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Hindawi Publishing Corporation Journal of Artificial Evolution and Applications Volume 2008, Article ID 654184, 10 pages doi:10.1155/2008/654184 Research Article A Simplified Recombinant PSO Dan Bratton and Tim Blackwell Department of Computing, Goldsmiths, University of London, New Cross, London SE14 6NW, UK Correspondence should be addressed to Dan Bratton, d.bratton@gold.ac.uk Received 20 July 2007; Accepted 3 December 2007 Recommended by Riccardo Poli Simplified forms of the particle swarm algorithm are very beneficial in contributing to understanding how a particle swarm opti- mization (PSO) swarm functions. One of these forms, PSO with discrete recombination, is extended and analyzed, demonstrating not just improvements in performance relative to a standard PSO algorithm, but also significantly different behavior, namely, a reduction in bursting patterns due to the removal of stochastic components from the update equations. Copyright © 2008 D. Bratton and T. Blackwell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION native topologies and parameter settings, running compar- isons over a more comprehensive test suite, deriving simpli- Originally conceived as a modification to the standard PSO fied variants of the algorithm, and subjecting the model to a algorithm for use on self-reconfigurable adaptive systems burst analysis. used in on-chip hardware processes, PSO with discrete re- The following section introduces PSO-DR (known here combination (PSO-DR) introduces several appealing and ef- as model 1) as originally defined by Pena ˜ et al. and summa- fective modifications, resulting in a simpler variant of the rizes the burst analysis of [2]. Section 3 describes a series of original [1]. It is one of the more interesting advances in PSO simplifications to PSO-DR (models 2 and 3) which are intro- research over the last few years because these simplifications duced in this paper. The motivations for these simplifications apparently do not degrade performance yet they remove var- are explained. Section 4 presents the results of performance ious issues associated with the stochasticity of the PSO accel- experiments of models 1–3, and for comparative purposes, eration parameters that hinder theoretical analysis of PSO. standard PSO. Following this, the paper proceeds with an empirical investigation of bursting patterns in recombinant Physical creation of hardware-based optimizers is a sub- stantially more intricate undertaking than software imple- PSO. The final section together draws together the experi- mentations, so fast, simple algorithms are desirable in or- mental results of this paper and advances some ideas for the der to minimize complexity. The comparative straightfor- immediate future of PSO research. wardness of PSO to many other evolutionary optimization algorithms makes it a good choice for this purpose, and fur- 2. PSO WITH DISCRETE RECOMBINATION ther modifications were applied by the authors of [1]inor- der to simplify it even further and to introduce concepts The velocity update for particle i in standard PSO (SPSO) in from recombinant evolutionary techniques. The resulting al- the inertia weight formalism is gorithm, which can be implemented using only addition and subtraction operators and a simple 1-bit random number φ φ generator, is well suited for dedicated hardware settings. t+1 t t t IW : v = wv + u p − x + u p − x ,(1) 1 id 2 nd id id id id 2 2 Despite this rather specific original design specification, PSO-DR has shown to be a robust optimizer in its own right, equalling or surpassing a more common PSO implementa- where d labels components of the position and velocity tion on a few tested benchmarks [1]. In this paper we ex- vectors, d = 1, 2,... , D, p is the personal best position tend the original work of Pena ˜ et al. by considering alter- achieved by i, p is the best position of informers in i’s social n 2 Journal of Artificial Evolution and Applications neighborhood and u ∼U (0, 1) [3]. After velocity update, which is equivalent to SPSO with the identification