each terato of the ACO phase we mprove each at depedetly wth a local Patter Search (PS) of eghborg solutos. Our patter deoted by D { e e... e d e d } s defed by the ut coordate axes where d s the dmeso of the problem. The PS s talzed by settg the step sze to a user specfed tal o. The PS explores the coordate axes for mprovg soluto for k teratos. If the search does ot fd a mprovemet for ay drecto the the step sze s reduced by half. The PS s stopped after a fxed umber of teratos. If a thread reaches covergece to a user specfed tolerace tol before the terato lmt reached ths research resets the step sze to the tal o to make use of computer resources that otherwse would be dle. By settg the umber of teratos the Patter Search the modeler ca cotrol the amout of resources dedcated to the ACO ad PS sectos of j. the search heurstc. The Patter Search compoet s llustrated Fg.. IV. PARALLELIZATION ANALYSIS AND IMPLEMENTATION A. Radom Numbers All the stochastc optmzato methods eed hgh qualty radom umbers. I tradtoal CPU-based ACOs radom umbers are typcally geerated as eeded. I the GPU all the radom umbers are geerated before the teratos starts ad these umbers are stored the texture memory space to eable faster access by the kerels. The radom umber geerator chose s Mersee Twster (Matsumoto ad Nshmura 008). B. Ftess Fuctos ad Feasblty Fuctos The purpose of a ftess fucto s to evaluate the cost of a objectve fucto. The purpose of a feasblty fucto s to make sure that the curret at meets all the costrats. I ths project all feasblty fuctos are oly to make sure they are betwee the mmum ad maxmum values. The parallelzato of ftess fuctos ad feasblty fuctos are straghtforward sce these evaluatos are depedet each thread. C. Geerato of New Ats The geerato of the ew ats.e. solutos s the most tme-cosumg part of the algorthm. It takes the Probablty Vector ad radom umbers as the put. We keep a copy of the Probablty Vector ad radom umbers the texture memory to eable faster access of these data durg ew ats geerato. Ths geerato process s totally depedet for each at ad thus s hghly parallelzable. The CPU-GPU commucato overhead s small sce the formato that eeds to be frequetly passed s oly the Probablty Vector. D. Patter Search The Patter Search s mplemeted as oe kerel fucto. The Patter Search process vokes frequet ftess fucto evaluatos ad feasblty checks. Ths task s deally suted for the GPU due to beg a large task wth o commucato betwee threads. E. GPU ad CPU Task Allocato We proposed the SIMT-ACO-PS algorthm as Fg. 3. It s a mxture of CPU ad GPU fucto calls. We put several steps to GPU wth each thread operatg to costruct ad mprove a sgle vector (soluto). A) Italze T ats each of whch has P dmeso varables ( CPU); B) Evaluate the cost ad costrats of all the ats ( GPU); C) Sort the ats based o the cost ( CPU); D) Italze ACO ad PS parameters ( CPU); Whle (Not meetg termate crtero) E) Evaporate pheromoe wth () ( CPU); F) Use the best few ats to update the pheromoe vector wth () ( CPU); G) Update the Probablty Vector wth the pheromoe formato wth (3) ( CPU); 874 SMC 009

4 H) Geerate the ew at solutos based o the Probablty Vector ( GPU); I) Evaluate the cost f ( x ) ad costrats c( x ) of the ats ( GPU); J) Improve all the ats wth Patter Search ( GPU); K) Sort the ats based o the cost f ( x ) ( CPU); L) Collect the geerato result ( CPU); Ed Whle M) Collect fal result ( CPU). Fgure 3. Procedure of the SIMT - At Coloy Patter Search V. COMPUTATIONAL EXPERIMENTS AND RESULTS The proposed Sgle Istructo Multple Thread At Coloy Patter Search (SIMT-ACO-PS) algorthm for the geeral optmzato fuctos has bee mplemeted by the author. The algorthm was mplemeted Vsual C 005 evromet wth the CUDA TM evromet for programmg the GPU. The computatoal