Parallel Learning of Large Fuzzy Cognitive Maps

Transcription

1 g Proceedigs of Iteratioal Joit Coferece o Neural Networks, Orlado, Florida, USA, August 2-7, 07 Parallel Learig of Large Fuzzy Cogitive Maps Wojciech Stach, Lukasz Kurga, ad Witold Pedrycz Abstract Fuzzy Cogitive Maps (FCMs) are a class of discrete-time Artificial Neural Networks that are used to model dyamic systems. A recetly itroduced supervised learig method, which is based o real-coded geetic algorithm (RCGA), allows learig high-quality FCMs from historical data. The curret bottleeck of this learig method is its scalability, which origiates from large cotiuous search space (of quadratic size with respect to the size of the FCM) ad computatioal complexity of geetic optimizatio. To this ed, the goal of this paper is to explore parallel ature of geetic algorithms to alleviate the scalability problem. We use the global sigle-populatio master-slave parallelizatio method to speed up the FCMs learig method. We ivestigate the ifluece of differet hardware architectures o the computatioal time of the learig method by executig a wide rage of sythetic ad real-life bechmarkig tests. We aalyze the quality of the proposed parallel learig method i applicatio to both dese ad sparse large FCMs, i.e. maps that cosist of several dozes of cocepts. The parallelizatio is show to provide substatial speed-ups, allowig doublig the size of the FCM that ca be leared by parallelizatio with 8 processors. A. Fuzzy Cogitive Maps I. INTRODUCTION ) Overview roposed i 986 by Bart Kosko [4], Fuzzy Cogitive P Maps (FCMs) form a class of discrete-time Artificial Neural Networks. They represet kowledge i a symbolic maer ad relate states, variables, evets, outputs ad iputs usig a cause ad effect approach. FCMs, whe compared with eural etworks, have several importat advatages such as relative easiess to represet structured kowledge, ad simplicity of the iferece that is computed by umeric matrix operatios [7]. The FCM structure is similar to a recurret artificial eural etwork (RNN), where cocepts are represeted by euros ad causal relatioships by weighted edges coectig the euros. However, i cotrast to FCMs, RNN euros have exteral iputs, whereas i FCMs, the (euros) are oly iterally itercoected. Each of FCM s edges is associated with a weight value that reflects the stregth of the correspodig relatio. This value is usually ormalized to the iterval [,]. Positive values This work was supported i part by the Walter Karplus Summer Research Grat, by the Alberta Igeuity, ad by the Natural Scieces & Egieerig Research Coucil of Caada (NSERC) W. Stach, L. Kurga, ad W. Pedrycz are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Alberta, Edmoto, Caada ( {wstach, lkurga, pedrycz}@ece.ualberta.ca) reflect promotig effect, while egative oes describe ihibitig effect. The value of represets full egative, + full positive ad 0 deotes eutral relatio. Other values correspod to differet itermediate levels of causal effect. The graph represetatio is equivalet to a square matrix, called coectio matrix, which stores all weight values of edges betwee correspodig cocepts. Figure shows a example of FCM that models city health issues [6]. N N2 N3 N4 N5 N6 N7 N N N N N N N Fig.. Example of FCM model ad its equivalet coectio matrix I FCMs, each ode has a value that reflects the degree to which the correspodig cocept i the system is active at a particular iteratio. This value, called activatio level, is a floatig-poit umber betwee 0 (iactive) ad (active). For a give cocept, this value is calculated by takig ito accout the activatio levels at the previous iteratio of all the cocepts that exert ifluece o it: N j {,..., N}, C ( t + ) = f e C () j ij i i= where: C j (t) activatio level of cocept j th at iteratio t e ij stregth of relatio from cocept C i to cocept C j f trasformatio fuctio The trasformatio fuctio is used to reduce ubouded weighted sum to a certai rage. The ormalizatio hiders quatitative aalysis, but, at the same time, it allows for comparisos betwee activatio levels of differet cocepts. A sapshot of activatio levels of all at a particular iteratio defies the system state. It ca be coveietly represeted by a vector, called state vector, which cosists of the activatio values. Iitial state vector refers to the system state at the first iteratio. Successive states are calculated by iterative applicatio of the formula (). Fuzzy Cogitive Maps have bee successfully applied i various domais, icludig egieerig [2][23], medicie [], political sciece [2], ecoomics [], Earth ad evirometal scieces [6], etc X/07/$ IEEE