a(t) = 1,2 the particle position is adjusted: (φ/2)(u + u )− w− 1, b = w,and c(t) = (φ/2)(u p + u p ) 1 2 1 1 2 2 for fixed attractors p . Since max(|a|) > 0, amplification of 1,2 t+1 t+1 t x = v + x . (2) id id id x(t) can occur through repeated multiplication of x(t)by a despite the second order reduction by multiplication by the Pena ˜ et al. introduced a recombinant version of PSO by constant b. Interestingly, the distribution tail of |x|,byvirtue replacing either the personal best or the neighborhood best of the bursts that become increasingly less probable for in- position by the recombinant position [1]. We focus here on creasing size, is fattened compared to an exponential falloff the former for reasons of improved performance and the as provided by, for example, a Gaussian. A theoretical justi- more interesting social aspect. A recombinant position vec- fication of these power laws and some empirical tests can be tor r is defined by found in [2]. PSO bursts differ from the random outliers generated by r = η p + 1 − η p,(3) id ld rd d d PSO models which replace velocity by sampling from a dis- tribution with fat tails such as a Richer and Blackwell [5]. In where η = U{0, 1} and p are immediate left and right l,r contradistinction to the outliers of these “bare bones” for- neighbors of i in a ring topology. While separate random mulations [6], the outliers from bursts occur in sequence, numbers η are used for separate dimensions d,asingle value and they are one dimensional. Bursting will therefore pro- is generated for each single dimension and used for both oc- duce periods of rectilinear motion where the particle will currences of η in that dimension. This places r at a corner have a large velocity parallel to a coordinate axis. Further- of the smallest D-dimensional box which has p and p at its l r more, large bursts may take the particle outside the search corners. space. Although this will not incur any penalty in lost func- The authors of [1], in a search for a very efficient imple- tion evaluations if particles that exit the feasible bounds of mentation, argued for the removal of the random numbers the problem are not evaluated, as is the common approach to u from (1) and parameter settings φ = 2and w = 0.5. The 1,2 this situation, they are not contributing to the search while in velocity update for the original form of PSO-DR is outer space. PSO-DR, which is predicted not to have bursts φ φ [2], therefore provides a salient comparison. t+1 t t t DR : v = wv + r − x + p − x . (4) id nd id id id id 2 2 3. SIMPLIFYING RECOMBINANT PSO The choice of φ was based on the observation that φ ≈ 4.0 in standard PSO, but, since u are uniform in [0, 1], the ex- 1,2 This section details the two new recombinant models that are pectation value of φu is 2.0. Furthermore, the multipli- 1,2 being proposed in this paper. To begin, an investigation into cation by w = 0.5 can be implemented in hardware by a PSO-DR reveals more interesting properties of the formula- right shift operation. While optimal efficiency is desirable for tion. Performance plots for a sweep through parameter space hardware implementations, this issue does not concern us to to find an optimal balance between the inertia weight coef- the same degree in this study of (4) and it is one aim of this ficient w and the φ coefficients show that while the optimal paper to study PSO-DR for arbitrary parameter values. region is spread across the parameter space, it also intersects Although (4) contains a random element in the recom- the axis for the w term (see Figure 1 for results on selected binant position, the acceleration parameters are constant. In functions from Table 1). This demonstrates that the system other words, the update rule has additive rather than mul- is able to obtain good optimal results even at w = 0.0and tiplicative stochasticity [2]. This has two ramifications; first, there is no inertia term in the velocity update equations. a stability condition can be computed based on the theory Model 2 PSO-DR sets w = 0, with a velocity update, of second order, fixed parameter, difference