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DriP PSO- A fast and inexpensive PSO for drifting problem spaces
- 1. Zubin Bhuyan Sourav Hazarika Tezpur University, Assam, INDIA International Conf. on Science, Engineering & Technology- 2012, Trichy, India Full paper: http://zubinb.com/papers/T1_EC-346.pdf
- 2. Outline PSO basics The PSO Algorithm Dynamic systems Proposed PSO Models Drift Predictive PSO Experimental Results Conclusion and Future Work
- 3. Swarm Intelligence Swarm intelligence collective behavior of simple rule-following agents overall behavior of the entire system appears intelligent In Nature such behavior is seen in bird flocks, fish schools, ant colonies and animal herds Particle Swarm Optimization is a class of stochastic, population based optimization techniques
- 4. PSO Basics PSO was developed in 1995 by James Kennedy (social-psychologist) and Russell Eberhart (electrical engineer). † PSO is inspired from the concept of social interaction and is used for problem solving. A swarm of n agents or particles flies around in the search space looking for the best solution Particles communicate directly or indirectly with one another to determine its search direction. † Kennedy, J. and Eberhart, R. (1995). “Particle Swarm Optimization”, Proceedings of the 1995 IEEE International Conference on Neural Networks, pp. 1942-1948, IEEE Press.
- 5. PSO Basics pBest: Best value obtained so far by an individual particle Each particle has its own pBest gBest: Best of all pBest The basic concept of PSO is to accelerate each particle toward its own pBest, AND the gBest locations Usually with a random weighted acceleration at each time step
- 6. PSO Basicsᵵ sk vk vpbest vgbest sk+1 vk+1 sk vk vpbest vgbest sk+1 vk+1 Concept of modification of a searching point by PSO x y sk : current searching point. sk+1: modified searching point. vk: current velocity. vk+1: modified velocity. vpbest : velocity based on pbest. vgbest : velocity based on gbest ᵵSlide taken from Varadarajan Komanduri, Research Assistant, ECE Dept.,Villanova University http://www23.homepage.villanova.edu/varadarajan.komanduri/PSO_meander-line.ppt
- 8. PSO Algorithm 1. Initialize a population of particles randomly over a problem space with random velocities. 2. Evaluate fitness of each particle. 3. If current fitness of particle is better than pbest, then set pbest value equal to current fitness. Set pbest location to current location. 4. If current fitness is better than gbest, reset gbest to current fitness value. Set new gbest location to current location. 5. Change velocity according to the equation: vvid = w *vid + c1 * rand() * (pid -xid) + c2 * rand() * (pgd -xid) 6. Change the position according to equation: xid = xid + vid Here w is inertia weight, c1 and c2 are acceleration constants, and rand() is a random number generator function. 7. Loop back to Step 2 until end criterion is satisfied, or maximum number of iterations is completed.
- 9. PSO Algorithm vid = w *vid + c1 * rand() * (pid -xid) + c2 * rand() * (pgd -xid) w: Inertia weight of current velocity c1 : Acceleration component of cognitive part c2 : Acceleration component of social part xid = xid + vid
- 10. Dynamic Systems
- 11. Dynamic Systems Practical/Real world problems are time-varying or dynamic Problem space changing its state over time Optima changes continuously Changes may occur: periodically in some predefined sequence continuously in random fashion
- 12. Dynamic Systems ‡ Hu, et al, defines in [2] three types of basic dynamic systems: The location of the optimum value can change The location can remain constant but the optimum value may vary Both the location and the value of the optimum can vary ‡ X. Hu, and R. C. Eberhart, “Adaptive Particle Swarm Optimization: Response to Dynamic Systems” Proceedings of the 2002 Congress on Evolutionary Computation, 2002.
- 13. Dynamic Systems Particles might lose its global exploration ability Redundant pBest, gBest Leads to unsatisfactory, unacceptable and sub-optimal results
- 14. PSO for Dynamic Systems Several propositions Eberhart and Hu, 2002 “fixed gBest-value method” : If these two values do not change for certain number of iterations then a possible optimum change is declared increase the accuracy and prevent false alarms Charged-PSO: Blackwell and Bentley, 2002 Main idea: good balance between exploration and exploitation results in continuous search for better solution
- 15. PSO for Dynamic Systems Cooperative split PSO: Rakitianskaia, et al, 2008 modified the charged-PSO search space is divided into smaller subspaces, with each subspace being optimised by a separate swarm Cellular PSO: Hashemi , et al, 2009 hybrid model of particle swarm optimization and cellular automata population of particles is split into different groups across cells of cellular automata by imposing a restriction on number of particles in each cell further modified by introducing temporary quantam particles
- 16. DRIFT PREDICTIVE PSO MODEL Full paper: http://zubinb.com/papers/T1_EC-346.pdf
- 17. Drip-PSO Specifically designed for the scenario where the problem-space drifts in an unknown direction ASSUMPTION: Amount of drift is assumed to be gradual Most practical transitions are “not abrupt” Change can be determined by LOCALITY searching AIM: To determines the approximate direction in which the problem-space is drifting Adjust particle velocities accordingly in the subsequent iteration of the algorithm
- 18. Drip-PSO Idea: Add “Adjustment” to the velocity of each particle.