2 2) Developmet There are two mai groups of approaches to develop Fuzzy Cogitive Maps: () deductive modelig (i.e., it uses a expert kowledge from the domai of applicatio); ad (2) iductive modelig (i.e., it uses learig algorithms to establish FCMs from historical data) A comprehesive overview of these methods ca be foud i [22]. Figure 2 shows a high-level diagram of the FCMs learig method based o real-coded geetic algorithms (RCGA), which is used i this paper. This method has bee recetly itroduced ad thoroughly tested [9][]. RCGA is a floatig-poit extesio to geeric geetic algorithms. Sectio B.2 gives more iformatio o both geetic algorithms ad RCGA. Iitialize ad evaluate populatio Stop? Yes No Mutatio New populatio Fig. 2. High-level diagram of RCGA learig method for FCMs Recombiatio Evaluatio Selectio The RCGA learig method uses iput data to develop a FCM (cadidate FCM), which is capable to mimic the historical data [9]. The iput data is give as time series ad cosist of a sequece of state vectors that describe a give system at differet time poits. Sice FCM ca be fully represeted by its coectio matrix, the learig goal is to establish N 2 parameters, where N deotes the umber of cocepts. They correspod to the weighted values of mutual relatios betwee the cocepts. The RCGA algorithm exploits the iput data to fid these values. The learig objective is to geerate the same state vector sequece from the cadidate FCM for the same iitial state vector, as it is defied i the iput data. At the same time, the cadidate FCM geeralizes the iter-relatios betwee cocept, which are iferred from the iput data. Therefore, it is suitable to perform simulatios from differet iitial state vectors ad, based o their results, to draw coclusios about the modeled system. B. Parallelizatio of Geetic Algorithms ) Itroductio to parallel computig Parallel computig is oe of the popular techiques to speed up the process of solvig complex computatioal problems [8]. It assumes simultaeous executio of the same task o multiple processors to obtai the results faster. The uderlyig assumptio is that a problem beig solved ca be divided ito smaller tasks, which ca be executed simultaeously with a certai level of coordiatio. I parallel computig, it is crucial to use parallel algorithms to take advatage of hardware systems [25]. Parallel algorithms, i cotrary to sequetial algorithms, ca be divided ito parts performed i parallel. Subsequetly, the partial results are put back together to obtai the fial result. The task of fidig a efficiet parallel algorithm to solve a give problem ca be very challegig as we ofte deal with sequetial costrais. For istace, recursive solutios that require a result from previous iteratio to calculate result i ext iteratio are very difficult to parallelize. While implemetig parallel algorithms, commuicatio amog the processors to coordiate the executio of subtasks must be cosidered. Hece, i practice, liear time speed-up vs. the umber of processors is very difficult to obtai. The commuicatio is usually realized either usig shared memory or message passig approach [5]. 2) Overview of geetic algorithms Geetic algorithms (GAs) are a search techique to fid solutios to optimizatio ad search problems [5][7]. They have may advatages, which iclude broad applicability, ease of use, ad global perspective, to ame a few. GAs origiate from evolutioary biology ad use geetically ispired mechaisms, such as mutatio, selectio, ad crossover. These operators are applied to a populatio of chromosomes that is maitaied throughout the etire searchig process. Each chromosome represets a solutio to the problem, ad its quality is quatified by a fitess value that is calculated from the fitess fuctio. The GAs usually start from a radomly geerated populatio ad modify it i subsequet geeratios. I each geeratio, a ew populatio is formed by usig geetic operators, ad by exploitig fitess values of the chromosomes. The idea is to produce better solutios to the problem over the evolvig geeratios. GAs ca be successfully applied to large, complex, ad poorly uderstood search spaces, i which classical optimizatio tools are ofte iappropriate. The detailed iformatio about GAs ca be foud i [5][7][24]. Several extesios to the geeric GAs have bee itroduced. I this paper, a real-coded geetic algorithms (RCGA) approach is used as a part of the method for FCMs learig. RCGA is a floatig-poit extesio to GAs. I RCGA chromosomes are represeted as floatig-poit vectors i cotrast to GAs that use biary vectors. This exteded chromosome represetatio makes RCGA more