equations and second, recombinant PSO is predicted not to exhibit parti- φ φ t+1 t t cle velocity bursts. The details of these results are to be found DR2 : v = r − x + p − x . (7) id nd id id id 2 2 in [2]. The stability condition is Velocity now serves as a dummy variable in the update |w| < 1, 0 <φ < 2(1 + w). (5) equations (1)and (2) and model 2 can be represented as a It is known that PSO at stagnation, that is, when no im- single, velocity-free rule provements to personal bests are occurring, and the particles φ φ effectively decouple, exhibits bursts of outliers [4]. These are t+1 t t t DR2 : x = x + r − x + p − x . (8) id nd id id id id temporary excursions of the particle to large distances from 2 2 the attractors. A burst will typically grow to a maximum and At this point, the two φ terms were detached and another then return through a number of damped oscillations to the sweep through parameter space to find an optimal combina- region of the attractors. The origin of bursts, and of the con- tion of the recombinant component via its coefficient φ and comitant fattening of the tails of the position distribution at the neighborhood best component via its coefficient φ was stagnation, can be traced to the second-order stochastic dif- 2 performed. Surprisingly, results again showed that the opti- ference equation mal region intersects an axis, this time for the neighborhood term (p − x ) (see Figure 2 for selected results). x(t +1)+ a(t)x(t)+ bx(t − 1) = c(t)(6) gd id D. Bratton and T. Blackwell 3 Contour plot for PSODR performance on f Contour plot for performance on f 1 3 5 5 4 4 3 3 2 2 1 1 0 0 00.25 0.50.75 1 00.25 0.50.75 1 w w (a) f (b) f 1 3 Contour plot for PSODR performance on f Contour plot for PSODR performance on f 5 12 5 5 4 4 3 3 2 2 1 1 0 0 00.25 0.50.75 1 00.25 0.50.75 1 w w (c) f (d) f 5 12 Figure 1: Optimal regions for w versus φ in PSO-DR model 1, found empirically through 30 runs for each combination of parameters w = 0.0,... ,1.0, φ = 0.0,... ,5.0 at a granularity of 0.1. Each contour line represents a 10% improvement in performance with the region within the innermost line representing the best performing 10% of possible combinations of w and φ. This allows a further simplification to the update equa- randomly chosen neighbor to fully influence the particle in tion (4), down to PSO-DR model 3: each dimension. This gives the particle an updated position that is a combination of the best positions of all of its neigh- t+1 t t DR3 : x = x + φ r − x (9) id id id id bors throughout all dimensions. The following section presents evidence that PSO-DR3 is which is clearly a substantial reduction of the original PSO- a viable alternative to standard PSO by reporting on perfor- DR equation.This PSO variant, if it proves to be viable, mance results for all three models of PSO-DR over a number would raise a couple of interesting questions.To what ex- of commonly used test functions. tent is velocity a necessary component, or is it a relic of the biological origins of PSO [6]? Secondly, how important is the neighborhood component drawn from the single best 4. PERFORMANCE EXPERIMENTS neighbor? The optimization process of Model 3 is entirely driven by the recombinant component; this idea is reminis- Algorithms were tested over a series of 14 benchmark func- cent of fully informed particle swarms (FIPS) [7], where the tions chosen for their variety, shown in Tables 1 and 2.Func- entire neighborhood influences particle behavior. However, tions f − f are unimodal functions with a single mini- 1 3 whereas FIPS allows every neighbor to influence a particle’s mum, f − f are complex high-dimensional multimodal 4 9 behavior in every dimension, Model 3 allows only a single problems, each containing many local minima and a single φ (phi) φ (phi) φ (phi) φ (phi) 4 Journal of Artificial Evolution and Applications Table 1: Benchmark function equations. Equation f = x i=1 D i f = x 2 j i=1 j=1 D−1 2 2 f = 100 x − x + x − 1 3 i+1 i i=1 f =− x sin x 4 i i i=1 f = x − 10 cos 2πx +10 5 i i=1 1 1 f =−20 exp −0.2 x − exp cos 2πx +20+ e 6 i i=1 D D i=1 D D 1 x f = x − cos +1 4000 i i=1 i=1 D−1 2 2 2 2 f = 10 sin πy + y − 1 1+10 sin πy + y − 1 8 i i i+1 D i=1 + μ x , 10, 100, 4 i=1 y = 1+ x +1 i i 4 ⎧ ⎪ k x − a x >a i i μ x , a, k, m = i 0 −a ≤ x ≤ a ⎪ m k − x − a x < −a i i D−1 2 2 2 2 f = 0.1 sin 3πx + x − 1 1+sin 3πx + x − 1 9 i i i+1 D i=1 × 1+sin 2πx + μ x , 5, 100, 4 D i i=1 2 4 6 2 4 f = 4x − 2.1x + x + x x − 4x +4x 10 1 2 1 1 1 2 2 2 2 f = 1+ x + x +1 19 − 14x +3x − 14x +6x x +3x 11 1 2 1 2 1 2 1 2 2 2 × 30 + 2x − 3x 18 − 32x +12x +48x − 36x x +27x 1 2 1 1 2 1 2 2 −1 5 4 f =− x − a + c 12 j ij i i=1 j=1 −1 7 4 f =− x − a + c 13 j ij i i=1 j=1 −1 10 4 f =− x − a + c 14 j ij i i=1 j=1 global optimum, and f − f are lower-dimensional mul- of each dimension was defined, into which all particles were 10 14 timodal problems with few local minima and a single global placed with uniform distribution. This method ensures that optimum apart from f , which is symmetric about the ori- the swarm will not be initialized within the same area for ev- gin with two global optima. ery optimization run, but will still be confined to an area at Particles were initialized using the region scaling tech- most 0.25 of the search space, making the chance of ini- nique where initialization takes place in an area of the search tialization directly on or near the global optimum extremely space known not to contain the global optimum [8]. To unlikely. In instances where the global optimum was located avoid initializing the entire swarm directly within a local at the center of the search space (i.e., f , f , f − f ), the func- 1 2 5 7 minimum, as could be possible with f − f if initializa- tion was shifted by a random vector with maximum mag- 12 14 tion takes place in the bottom quarter of the search space nitude of a tenth of the size of the search space in each di- in each dimension (as is common), an area of initialization mension for each run to remove any chance of a centrist bias composed of the randomly chosen top or bottom quarter [9]. φ (recombinant term) φ (recombinant term) D. Bratton and T. Blackwell 5 3D plot for performance on f 3D plot for performance on f ×10 5 30 0 20 5 0 10 20 15 15 0 25 15 5 30 40 40 0 0 45 (a) f (b) f 1 5 Figure 2: Performance plots for φ and φ in PSO-DR model 2, found empirically through 30 runs for each combination of parameters 1 2 φ = 0.0,... ,4.5, φ = 0.0,... ,4.5 at granularity 0.1. 1 2 Table 2: Benchmark function details. Function Name D Feasible bounds f Sphere/parabola 30 (−100, 100) f Schwefel 1.2 30 (−100, 100) f Generalized Rosenbrock 30 (−30, 30) f Generalized Schwefel 2.6 30 (−500, 500) f Generalized Rastrigin 30 (−5.12, 5.12) f Ackley 30 (−32, 32) f Generalized Griewank 30 (−600, 600) f Penalized function P8 30 (−50, 50) f Penalized function P16 30 (−50, 50) f Six-hump camel-back 2 (−5, 5) f Goldstein-price 2 (−2, 2) f Shekel 5 4 (0, 10) f Shekel 7 4 (0, 10) f Shekel 10 4 (0, 10) This investigation tested PSO-DR model 1 using both velocity update equation global (as used in the originally proposed algorithm) and lo- φ φ cal ring topologies for selecting the neighborhood operator v = χ v + u p − x + u p − x (11) t+1 t 1 i t 2 g t 2 2 p . The parameter settings were Pena’s, giving a velocity up- date with the form with φ = 4.1, χ = 0.72984 and with 50 particles [3]. All PSO- DR model tests were carried out using 50 particles as well. t+1 t t t v = 0.5v + r − x + p − x . (10) id nd Algorithm performance was measured as the minimum error id id id id | f (x)− f (x )| found over the trial where f (x ) is the fitness Results shown for PSO-DR model 2 use the value φ ≈ at the global optimum for the problem. Results were averaged 1.6, while those for PSO-DR model 3 use φ ≈ 1.2. These val- over 30 independent trials, and are displayed, with standard −15 ues were empirically determined to be optimal for these algo- error, in Table 3. Values less than 10 have been rounded to rithms; an analytical determination is the subject of current 0.0. research. Results for both models 2 and 3 are shown for runs Performance results in Table 3 for all models of PSO-DR using a ring topology, which showed superior