The Essential Particle Swarm. James Kennedy Washington, DC

Transcription

1 The Essential Particle Swarm James Kennedy Washington, DC

2 The Social Template Evolutionary algorithms Other useful adaptive processes in nature Social behavior Social psychology Looks at the individual in a social context People learn from one another Social influence, norm formation, culture Social dynamics are adaptive Latané Social Impact Theory Human subjects research Computer simulations

3 The Particle Swarm A kind of program comprising a population of very simple individuals that interact with one another according to a very simple set of rules in order to solve problems, which may be very complex.

4 Applications You engineers will be glad to know that the particle swarm has been used in dozens, if not thousands, of applications: Cockshott A. R., Hartman B. E., "Improving the fermentation medium for Echinocandin B production. Part II: Particle swarm optimization", Process biochemistry, vol. 36, 2001, p He Z., Wei C., Yang L., Gao X., Yao S., Eberhart R. C., Shi Y., "Extracting Rules from Fuzzy Neural Network by Particle Swarm Optimization", IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, USA, Secrest B. R., Traveling Salesman Problem for Surveillance Mission using Particle Swarm Optimization, AFIT/GCE/ENG/01M-03, Air Force Institute of Technology, Yoshida H., Kawata K., Fukuyama Y., "A Particle Swarm Optimization for Reactive Power and Voltage Control considering Voltage Security Assessment", IEEE Trans. on Power Systems, vol. 15, 2001, p and very many more, in many fields, in many countries.

5 A Particle Has current position: x id This set of variables can be thought of as coordinates of a point in the search space to test a candidate problem solution Has previous best position p id good point in the search space This records a relatively Has velocity v id a new point This is adjusted, and moves the particle to Has previous best function result pbest i And the particle has neighbors who have these same properties

6 Neighborhoods: Population topology Gbest Lbest This is how it works: particles learn from one another. Their communication structure makes a big difference. (All N=20)

7 The Original Particle Swarm For each population member i do Identify best neighbor g For each dimension d do v id = v id + rand() W (p id x id ) + rand() W (p gd x id )) Limit v id to ± Vmax x id =x id + v id Next d If eval(i) < pbest i then do For each dimension d do p id =x id End pbest i =eval(i) End Next i

8 The Canonical Particle Swarm (w/ constriction coefficient) For each dimension d do v id = khi (v id + rand() (phi/2) (p id x id ) + rand() (phi/2) (p gd x id )) x id =x id + v id Next d where khi=0.729 (approx.), phi=4.1 Sometimes g is the global best particle = gbest Sometimes g is the best in i s topological neighborhood=lbest Note no Vmax also the inertia weight version For each dimension d do v id = W1 v id + rand() W2 (p id x id ) + rand() W2 (p gd x id ) xid=xid + vid Next d

9 Test Functions Rosenbrock Sphere Griewank Rastrigin

10 Example: Test Functions in Action

11 Step-Size Movement of the particle through the search space is centered on the mean of p i and p g on each dimension, and its amplitude is scaled to their difference. Exploration vs. exploitation: automatic p i =0 p g =0 p i =+2 p g =-2 p i =+0.1 p g =-0.1

12 phi= phi= phi= phi= phi= phi= phi= phi= phi= Particle Trajectories v=v+phi (p-x) x=x+v

13 Particle Interactions Individual trajectories very weak. Optimization is a function of interparticle interactions. The swarm as a whole, and as an aggregation of subpopulations Effect on trajectory when new bests are found

14 Search distribution Q: What is the distribution of points that are tested by the particle? A: It is a symmetric, bell-shaped distribution around the mean of the previous bests, with s.d. a function of their difference.

15 Bare Bones particle swarm x = G((p i + p g )/2, abs(p i p g )) G(mean, s.d.) is Gaussian RNG Simplified (!) Works pretty well, but not as good as canonical.

16 Woops -- Kurtosis Peaked -- fat tails Tails trimmed Empirical observations with p s held constant Not trimmed

17 Bursts of Outliers Volatility clustering seems to typify the particle s trajectory

18 Adding Bursts of Outliers to Bare Bones PSO Center = (p id + p gd )/2 SD = p id -p gd x id = G(0,1) Sphere Griewank if Burst = 0 and U(0,1)< PBurstStart then Burst = U(0, maxpower) Else If Burst > 0 and U(0,1)< PBurstEnd then Burst = 0 End If If Burst > 0 then x id = x id ^ Burst x id = Center + x id * SD Rosenbrock Rastrigin Griewank10 f (Bubbled line is canonical PS)

19 The Box Where the particle goes next depends on which way it was already going, the random numbers, and the sign and magnitude of the differences. v id = khi (v id + rand() (phi/2) (p id x id ) + rand() (phi/2) (p gd x id )) x id =x id + v id In fact, the area where it can go it crisply bounded. But the probability inside the box is not uniformly dense.