3 effective to tackle optimizatio problems with cotiuous variables. Although the geetic operators i RCGA are revised i order to deal with floatig-poit values, the mai priciples of both GA ad RCGA are the same. Comprehesive descriptio of the RCGA is give i [9]. The geetic algorithms ca be parallelized i several differet ways that are compatible with the RCGA learig. Parallelizatio ca be implemeted based o a sigle populatio, or by splittig the populatio ito subpopulatios. I this paper, a sigle populatio method (global sigle-populatio master-slave) is chose. This approach parallelizes the evaluatio of fitess fuctio, which is usually the most time-cosumig part of the GA s optimizatio. The implemetatio is usually doe as a master-slave structure, i which the master process stores the populatio ad the slaves evaluate the fitess. With this feature i mid, the etire populatio is split up ito subsets, which are the assiged to differet available processors. Commuicatio betwee the processes occurs oly whe slaves receive a subset of idividuals to evaluate ad whe they retur the fitess values to the master process. This parallelizatio architecture does ot affect the behavior of the geetic algorithm, sice o additioal restrictios are imposed o the geetic operators such as crossover or selectio. This simplicity motivated our selectio of this parallelizatio approach. Additioal iformatio o differet approaches to parallelizatio of geetic algorithms ca be foud i [][3][4][3]. II. MOTIVATION AND METHODS The bottleeck of the RCGA learig method for FCMs is its scalability, as the umber of parameters to be established grows quadratically with the map size (umber of cocepts). I additio, geetic optimizatio is time cosumig whe employed to problems with large umber of variables. As a result, RCGA method has bee applied to FCMs which cosist of up to cocepts. This restrictio substatially limits the applicability of this learig approach. Sice there are o other quality iductive techiques, learig of larger size FCMs from data was ot possible. At the same time, i some areas such as systems biology, the uderlyig etworks that could be modeled with FCMs are large ad cosist of at least several dozes of. Moreover, recetly proposed extesio to geeric FCMs, called higher-order fuzzy cogitive maps [8], requires eve more parameters to be established durig the learig process. These issues call for developmet of learig approaches for FCMs that would be fast eough to be applied to larger systems. Therefore, this paper has two mai goals: to propose a time-efficiet learig method for FCMs based o RCGA approach. To satisfy this goal, we use a parallelized approach to real-coded geetic algorithm i order to achieve boost i executio time. I this paper, the global sigle-populatio master-slave method for parallelizatio of GAs is used due to its simplicity. to test the proposed parallelized RCGA learig method o a set of large ad diverse FCMs to assess accuracy of the developed FCMs ad the amout of obtaied speedup (whe compared with sequetial learig). This paper icludes results of experimets with sythetic FCMs that cosist of,,, ad cocepts. Additioally, the tests are carried out with a large real-life map. III. EVALUATION A. Data Sets Similarly to experimets reported i [9], we used both sythetic ad real-life data i our experimets. ) Sythetic data The sythetic data for our experimets were obtaied by simulatig radomly geerated FCMs from radom iitial vectors. Four groups of experimets have bee performed with maps that cosist of,,, ad cocepts, respectively. Additioally, for each group we geerated two series of data with two differet map desities (defied as the ratio of the o-zero relatios weights to the total umber of weights), % ad %, respectively. Thus, as a result, we applied 8 differet experimetal setups. Additioally, for each setup, 5 idepedet maps were geerated to assure statistical validity of the results. 2) Real-life data We selected oe of the largest FCMs foud i literature to prepare the iput data. I this case, the FCM was predefied by the domai experts ad cocered factors that affected slurry rheology [2]. The iput map icluded 3 cocepts: gravity, mechaical properties of particles, physiochemical iteractio, hydrodyamic iteractio, effective particle cocetratio, particle-particle cotact, liquid viscosity, effective particle shape, effective particle size, temperature, iter-particle attractio, floc/structure, ad shear rate. The actual FCM ca be foud i [2]. Its desity is 38.5%. We geerated the iput data from the iitial vector, which was used to aalyze the map i the origial paper. B. Evaluatio Criteria The evaluatio measures of the proposed method are threefold: executio time time eeded to complete the learig i-sample error differece betwee the iput data, ad data geerated by simulatig the cadidate FCM from the same iitial state vector as for the iput data. The criterio is defied as a ormalized average error betwee correspodig cocept values at each iteratio betwee the two state vector sequeces [9]. K N error _ iitial = C ˆ C (2) ( K ) N t= = where C (t) is the value of a ode at iteratio t i the