performance versus SPSO clearly indicate that it is a competitive variant, in testing. especially on highly complex problems such as f (Rastrigin). For comparison, results are presented for a standard Statistical tests were performed on these results to determine PSO algorithm (SPSO), which operates using the constricted the significance of the performance differences between the φ (neighborhood best) φ (neighborhood best) Best found fitness Best found fitness 6 Journal of Artificial Evolution and Applications Table 3: Mean error after 30 trials of 300 000 evaluations. Necessary function evaluations are shown where 0.0 error was attained. SPSO SPSO PSO-DR PSO-DR PSO-DR PSO-DR Ring Global M1 Ring M1 Global M2 M3 0.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.0 97063 ± 377 46897 ± 421 59322 ± 125 33290 ± 170 60063 ± 41 75810 ± 322 0.12 ± 0.01 0.0 ± 0.0 0.01 ± 0.002 0.0 ± 0.0 3.7E-7 ± 7.5E-8 5.14 ± 1.27 — 297800 ± 928 — 168852 ± 1205 — — 6.18 ± 1.07 8.37 ± 2.26 16.79 ± 0.49 0.80 ± 0.29 34.57 ± 5.46 18.64 ± 4.45 ——— — — — 3385 ± 40 3522 ± 32 2697 ± 36 3754 ± 48 2418 ± 27 1830 ± 46 ——— — — — 163.50 ± 5.64 140.16 ± 5.87 44.64 ± 2.71 115.51 ± 7.03 35.21 ± 2.13 9.88 ± 0.86 ——— — — — 18.28 ± 0.85 12.93 ± 1.59 0.68 ± 0.67 18.51 ± 0.90 0.0 ± 0.00.0 ± 0.0 — — — — 287220 ± 2105 248160 ± 1945 0.0 ± 0.0 0.019 ± 0.004 0.0 ± 0.0 0.008 ± 0.002 0.0 ± 0.00.0 ± 0.0 110616 ± 3320 — 101526 ± 9227 — 81226 ± 6560 70348 ± 2954 0.004 ± 0.003 0.15 ± 0.05 0.0 ± 0.0 0.05 ± 0.02 0.0 ± 0.00.0 ± 0.0 — — 61370 ± 249 — 85101 ± 581 95810 ± 655 0.0 ± 0.0 0.003 ± 0.001 0.0 ± 0.0 0.002 ± 0.0007 0.0 ± 0.00.0 ± 0.0 106163 ± 537 — 61793 ± 221 — 86031 ± 377 92416 ± 437 0.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.0 9348 ± 190 11808 ± 445 44577 ± 7608 40015 ± 4483 6103 ± 104 5918 ± 75 0.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.00.0 ± 0.0 8258 ± 104 7080 ± 108 4772 ± 47 3968 ± 27 6720 ± 66 6846 ± 87 0.59 ± 0.33 4.61 ± 0.54 0.17 ± 0.17 4.34 ± 0.59 0.0 ± 0.00.0 ± 0.0 — — — — 59958 ± 43 16229 ± 855 1.09 ± 0.45 4.40 ± 0.60 0.0 ± 0.0 2.55 ± 0.62 8.1E-11 ± 1.8E-11 0.0 ± 0.0 — — 13433 ± 249 — — 14012 ± 764 0.96 ± 0.45 3.24 ± 0.66 0.0 ± 0.0 3.13 ± 0.66 6.6E-11 ± 1.7E-11 0.0 ± 0.0 — — 12760 ± 1386 — — 121031004 ± two algorithms. To avoid the problem of the probabilistic na- ous work (where only 30 k–60 k function evaluations might ture of t-tests potentially affecting results when conducting be performed), selected convergence plots are shown in multiple significance tests, a modified Bonferroni procedure Figure 3. These show that the standard PSO obtains supe- was applied to values of α for successive tests [10]. This pro- rior results at the very start of the optimization process, cedure involves inversely ranking observations by ascending up to 5000 function evaluations for the highest observed values of p, then setting value (Figure 3(b)). After the point at which this occurs, PSO-DR model 3 surpasses the standard algorithm in per- formance, and maintains this advantage to the end of the α = . (12) 300 k function evaluations on 7 of the 14 tested problems inverse rank ( f − f , f , f − f ). On problems for which both algo- 4 6 8 12 14 Results for these statistical tests on PSO-DR model 3 and rithms attained equal error levels of 0.0 ( f , f , f − f ), the 1 7 9 11 point at which this occurs, that is, when SPSO “catches up” SPSO are shown in Table 4 and confirm that the performance is significantly improved on 3 of the 14 tested functions, to PSO-DR model 3, can be observed in Table 3.Onav- equivalent for 10 functions, and worsened for 1 function for erage, SPSO took 25% more function evaluations to attain PSO-DR model 3 versus SPSO with ring topology. Perhaps the optimum than PSO-DR model 3 on these problems. Fi- the most impressive improvement comes for f (Rastrigin), nally, for the two problems on which SPSO outperformed PSO-DR model 3 ( f , f ), the same early performance is seen a notoriously difficult multimodal problem that PSO algo- 2 3 rithms perform poorly on some problems in high dimen- with PSO-DR model 3 surpassing SPSO in performance early sionality. in the