20 Empirical distribution of means from different ranges Simulate with uniform RNG, trim tails

21 TUPS: Truncated-Uniform Particle Swarm Start at current position: x(t) Move weighted amount same direction: W1 (x(t) x(t-1)) Find the length of the side of the box Find the center of the side Generate a uniformly distributed random number around the center, slightly less than the length of the side That s x(t+1)

22 TUPS: Truncated-Uniform Particle Swarm x(t+1)=x(t) + W1 (x(t) x(t-1)) + W2 ((U(-1,+1) (width/2.5)) + center) Sphere Canonical TUPS FIPS Rastrigin W1=0.729; W2=1.494 Width is difference between highest and lowest (p-x) Center is width/2 Generates a point less than Width/2 from the center Griewank Rosenbrock (Behavior is similar to canonical version.) Griewank f

23 FIPS -- The fully-informed particle swarm (Rui Mendes) v(t+1) = W1 v(t) + sum(rand W2 (p k x(t)))/k x(t+1)=x(t)+v(t+1) (K=number of neighbors, k=index of neighbor) Note that p i is not a source of influence in FIPS. Doesn t select best neighbor. Orbits around the mean of neighborhood bests. This version is more dependent on topology.

24 Aspects of Performance Red: Topologies with average degree in the interval (4, 4.25). Green: Topologies with average degree in the interval (3, 3.25) and clustering coefficient in the interval (0.1, 0.6). Blue: Topologies with average degree in the interval (3, 3.25) and clustering coefficient in the interval (0.7, 0.9). Light Blue; Topologies with average degree in the interval (5, 6) and clustering coefficient in the interval (0.025, 0.4). Black: All other topologies.

25 Deconstructing Velocity Because x(t+1) = x(t) + v(t+1) we know that on the previous iteration, x(t) = x(t-1) + v(t) So we can find v(t) v(t) = x(t) x(t-1) and can substitute that into the formula, to put it all in one line: x id (t+1)= x id (t) + W1(x id (t)- x id (t-1)) + Sum(rand() (W2/K) (p kd -x id (t)))

26 In Search of the Essential Particle Swarm We can generalize the canonical and FIPS versions: x id (t+1)= x id (t) + W1(x id (t)- x id (t-1)) + Sum(rand() (W2/K) (p kd -x id (t))) or in words NEW POSITION = CURRENT POSITION + PERSISTENCE + SOCIAL INFLUENCE

27 Social Influence has two components Central Tendency, and Dispersion NEW POSITION= CURRENT POSITION + PERSISTENCE + SOCIAL CENTRAL TENDENCY + SOCIAL DISPERSION Hmmm, this gives us something to play with!

28 Gaussian Essential Particle Swarm Note that only the last term has randomness in it the rest is deterministic do j=1 to &dimen; tem=x{i,j}; meanp=(p{i,j} + p{g,j})/2; disp=abs(p{i,j} - p{g,j})/2; x{i,j}= x{i,j} + &khi*(x{i,j}-xtm1{i,j}) + (&phi/2)*(meanp-x{i,j}) + rannor(0)*disp; NEW POSITION= CURRENT POSITION + PERSISTENCE + SOCIAL CENTRAL TENDENCY + SOCIAL DISPERSION xtm1{i,j}=tem; end;

29 Gaussian Essential Particle Swarm x{i,j}= x{i,j} + &khi*(x{i,j}-xtm1{i,j}) + (&phi/2)*(meanp-x{i,j}) + rannor(0)*disp; Sphere Canonical FIPS GDPS Rastrigin Griewank Rosenbrock Griewank f

30 In Sum There is some fundamental process Uses information from neighbors Seems to require tension between habit and influence Decomposing a version we know is OK We can understand it We can improve it Particle swarms are like an evolutionary algorithm Populations, randomness Members interact over time They are unlike EA s Cooperation, not competition No selection Moves around the target, not toward it Unsuccessful trials are platform for next iteration

31 Send me a note Jim Kennedy Kennedy.Jim@gmail.com