Natural Computing. Lecture 13: Particle swarm optimisation INFR /11/2010

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1 Natural Computng Lecture 13: Partcle swarm optmsaton Mchael Herrmann phone: Informatcs Forum 1.42 INFR /11/2010

2 Swarm ntellgence Collectve ntellgence: A super-organsm emerges from the nteracton of ndvduals The super-organsm has abltes that are not present n the ndvduals ( s more ntellgent ) The whole s more than the sum of ts parts Mechansms: Cooperaton and competton self-organsaton, and communcaton Examples: Socal anmals (ncl. ants), smart mobs, mmune system, neural networks, nternet, swarm robotcs Ben, G., Wang, J.: Swarm Intellgence n Cellular Robotc Systems, Proc. NATO Adv. Workshop on Robots and Bologcal Systems, Tuscany, Italy, 26 30/6 (1989)

3 Swarm ntellgence: Applcaton areas Bologcal and socal modelng Move effects Dynamc optmzaton routng optmzaton structure optmzaton data mnng, data clusterng Organc computng Swarm robotcs

4 Swarms n robotcs and bology Robotcs/AI Man nterest n pattern synthess Self-organzaton Self-reproducton Self-healng Self-confguraton Constructon Bology/Socology Man nterest n pattern analyss Recognzng best pattern Optmzng path Mnmal condtons not what, but why Modelng Dumb parts, properly connected nto a swarm, yeld smart results. Kevn Kelly

5 Complex behavour from smple rules Rule 1: Separaton Avod Collson wth neghborng agents Rule 2: Algnment Match the velocty of neghborng agents Rule 3: Coheson Stay near neghborng agents

6 Towards a computatonal prncple Evaluate your present poston Compare t to your prevous best and neghborhood best Imtate self and others Hypothess: There are two major sources of cognton, namely, own experence and communcaton from others. Leon Festnger, 1954/1999, Socal Communcaton and Cognton

7 Partcle Swarm Optmzaton (PSO) Methods for fndng an optmal soluton to an objectve functon Drect search,.e. gradent free Smple and quas-dentcal unts Asynchronous; decentralzed control Intermedate number of unts: ~ <<23 Redundancy leads to relablty and adaptaton PSO s one of the computatonal algorthms n the feld of swarm ntellgence (another one s ACO) J. Kennedy, and R. Eberhart, Partcle swarm optmzaton, n Proc. IEEE. Int. Conf. on Neural Networks, Pscataway, NJ, pp , 1995.

8 PSO algorthm: Intalzaton Ftness functon Number of partcles n = 20,, 200 Partcle postons Partcle veloctes Current best of each partcle ( smple nostalga ) Global best ( group norm ) Intalze constants f : R m R m x R, = 1,, n v R m, = 1, n, xˆ ĝ ω, α 1/ 2

9 The canoncal PSO algorthm For each partcle For all members of the swarm,.e. create random vectors r 1, r 2 wthcomponents drawn fromu update veloctes v update postons x x + ω v update local bests update global best ( x ) f ( gˆ ) gˆ x f f < v [0,1] 1 n ( x x ) + r ( gˆ x ) ˆ α r α 1 1 xˆ x f f < ( x ) f ( xˆ ) componentwse multplcaton mnmzaton problem!

10 Comparson of GA and PSO Generally smlar: 1. Random generaton of an ntal populaton 2. Calculaton of a ftness value for each ndvdual. 3. Reproducton of the populaton based on ftness values. 4. If requrements are met, then stop. Otherwse go back to 2. Modfcaton of ndvduals In GA: by genetc operators In PSO: Partcles update themselves wth the nternal velocty. They also have memory. Sharng of nformaton Mutual n GA. Whole populaton moves as a group towards optmal area. One-way n PSO: Source of nformaton s only gbest (or lbest). All partcles tend to converge to the best soluton quckly. Representaton GA: dscrete PS: contnuous

11 PSO as MBS As n GA the model s actually a populaton (whch can be represented by a probablstc model) Generate new samples from the ndvdual partcles of the prevous teraton by random modfcatons Use memory of global, neghborhood or personal best for learnng