optimization process; in these cases, SPSO eventu- ally repasses the other algorithm by 50 k function evalua- Due to the high number of function evaluations that were performed to obtain these results relative to previ- tions. D. Bratton and T. Blackwell 7 Table 4: Significance for SPSO versus PSO-DR model 3 with ring topologies. Function p-value Inverse rank α Significant f 0 14 0.003 571 Yes f 0 13 0.003 846 Yes f 0 12 0.004 167 Yes f 2.11e-11 10 0.005 Yes f 0.0086 11 0.004 545 No f 0.02 9 0.005 556 No f 0.04 8 0.00 625 No f 0.2663 6 0.08 No f 0.3215 7 0.007 143 No f 1 5 0.01 No f 1 4 0.0125 No f 1 3 0.016 667 No f 1 2 0.025 No f 1 1 0.05 No ×10 f PSODR model 3 versus standard PSO convergence f PSODR model 3 versus standard PSO convergence 1 5 7 450 0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Function evaluations Function evaluations PSODR m3 PSODR m3 SPSO SPSO (a) f (b) f 1 5 Figure 3: Convergence plots for SPSO and PSO-DR model 3 early in the optimization process. A potential explanation for this behavior lies in the tialization as well, but this is inherent to the swarm be- diversity of the swarms at this point in the optimization havior and influenced only by the size of the entire search process. Figure 4 shows the mean Euclidean distance be- space. tween particles for the corresponding convergence plots of As can be seen in the plots of Figure 4, neither swarm Figure 3. It should be noted that uniform initialization was type begins converging immediately following initialization used in the trials used to generate these plots; relative per- but rather they maintain their diversity or expand slightly. formance between the algorithms was unaffected, and ini- On a comparative basis, the standard PSO swarm expands tializing particle positions uniformly throughout the search substantially more than the PSO-DR model 3 swarm; for ex- space removes an unrelated phenomenon in subspace ini- ample Figure 4(c) shows that after the first 100 function eval- tialization wherein the swarm expands greatly beyond the uations, the mean distance between particles in the standard relatively small initialization region at the start of the op- PSO swarm increases from 23 to 31.5, while the PSO-DR timization process to explore the search space. Expansion swarm diversity increases only from 23 to 24.5. Similar dis- is common in the first few iterations using uniform ini- parities were observed for all other tested problems. Error Error 8 Journal of Artificial Evolution and Applications f PSODR model 3 versus standard PSO diversity f PSODR model 3 versus standard PSO diversity 0 10000 20000 30000 40000 50000 0 1000 2000 3000 4000 5000 Function evaluations Function evaluations PSODR m3 PSODR m3 SPSO SPSO (a) f (b) f 1 5 f PSODR model 3 versus standard PSO diversity 0 500 1000 Function evaluations PSODR m3 SPSO (c) f (zoomed) Figure 4: Diversity plots for SPSO and PSO-DR model 3 early in the optimization process. It is reasonable to gather from these results that the means of the multiplicative stochasticity of the algorithm higher swarm diversity for the standard PSO algorithm early [2]. In order to investigate bursting behavior in PSO-DR and in the optimization process demonstrates a wider spread of SPSO an empirical measure was devised. particle dispersion, and hence an improved probability of This bursting measure was implemented to highlight finding and starting to explore the basin of attraction for when a particle had a velocity in a single dimension that was global or good local optima. PSO-DR model 3 expands very considerably higher than the next highest dimensional veloc- little, if at all, early in the optimization process, resulting in ity. Bursting patterns of behavior were detected by reporting delayed acquisition of optimal regions of the search space. that every time particle velocity in a single dimension was a set amount λ times higher than velocity in the next highest dimension. Bursting behavior is demonstrated in Figure 6, 5. EXAMINATION OF BURSTING where the velocity of a single particle in a 10-dimensional Bursts in the velocities of particles