expermets were executed o a Dell Precso 7400 Workstato computer wth a Itel Xeo E540 CPU GB memory ad a Vda GeForce GTX 80 GPU. For bechmarkg the algorthm was also mplemeted wth CPU-based oly fuctos to compare the computato speed to the GPU-accelerated mplemetato. The same computer was used testg both CPU ad GPU versos. Twelve bechmark fuctos have bee selected for these computatoal expermets [8]. Some of them are lsted the Appedx. ACO performace s affected by ts parameter settg [3]. Based o a set of tal testg wth several problems of dfferet szes we selected the followg parameter settgs: Pheromoe update costat Q: 0; Explorato costat: 0.4; Global pheromoe decay rate: 0.7; Pheromoe sestvty :.0; Pheromoe vector legth: 000; Ital values of the Pheromoe Vector:.0E-6; Patter Search: 0 teratos wth 0 set to the parameter rage / 0. m s set to be 0 / 6. A. Computatoal Performace Aalyss Table shows the comparso of computato tmes o the Ackley Fucto wth dmeso 30 betwee the proposed SIMT-ACO-PS ad CPU mplemetato of the same algorthm. The speedup values are calculated by dvdg the CPU-oly algorthm tmes by the GPU-accelerated algorthm tmes. Table (a) shows the speedup result of the ACO compoet. Table (b) shows the speedup result of the PS compoet. Table (c) shows the speedup result of the hybrd ACO-PS algorthm. The speedup creases as the umber of threads (ats) creases. Note that Table (b) t s 50 repettve rus of 0-terato Patter Search whle Table (c) total steps of the Patter Search s equal to (50 ACO geeratos x 0 PS teratos). Wth a larger umber of ats the heavy computato tasks are prmarly executed o the GPU greatly reducg ru tme compared to the CPU-oly mplemetato. The peak performace s expected at 5360 threads whch s orgazed to 60 blocks wth 56 threads per block. For our GPU kerels each mult-processor ca take blocks due to the regster costrats GPU. It meas each mult-processor ca ru blocks of 56 threads that s 30 mult-processor (of GeForce GTX 80 GPU) tmes blocks tmes 56 threads s equal to 5360 threads [6]. Therefore 30 mult-processors of a GeForce GTX 80 ca take totally 60 blocks or ts multples to reach peak performace. TABLE. AVERAGE RUN TIME COMPARISON BETWEEN CPU AND GPU IMPLEMENTATION IN MILLISECONDS ON THE ACKELY FUNCTION WITH 30 VARIABLES 50 ACO GENERATIONS AND 0 PS ITERATIONS FOR 0 RUNS TIMED IN M-SEC a) The speedup result of the ACO compoet Threads SIMT-ACO ACO Speedup b) The speedup result of the PS compoet Threads SIMT-ACO ACO Speedup c) The speedup result of the ACO-PS Threads SIMT-ACO ACO Speedup B. Speedup Comparso betewee CPU ad GPU Implemetatos Table shows the comparso of computato tmes betwee the CPU ad GPU mplemetato of the ACO-PS algorthm for all the bechmark fuctos. We coduct a test wth 5360 threads to determe the tmes spet o solvg each bechmark problem. The average results of 0 rus were collected ad summarzed. C. Soluto Qualty uder Tme Lmts Tme to soluto s a crtcal performace measure for a optmzato procedure. Gve eough tme both CPU ad GPU versos ca gve satsfactory results. Whe tme s lmted the result obtaed wth GPU-accelerated algorthm s otably better. Table 3 shows the mea best solutos ad stadard devato for 0 rus of At Coloy Patter Search ad At Coloy plus Patter Search o CPU ad GPU hardware. All the tests are lmted to oe secod. The 875 SMC 009