12 Intalzaton # Intalze the partcle postons and ther veloctes X = lower_lmt + (upper_lmt - lower_lmt) * rand(n_partcles, m_dmensons) assert X.shape == (n_partcles, m_dmensons) V = zeros(x.shape) # Intalze the global and local ftness to the worst possble ftness_gbest = nf ftness_lbest = ftness_gbest * ones(n_partcles) w=0.1 # omega range a1=a2=2 # alpha range 0 4, both equal n=25 # range max velocty # no larger than: range of x per step or 10-20% of ths range Man loop (next page)

13 for k = 1.. T_teratons: # loop untl convergence ftness_x = evaluate_ftness(x) # evaluate ftness of each partcle for I = 1.. n_partcles: # update local bests f ftness_x[i] < ftness_lbest[i]: ftness_lbest[i] = ftness_x[i] for J = 1.. m_dmensons: X_lbest[I][J] = X[I][J]; end J; end l; mn_ftness_ndex = argmn(ftness_x) # update global best mn_ftness = ftness_x[mn_ftness_ndex] f mn_ftness < ftness_gbest: ftness_gbest = mn_ftness; X_gbest = X[mn_ftness_ndex,:] for I = 1.. n_partcles: # update veloctes and postons for J = 0.. m_dmensons: R1 = unform_random_number() R2 = unform_random_number() V[I][J] = (w*v[i][j] + a1*r1*(x_lbest[i][j] - X[I][J]) + a2*r2*(x_gbest[j] - X[I][J])) X[I][J] = X[I][J] + V[I][J] end I, end J, end k;

14 Illustratve example Marco A. Montes de Oca PSO Introducton

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16 Exploratory behavour: Search a broad regon of space Explotatve behavour: Locally orented search to approach a (possbly local) optmum Parameters to be chosen to properly balance between exploraton and explotaton,.e. to avod premature convergence to a local optmum yet stll ensure a good rate of convergence to the optmum. Convergence How does t work? Exploraton: Swarm collapses (or rather dverges, oscllates, or s crtcal) Explotaton: Global best approaches global optmum (or rather, for a collapse of the swarm, a local optmum) Mathematcal attempts (typcally oversmplfed): Convergence to global optmum for a 1-partcle swarm after nfnte tme (F. v. d. Bergh, 2001) see PSO at en.wkpeda.org

17 Repulsve PSO algorthm For each partcle 1 n create random vectors r 1, r 2, r 3 wth components drawn from U[0,1] update veloctes ŷ best of random neghbors, α 2 <0 z random velocty componentwse multplcaton update postons etc. Propertes: sometmes slower, more robust and effcent

18 Constrcton factor n canoncal PSO Introduced by Clerc (1999) Smplest form: May replace ntera ω Meant to mprove convergence by an enforced decay (more about ths later)

19 Topology: Restrcted competton/coordnaton Topology determnes wth whom to compare and thus how solutons spread through the populaton Tradtonal ones: gbest, lbest Global verson s faster but mght converge to local optmum for some problems. Local verson s a somewhat slower but not easy to be trapped nto local optmum. Combnaton: Use global verson to get rough estmate. Then use local verson to refne the search. For some topologes analogous to slands n GA

20 Innovatve topologes Specfed by: Mean degree, clusterng, heterogenety etc.

21 Comparson of GA and PSO Generally smlar: 1. Random generaton of an ntal populaton 2. Caclulate of a ftness value for each ndvdual. 3. Reproducton of the populaton based on ftness values. 4. If requrements are met, then stop. Otherwse go back to 2. Modfcaton of ndvduals In GA: by genetc operators In PSO: Partcles update themselves wth the nternal velocty. They also have memory. Sharng of nformaton Mutual In GA. Whole populaton moves as a group towards optmal area. One-way n PSO: Source of nformaton s only gbest (or lbest). All partcles tend to converge to the best soluton quckly. Representaton GA: dscrete PS: contnuous

22 Lterature on swarms Erc Bonabeau, Marco Dorgo, Guy Theraulaz: Swarm Intellgence: From Natural to Artfcal Systems (Santa Fe Insttute Studes on the Scences of Complexty) OUP USA (1999) J. Kennedy, and R. Eberhart, Partcle swarm optmzaton, n Proc. of the IEEE Int. Conf. on Neural Networks, Pscataway, NJ, pp , Y Sh, RC Eberhart (1999) Parameter selecton n partcle swarm optmzaton. Sprnger. Eberhart Y. Sh (2001) PSO: Developments, applcatons ressources. IEEE. Tutorals: Bblography: cdweb.cc.purdue.edu/~hux/pso.shtml