are commonly observed problem is shown. On the plot of the multidimensional ve- using the standard PSO algorithm. These are generated by locity of the SPSO particle, it can be seen that velocity in a Mean distance between particles Mean distance between particles Mean distance between particles D. Bratton and T. Blackwell 9 Percentage of updates showing burst patterns of behavior on f Bursting behavior for SPSO on 10-dimensional Rastrigin 9 20 8 18 5 12 4 10 3 8 1 4 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 0 50 100 150 200 250 300 350 400 λ Iterations (a) SPSO SPSO PSODR Bursting behavior for PSODR on 10-dimensional Rastrigin Figure 5: Frequency of updates showing burst behavior for values of λ. single dimension increases suddenly and dramatically while remaining relatively level and low in all other dimensions. This is an example of a velocity burst. While the figure shows velocity for a single particle on a single run, examination of velocity plots for hundreds of particles over dozens of runs confirmed this to be representative of general particle behav- 4 ior. Velocity for a PSO-DR particle is also shown in Figure 6, and demonstrates the absence of bursts. Similarly to the 0 50 100 150 200 250 300 350 400 SPSO plot, examination of a large number of plots confirmed Iterations this to be representative of general behavior for PSO-DR. (b) PSO-DR Examination of these empirical analyses show that PSO- DR clearly does not contain bursting behavior on the scale Figure 6: Representative particle velocities for SPSO and PSO-DR of SPSO while demonstrating equal or superior performance on 10D Rastrigin. on 13 of the 14 benchmark functions, leading to the hypoth- esis that bursts are not, in fact, integral to the successful op- eration of particle swarm algorithms. The fact that a very few 0.91% improve g for SPSO compared with 0.02% for PSO- bursts do occur with PSO-DR indicates that it is a highly im- DR on f for λ = 100. probable feature of DR dynamics. Analysis performed on statistics of several functions 6. CONCLUSIONS shows that particle updates involving bursts are far less effec- tive than more common nonbursting updates. For example, Simplification of the standard PSO algorithm is an impor- results showed that for SPSO on f with λ = 100, on aver- tant step toward understanding how and why it is an effec- age 20.1% of all particle, updates involve an improvement tive optimizer. By removing components of the algorithm to the particle’s best found position p , whereas only 1.8% and seeing how this affects performance, we are granted in- of updates involving bursts result in an improvement to p . sight into what those components contribute to overall par- Likewise, on average 0.9% of all particle, updates improve ticle and swarm behaviors. the best found swarm position g,asopposed to only 0.01% In particular, this paper has proposed a very simple PSO for bursting particles. Burst frequencies for values of λ from t+1 t t 10 to 150 are shown in Figure 5. DR3 : x = x + φ r − x (13) id id id id It is also interesting to note that far fewer total updates result in an improved p or g for PSO-DR when compared to which offers competitive performance to standard PSO, but SPSO, for example, results showed that 20.1% of all updates removes multiplicative randomness, inertia, and the personal improve p for SPSO compared with 0.64% for PSO-DR, and memory term p from the position update. i i (%) Velocity Velocity 10 Journal of Artificial Evolution and Applications There is still much to be done before questions concern- gence Symposium (SIS ’07), pp. 120–127, Honolulu, Hawaii, USA, April 2007. ing PSO behavior can be completely answered, and it is ex- [4] J. Kennedy, “Probability and dynamics in the particle swarm,” pected that the next decade of PSO research will be focused in Proceedings of the Congress on Evolutionary Computation on understanding the basic algorithm that powers both the (CEC ’04), vol. 1, pp. 340–347, IEEE Press, Portland, Ore, standard implementation and its variants. USA, June 2004. In that light, the PSO-DR variant is important not