4 iput data, Cˆ is the value of a ode at iteratio t from simulatio of the cadidate FCM, K is the iput data legth, ad N is the umber of out-of-sample error evaluatio of the geeralizatio capabilities of the cadidate FCM. To compute this criterio, both the iput model ad the cadidate FCMs are simulated from te radomly chose iitial state vectors. Subsequetly, the error_iitial value is computed for each of the simulatios to compare state vector sequeces geerated by the iput ad the cadidate FCM, ad a average of these values is computed [9]. P K N p p error _ behavior = C Cˆ (3) P ( K ) N p= t = = where C p (t) is the value of a ode at iteratio t for data geerated by iput FCM started from p th iitial state vector, Cˆ p is the value of a ode at iteratio t for data geerated by cadidate FCM started from p th iitial state vector, K is the iput data legth, ad N is the umber of, ad P is the umber of differet iitial state vectors. The first criterio is used to test the speed-up i executio time, whereas the other two are used to evaluate quality of the developed map. They are cosistet with the criteria reported i [9]. C. Experimetal Setup The hardware used to execute the experimets was a state-of-the-art 2-way IBM p570 server with POWER5 processors. The code has bee writte i C++ usig OpeMP, which is a applicatio programmig iterface that supports multi-platform shared memory multiprocessig. We repeated each experimet with differet cofiguratio of our hardware usig the followig sceario. We started by performig simulatio o a sigle processor, ad the, we doubled the umber of processors i the subsequet experimets, up to eight processors. As a result, we carried out four idepedet simulatios for each experimet. IV. RESULTS Table I summarizes the experimetal results for the sythetic data. The reported values have bee calculated as averages obtaied from 5 idepedet experimets (with differet models) for each setup. The colums correspod to differet experimetal setups i terms of maps sizes (,,, ad ) ad desities (% ad %), the rows correspod to differet hardware cofiguratios (the first colum defies the umber of processors), ad the value i each cell expresses the average value of a correspodig criterio followed by the stadard deviatio. The criteria are labeled o the right had side of the table. TABLE I EXPERIMENTAL RESULTS FOR SYNTHETIC DATA # % % % % % % % % ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.04 Time [s] I-sample error Out-of-sample error Table II summarizes the experimetal results for the reallife FCM. The reported values have bee calculated as averages obtaied from 5 idepedet experimets for each setup. TABLE II EXPERIMENTAL RESULTS FOR REAL-LIFE DATA Time # [s] 89.6 ± ± ± ±8.62 I-sample error ± ± ± ±0.009 Out-ofsample error 0.66 ± ± ± ±0.059 Aalysis of the results for both sythetic ad real-life data is preseted i the two followig subsectios. A. Executio time Results i Table I show o sigificat ifluece of desity o the executio time. The ifluece of the map s size ad the umber of processors o the executio time is show i Figure 3. The executio time for the two map desities has bee averaged out for each map size. Additioally, the colums are liearly ormalized to the maximum executio time for each setup (which correspods to the sequetial learig). A umber above each bar shows the executio time (i secods).

5 time 0% % 60% % % 0% processor 2 processors 4 processors 8 processors Fig. 3. Executio time vs. experimetal setup I-sample error % % % processor 2 processors 4 processors 8 processors Fig. 4. I-sample error vs. experimetal setup I geeral, parallel learig o eight processors allows doublig the size of the FCM withi virtually the same time that is required for the sequetial learig. Compariso of the results obtaied from simulatios o a sigle processor (traditioal, sequetial learig) to those o two processors shows almost two-fold time-reductio for each setup (time gai varies betwee 43% ad 47%). However, doublig the umber of processors agai does ot result i the correspodig two-fold time reductio. Whe compared with sequetial learig with a sigle processor, the time gai is, o average, 60% ad 69% for four ad eight processors, respectively. This is due to the fact that the executio time of the sequetial costraits of performig geetic operatios o populatio of chromosomes becomes a more sigificat part of the total executio time. This happes as the fitess fuctio evaluatios performed i parallel are completed faster o larger umber of processors. Therefore, i this RCGA parallelizatio approach we observe decreasig returs alog with icreasig the umber of processors. B. Learig quality Figure 4 shows relatio betwee i-sample error ad the size ad desity of FCMs, as well as the umber of processors. A few iterestig coclusios ca be draw from this figure. Firstly, the quality of learig gradually decreases alog with the icreasig iput map size. The isample error value chages from for FCMs with to for FCMs with. This is due to complexity of the optimizatio problem, which drastically icreases with the icrease of the size of maps. However, eve for the largest ivestigated maps the results are still sigificatly better from the baselie of this problem, which has bee experimetally foud at the value of 0.39 (see [9] for details). Secodly, the experimets with differet hardware setups, i.e. differet umber of processors for a certai map size ad desity, are cosistet i terms of the solutio quality. The stadard deviatio of the