- 19. Drip-PSO: Detecting the drift direction In each iteration a small number of stagnant particles are selected They do not change their positions for that particular round If a change is detected by them, Generate 4 sub-particles resting on a circular orbit of radius ρ the sub-particles will be placed at right angle to one another
- 20. Drip-PSO Pi has been selected as a stagnant particle. It detects change in its fitness (despite the fact that its position did not change) Pi expands its sub-particle orbit Two sub-particles, are selected such that previous fitness of the Pi lies between the fitness values of the two selected sub-particles
- 21. Drip-PSO: Calculating the Drift The approximate direction of drift, i.e. the direction in which the adjustment is required, is given by
- 22. Drip-PSO: Calculating the Drift The approximate direction of drift, i.e. the direction in which the adjustment is required, is given by is the angle representing the direction of adjustment of α is the previous fitness value of Pi α ∊ [ Sk, Pi, Sj,Pi ] are the angles at which the selected sub-particles are oriented are fitness values
- 23. Drip-PSO: Calculating the Drift is calculated by all stagnant particles. Weighted average of all ξ is taken and added as an extra term to the velocity equation as shown Weight for a particular ξ is calculated using is the number of times the value occurs, n is the total number of stagnant particles.
- 24. Drip-PSO: Adjusting the Drift Adding “Adjustment” component to the velocity
- 26. Experimental Setup Test tool for the the proposed model was implemented in C# WPF (.Net Framework 4.0) Functions used for testing: Sphere, Step, Rastrigin, Rosenbrock and an arbitrary peak function Screen shot of PSO Test Tool
- 27. Experimental Setup Sphere function f(x, y) = x2 + y2 Arbitrary Peaks function f(x, y) = 1 – [3(1-x)2 e-x2 – (y+1)2 + 10(x/5 – x3 – y5 )e- (x2+y2) – 1/3e-(x+1)2-y2 Step function f(x, y) = |x| + |y| Test tool for the the proposed model was implemented in C# WPF (.Net Framework 4.0) Functions used for testing: Sphere, Step, Rastrigin, Rosenbrock and an arbitrary peak function
- 28. We simulate a dynamic system the test tool drifts the problem space in any direction, by applying an offset, λ, in every dimension ft+1 = ft(x - λ, y - λ) Offset is varied in the range [0.01, 0.09] The range of x and y is [-3, 3] c1 and c2 are set at 1.49618. Swarm size = 25 and 35 Experimental Setup
- 29. Experimental Results Percent error in finding global minima Function Standard PSO Drift Predictive PSO Sphere 6.799% 2.571% Step 9.847% 2.091% Rastrigin 29.900% 9.143% Rosenbrock 24.616% 3.592% Arbitrary Peaks 27.629% 5.126% RESULTS OF DRIP-PSO IN DYNAMIC SCENARIO USING 25 PARTICLES
- 30. Experimental Results RESULTS OF DRIP-PSO IN DYNAMIC SCENARIO USING 35 PARTICLES Percent error in finding global minima Function Standard PSO Drift Predictive PSO Sphere 5.021% 1.871% Step 8.268% 1.438% Rastrigin 25.728% 7.895% Rosenbrock 21.616% 2.332% Arbitrary Peaks 25.744% 4.661%
- 31. Conclusion Drip-PSO gives more accurate result for dynamic systems Less computational cost Only few particles need to perform extra calculation Implemented in Atmega32
- 32. Future Work Can be modified to detect several probable local optima then explore by splitting the entire swarm into sub-swarms Comparison with other PSOs
- 33. Acknowledgement Gunther Maurice Helped us in designing the class structure Tuhin Bhuyan, JEC, Assam Gave us the idea of making the test tool multi- threaded by using .Net Framework ThreadPool
- 34. Thank You! Full paper: http://zubinb.com/papers/T1_EC-346.pdf
Editor's Notes
- The first papers on the topic, by Kennedy and Eberhart, were presented in 1995; since then more than ten thousand articles and papers have been published on particle swarms.Kennedy also worked as a professional musician for twenty years and currently plays in a band called The Colliders!