only [5] T. J. Richer and T. Blackwell, “The Levy ´ particle swarm,” in because of its improved performance on several benchmark Proceedings of the IEEE Congress on Evolutionary Computation functions, but also because its simplified state allows us to ex- (CEC ’06), pp. 808–815, IEEE Press, Vancouver, BC, Canada, amine what happens to the standard algorithm when pieces July 2006. are modified or removed. Based on the results presented [6] J. Kennedy, “Bare bones particle swarms,” in Proceedings of here, it can be argued that large bursts are not generally IEEE Swarm Intelligence Symposium (SIS ’03), pp. 80–87, In- beneficial or integral to PSO performance, and may possi- dianapolis, Ind, USA, April 2003. [7] R. Mendes, J. Kennedy, and J. Neves, “The fully informed par- bly be detrimental. Although the presence of particle out- ticle swarm: Simpler, maybe better,” IEEE Transactions on Evo- liers is demonstrably important for swarm optimization (as lutionary Computation, vol. 8, no. 3, pp. 204–210, 2004. demonstrated in bare bones analysis, [6]), bursts, which are [8] D. K. Gehlhaar and D. B. Fogel, “Tuning evolutionary pro- sequences of extreme particle positions, occurring along an gramming for conformationally flexible molecular docking,” axis and reaching outside the search space, remain a special in Proceedings of the 5th Annual Conference on Evolutionary feature of velocity-based swarms. This work, which compares Programming (EP ’96), L. Fogel, P. Angeline, and T. Back, standard PSO to a burst-free but comparable optimizer sug- Eds., pp. 419–429, MIT Press, San Diego, Calif, USA, Febru- gests that bursts are disadvantageous in general. (However, in ary 1996. the coincidence that the objective function has a rectangular [9] C. K. Monson and K. D. Seppi, “Exposing origin-seeking bias symmetry aligned with the axes, then bursting may actually in PSO,” in Proceedings of the Conference on Genetic and Evolu- be fortuitous.) tionary Computation (GECCO ’05), pp. 241–248, Washington, Further, the replacement of the direct personal influence DC, USA, June 2005. [10] J. Jaccard and C. K. Wan, LISREL Approaches to Interaction Ef- operator p from SPSO with the recombinant term r derived i i fects in Multiple Regression, vol. 114 of Quantitative Applica- from its neighborhood in PSO-DR strengthens the case for tions in the Social Sciences, Sage Publications, Thousand Oaks, PSO being mostly reliant on social interaction as opposed to Calif, USA, 1996. personal experience. This is further supported by the effec- tiveness of PSO-DR model 3, which lacks a cognitive term altogether. The social behavior occurring inside of a swarm is still a wide-open area in the field, and will hopefully con- stitute a great deal of the future research devoted to the devel- opment of a better understanding of this deceptively simple optimizer. Another property of PSO-DR resides in attractor jiggling that takes place even at stagnation (no updates to any p ) since r is never fixed. This jiggling will work against conver- gence and could propel the swarm onwards. This, and other matters concerning the nature of recombination within PSO, will be the subject of further study. ACKNOWLEDGMENTS The authors would like to acknowledge the support of EP- SRC XPS Project (GR/T11234/01). The authors also wish to thank Jim Kennedy for advice and references. REFERENCES [1] J. Pena, ˜ A. Upegui, and E. Sanchez, “Particle swarm opti- mization with discrete recombination: an online optimizer for evolvable hardware,” in Proceedings of the 1st NASA/ESA Con- ference on Adaptive Hardware and Systems (AHS ’06), pp. 163– 170, Istanbul, Turkey, June 2006. [2] T.Blackwell andD.Bratton,“Origin of bursts,” in Proceed- ings of the 9th Annual Conference on Genetic and Evolutionary Computation (GECCO ’07), pp. 2613–2620, London, UK, July [3] D. Bratton and J. Kennedy, “Defining a standard for particle swarm optimization,” in Proceedings of IEEE Swarm Intelli-
Journal of Artificial Evolution and Applications – Hindawi Publishing Corporation
Published: Feb 19, 2008
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