5 compoet or algorthm s allowed to fsh ay terato started before the oe secod lmt. It ca be see that the hybrd ACO-PS geerally gets better results gve the oe secod tme lmt. However the GPU platform PS-oly ca acheve better results some problems because t s able to complete more teratos tha ACO-PS gve the tme lmt. TABLE. COMPARISON OF AVERAGE COMPUTATION TIMES IN MILLISECONDS BETWEEN THE SIMT-ACO-PS AND THE ACO-PS ALGORITHMS FOR 0 MONTE CAROL RUNS.THE COMPUTATION SETTING WERE 5360 THREADS (ANTS) 50 ACO GENERATIONS30VARIABLES AND 0 PS ITERATIONS TIMED IN M-SEC CPU Verso GPU Verso GPU/CPU speedup ACOoly ACO- # Problems ACO-oly PS-oly ACO-PS PS-oly ACO-PS oly PS-oly ACO-PS Ackley Grewak Pealty Pealty Quartc Rastrg Rosebrock Schwefel Schwefel Schwefel Sphere Step VI. CONCLUSIONS I ths paper we preset a Sgle Istructo Multple Thread At Coloy Patter Search (SIMT-ACO-PS) algorthm for geeral boud costraed optmzato fuctos wth graphcs hardware accelerato. At Coloy Optmzato (ACO) s used the global search phase ad Patter Search (PS) s used the local search phase for mprovemet. Through parallelzato aalyss the classcal ACO algorthm ad the PS algorthm are adapted for dataparallel computg a GPU desktop parallel computg evromet. The computato tme was sgfcatly reduced ad the better optmzato results ca be obtaed more quckly wth massve ats ad Patter Search by leveragg GPU parallel computg. ACKNOWLEDGMENT Ths work was partally supported by the Lamar Research Ehacemet Grats to Dr. W. Zhu ad Dr. J. Curry at Lamar Uversty. Partal support for ths work was provded by a grat from the Texas Hazardous Waste Research Ceter to Dr. W. Zhu Dr. J. Curry ad Dr. H. Lou at Lamar Uversty. Partal support for ths work was provded by the Natoal Scece Foudato's Award No to Dr. W. Zhu. Ther support s greatly apprecated. ) Ackley Fucto: Appedx: Test Fuctos cos(πx ) f x 5 = = ( x) = 0 e 0e e x ) Grewak Fucto f =... ( x) = x cos( ) 4000 = = m( f ) = f (0... 0) 600 x 600 =... x m( f ) = f (0...0) 3) Geeralzed Pealzed Fucto = u ( x 0004) = = 0 [ 0s ( y )] ( y ) π f3 ( x) = 0s ( πy ) ( y ) π 30 = u ( x a k m) where m k( x a) x > a = 0 a x a y = x 4 m k( x a) x < a 0 ( ) 50 x 50 =... m( f3 ) = f3(... ) = 0 4) Geeralzed Pealzed Fucto 5) Quartc Fucto 6) Rastrg Fucto 876 SMC 009

6 7) Rosebrock Fucto 8) Schwefel s Problem. 9) Schwefel s Problem. 0) Schwefel s Problem. ) Sphere Fucto ) Step Fucto REFERENCES [] E. Alba Parallel Meta-heurstcs: a New Class of Algorthms edted by Erque Alba Joh Wley Ic. ISBN [] M. Dorgo Optmzato Learg ad Natural Algorthms Ph.D. Dssertato Poltecco d Mlao Italy 99. [3] Dorgo M. ad T. Stuetzle 004 At Coloy Optmzato MIT Press ISBN [4] M. Matsumoto; T Nshmura Mersee Twster. A 63-dmesoally equdstrbuted uform pseudoradom umber geerator. [5] Nguye H. Edtor 007. GPU Gems 3 New York: Addso-Wesley. [6] Vda 008 Cuda Programmg Gude Verso.. Vda Avalable at: [7] V. Torczo O the covergece of Patter Search algorthms SIAM J. Optma. 7 pp [8] X. Yao Y. Lu ad G. L Evolutoary Programmg Made Faster IEEE Trasactos o Evolutoary Computato (3) pp. 8-0 July 999. [9] W. Zhu J. Curry A. Marquez SIMT Tabu Search wth Graphcs Hardware Accelerato o the Quadratc Assgmet Problem the Iteratoal Joural of Producto Research 008 press. [0] W. Zhu J. Curry Partcle Swarm wth Graphcs Hardware Accelerato ad Local Patter Search o Boud Costraed Optmzato Problems IEEE Symposums Seres o Computatoal Itellgece Nashvlle TN USA March 30 ~ Aprl 009. [] W. Zhu J. Curry Mult-walk Parallel Patter Search o a GPU computg Platform Proceedgs of the IEEE Iteratoal Coferece o Computatoal Scece Bato Rouge LA USA May 5 ~ TABLE 3. MEAN BEST SOLUTIONS COMPARISON BETWEEN THE GPU AND CPU IMPLEMENTATIONS IN A ONE SECOND TIME LIMIT FOR TEN MONTE CARLO RUNS. COMPUTATION SETTING ARE 30 VARIABLES0PSITERATIONS AND 5360 THREADS a) SIMT-ACO-PS Mea Best Solutos Stadard Devato # Problems ACO-oly PS-oly ACO-PS ACO-oly PS-oly ACO-PS Ackley Grewak Pealty Pealty Quartc Rastrg Rosebrock Schwefel Schwefel Schwefel Sphere Step b) CPU ACO-PS Mea Best Solutos Stadard Devato # Problems ACO-oly PS-oly ACO-PS ACO-oly PS-oly ACO-PS Ackley Grewak Pealty Pealty Quartc Rastrg Rosebrock Schwefel Schwefel Schwefel Sphere Step SMC 009