i-sample error measured for differet umber of processors for a give setup is very small ad varies from ( %) to 0.00 ( %). This is i spite of the fact that the error values differ, which suggests that the learig method fids sub-optimal solutios of similar quality. The last observatio is also cosistet with the coclusios i [9]. Fially, the i-sample error does ot deped o the iput map s desity. For sparser maps, it is oly slightly lower for ad - FCMs, ad slightly higher for ad - FCMs. Figure 5 shows the relatioship betwee out-of-sample error ad the experimetal setup, i.e. the map size ad desity, ad the umber of processors. Out-of-sample error processor 2 processors 4 processors 8 processors 38.5% % % Fig. 5. Out-of-sample error vs. experimetal setup Figure 5 shows that the out-of-sample error slightly icreases as the size of maps icreases. However, it does ot icrease as rapidly for the larger maps as i case of the isample error. The out-of-sample error is, o average, equal to 0.9 for, 0.32 for, 0.44 for, ad 0.56 for. The error values for maps with are cosistet with values reported i [9]. Agai, we stress that the learig results are still of high quality with respect to the baselie value (0.39). The outof-sample error has larger stadard deviatio for experimets with the same map size ad desity performed o differet umber of processors whe compared with correspodig values for the i-sample error, i.e., they differs betwee ( %) ad ( %). This is due to the fact that sub-optimal solutios (FCM models) for the out-of-sample experimets may differ from each other. Thus, eve if all of them provide high quality results o previously see data (i-sample error); these models ca give differet simulatios for ew iitial vectors (out-of-sample error). Lastly, the relatioship betwee the iput map desity ad the out-of-sample error is more cosistet here tha i case of the i-sample error. O average, better quality is obtaied for deser maps i each case, i.e. for,,, ad.

6 We also aalyze relatio betwee the learig quality ad the umber of processors. The differece i the i-sample error betwee experimets repeated with the same setups (o 2, 4, ad 8 processors, respectively) with respect to the sequetial learig is very small (2-3% o average). I case of the out-of-sample error, these values are larger (6-2%), which is due to sub-optimal solutios foud by the RCGA (see the above paragraph). However, whe compared with the stadard deviatios obtaied for the sequetial setup, the differece i errors betwee the o-parallelized ad the parallelized implemetatios is ot sigificat. V. CONCLUSION This paper proposed a ovel parallel approach to learig of FCMs. The method is based o RCGA learig, which has bee previously reported as beig able to develop highquality FCMs from iput data. Our motivatio was to elimiate the mai drawback of the RCGA based method, which is icapability of dealig with larger maps due to high computatioal complexity. Our focus was to propose a solutio that would combie the quality of the RCGA method with a substatial decrease of the executio time. We have proposed, implemeted ad tested a solutio that parallelizes the geetic algorithm, which is the core of the RCGA approach. The experimetal results show that parallelizatio gives substatial improvemet i the executio time. The parallelized learig of FCMs o eight processors was reported to be up to four times faster tha the sequetial learig. The proposed method allows learig maps that iclude several dozes of cocepts i matter of few hours whe usig eight processors. The parallelized RCGA method for learig FCMs has bee tested o both sythetic ad real-life data. The experimetal results show that this method is able to provide high-quality solutios for large FCMs. Both i-sample ad out-of-sample errors icreased with the icreasig map size; however the error values were still substatially smaller tha the baselie error. VI. FUTURE WORK There are several future research directios based o this project. First, it would be iterestig to measure how much of the computatioal time is devoted to do the fitess evaluatio. Secod, a compariso betwee differet methods of RCGA parallelizatio to select the best approach will be ivestigated. Third, we pla to propose a alterative approach to speed up the learig process by exploitig iheret characteristics of FCMs, e.g. by dividig the iput data ito subsets, performig idepedet learig o each subset, ad, fially, mergig the sub-models. Last but ot least, we pla to apply oe of these learig methods to reallife problems, e.g. i the systems biology field. REFERENCES [] E. Alba, F. Lua, A. J. Nebro, ad J. M. Troya, Parallel heterogeeous geetic algorithms for cotiuous optimizatio, Parallel Computig, vol. 30, o. 5 6, pp , 04 [2] G. A. Baii, ad R. A. Bearma, Applicatio of fuzzy cogitive maps to factors affectig slurry rheology, It. Joural of Mieral Processig, vol. 52, o. 4, 998 [3] E. 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