We submit an implementation of an Estimation of Distribution Algorithm – specifically a variant of the Bayesian Optimisation Algorithm (BOA) – using GPGPU. Every aspect of the algorithm is executed on the device, and it makes effective of use multiple GPU devices in a single machine. As for other EDAs, our implementation is generic in that it may be applied to any problem for which solutions may be represented as binary strings. For the purpose of this paper, we apply it to a particular problem known to be difficult for metaheuristic algorithms due to high interdependency between variables: finding the lowest energy state of an Ising Spin Glass. We show that our GPU implementation demonstrates a speedup in excess of 80x compared with an equivalent CPU implementation. To our knowledge, this is the first EDA to be implemented fully on the GPU

Burles, Nathan John

Poulding, Simon Marcus

Staunton, Jan Peter

White Rose Research Online

This is a repository copy of Full Implementation of an Estimation of Distribution Algorithm on a GPU.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/88233/Version: Submitted VersionConference or Workshop Item:Poulding, Simon Marcus, Staunton, Jan Peter and Burles, Nathan John orcid.org/0000-0003-3030-1675 (2011) Full Implementation of an Estimation of Distribution Algorithm on aGPU. In: GECCO 2011, GPUs for Genetic and Evolutionary Computation Competition, 12-16 Jul 2011. eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Full Implementation of an Estimation of DistributionAlgorithm on a GPUCIGPU GPU Competition EntrySimon PouldingDepartment of ComputerScienceUniversity of YorkHeslington, York, UKsmp@cs.york.ac.ukJan StauntonDepartment of ComputerScienceUniversity of YorkHeslington, York, UKjps@cs.york.ac.ukNathan BurlesDepartment of ComputerScienceUniversity of YorkHeslington, York, UKnburles@cs.york.ac.uk1. INTRODUCTIONWe submit an implementation of an Estimation of Dis-tribution Algorithm – specifically a variant of the BayesianOptimisation Algorithm (BOA) – using GPGPU. Every as-pect of the algorithm is executed on the device, and it makeseffective of use multiple GPU devices in a single machine.As for other EDAs, our implementation is generic in that itmay be applied to any problem for which solutions may berepresented as binary strings. For the purpose of this paper,we apply it to a particular problem known to be difficult formetaheuristic algorithms due to high interdependency be-tween variables: finding the lowest energy state of an IsingSpin Glass. We show that our GPU implementation demon-strates a speedup in excess of 80x compared with an equiv-alent CPU implementation. To our knowledge, this is thefirst EDA to be implemented fully on the GPU.2. BACKGROUND AND PREVIOUS WORK2.1 Estimation of Distribution AlgorithmsEstimation of Distribution Algorithms (EDAs) are population-based probabilistic search techniques that search solutionspaces by learning and sampling probabilistic models [4].EDAs iterate over successive populations of candidate solu-tions to a problem. Each population is sometimes referred toas a generation. To construct a successor population, EDAsbuild a probabilistic model of promising solutions from thecurrent population and then sample that model to generatenew individuals. The newly generated individuals replaceindividuals in the current population to create the new pop-ulation.Algorithm 1 Pseudocode for basic EDAP = InitialPopulation();while not(termination criterion) doevaluate(P);S = SelectPromisingSolutions(P);M = UpdateModelUsingSolutions(S);N = SampleFromModel(M);P = ReplaceIndividuals(N);end whileThe pseudocode of a basic EDA algorithm is shown in Al-gorithm 1. Readers who are familiar with Genetic Algorithm(GA) literature can view EDAs as similar to a GA with thecrossover and mutation operators replaced with the modelbuilding and sampling steps. EDAs can be seen as strate-gically sampling the solution space in an attempt to find a“good” solution whilst learning a model of “good” solutionsalong the way. EDAs are sometimes referred to as Prob-abilistic Model-Building Genetic Algorithms (PMBGAs), afull overview of which can be found in [4].For this entry, we have implemented a variant of the BayesianOptimisation Algorithm (BOA) [3] on the GPU. BOA usesBayesian networks to model the relationship between valuesin binary string solutions. In order to construct the Bayesiannetwork given a set of desirable binary strings, a greedy net-work construction algorithm is used that adds edges betweenparents and child nodes (where each node represents a singlebinary value). Our implementation makes use of K2 metric[1] to determine the next edge to add in order to derive amodel that best represents the “good” solutions in the cur-rent population.Restricted Tournament Replacement (RTR) is employedas part of our implementation to encourage diversity in thepopulation during the course of any given search. To replacea solution with a new solution i in a population P duringthe replacement phase of our EDA, RTR chooses a randomsubset S from P , then finds the “most similar” solution jto i in S, where similarity is determined by the Hammingdistance of the binary string representations. If i is betterthan j, then replace j with i in P .2.2 Parallelisation of EDAsExisting work that parallelises EDA model building acrossCPUs makes use of an ancestral node ordering to allowthe model to be built independently for each node in theBayesian network [2]. The ordering is generated randomlyat each iteration and specifies which nodes can be parentsof a given child node, and thus avoid cycles in the networkwhen the model is constructed in a distributed manner. Wemake use of this ancestral node ordering in our implemen-tation.2.3 Ising Spin GlassesOur chosen example problem are Ising Spin Glasses (ISG),which are often used as benchmark problems for sophisti-cated EDAs such as BOA [5]. ISGs are a mathematicalmodel used in statistical physics. A number of particles ex-ist in a spin-up (Si = +1) or spin-down (Si = −1) state. Theparticles are typically arranged in a lattice: our implemen-tation considers toroidal 2D and 3D lattices. Each particleis coupled with its immediate neighbours in the lattice ac-cording to coupling constants, J , which also take values of−1 or +1. The free energy, E, of the particles is derivedusing the equation:E = −∑i,jJijSiSj (1)where sum occurs over all neighbouring particles Si and Sj .The objective is to find the lowest energy configurationof particle spins given a set of coupling constants (which inour case are randomly assigned). This problem is particu-larly difficult for many optimisation techniques owing to thehigh interdependency of particle spins, and thus the non-decomposability of the problem.3. IMPLEMENTATIONThe EDA algorithm is implemented entirely on the GPUdevice under the control of a CPU process. The chosen fit-ness function for our chosen example of Ising Spin Glasses isalso implemented on the GPU, with all processing occurringon the GPU under the control of a CPU process. The algo-rithm is implemented using the CUDA 4.0 SDK, and makesuse of the Thrust library for standard operations such assorting.The model building phase of the algorithm is by far mostcomputationally intensive step for large problem sizes. It isthis step that can be distributed to multiple GPUs, allowingfor near linear speed up over a single GPU card implemen-tation. The work is distributed proportionally according tothe computational power of the GPGPU capable cards, soas to avoid a slow card limiting overall performance.This step takes advantage of the ancestral node orderingdescribed in section 2.2 to enable the model to be built in-dependently for each node in the Bayesian network. In theGPU implementation, each CUDA block is assigned to themodel building for one node, resulting in significant paral-lelisation of this step and thereby effective utilisation of theGPU.As well as scaling to use multiple GPU devices, we alsobelieve the implementation scales to different cards architec-tures since parameters, such as grid and block topologies, arederived based on queried device properties.4. EVALUATIONIn order to show the scaling properties of the GPU imple-mentation, we compare to an equivalent CPU implementa-tion. The problem used is the 3D Ising Spin Glass problemdescribed in section 2.3. We performed experiments on twoparameters, the problem size and the population size. Theresults are shown in the tables below, and are calculatedfrom an average of 10 trials (runs) for each parameter combi-nation. The CPU implementation used 4 threads, whilst theGPU implementation used 4 Tesla cards. Any results with a* next to them are the result of one experiment since it wasimpractical to run these lengthy trials multiple times. Thepertinent parameters are as follows unless specified in thetables: Spin Glass Size: 16x16x16, Population Size: 1024,Maximum Parents: 4, Iterations: 20, Best Population Size:18Population, Number Replaced: 14Population, RTR Tour-nament Size: 16.Table 1: Problem Size ResultsProblem Size ResultsProblem Size 8 12 16 24GPU 0.3s 1.08s 4.45s 41.14sCPU 2.37s 23.46s 217.22s* 3391.25s*Speed up 7.9x 21.72x 48.81x 81.82xTable 2: Population Size ResultsPopulation Size ResultsMax Parents 512 1024 2048 4096GPU 2.75s 4.55s 4.98s 9.98sCPU 77.16s 208.56s 307.64s -Speed up 28.05x 45.84x 61.78x -As can be seen, the speedup provided by GPU implemen-tation increases from 8x to more than 80x as the problemsize increases to spin glasses of size 24x24x24. Similarly, theGPU speedup increases with population size.5. CONCLUSIONTo conclude, we have implemented an EDA fully on theGPU. We have shown that the GPU implementation can beup to 80 times faster in some cases, and scales well withrespect to the problem size and population size. The imple-mentation can use multiple cards on a single host and usespreprocessing in order to make the GPU implementationmore efficient. To our knowledge, this is the first sophisti-cated BOA-like EDA to be fully-implemented on the GPU.Please find more details in the help files of the implemen-tation package submitted with this report. Our future aimsare to use this implementation in order to perform researchinto larger practical problems.6. REFERENCES[1] C. Borgelt and R. Kruse. An empirical investigation ofthe K2 metric. In Symbolic and QuantitativeApproaches to Reasoning with Uncertainty, volume2143 of Lecture Notes in Computer Science, pages240–251. 2001.[2] J. Očenášek and J. Schwarz. The distributed bayesianoptimization algorithm for combinatorial optimization.In Proc. EUROGEN 2001, Evolutionary Methods forDesign, Optimisation and Control with Applications toIndustrial Problems, pages 115–120, 2001.[3] M. Pelikan, D. Goldberg, and E. Cantu-Paz. BOA: TheBayesian optimization algorithm. In Proceedings of theGenetic and Evolutionary Computation ConferenceGECCO-99, volume 1, pages 525–532, 1999.[4] M. Pelikan, D. Goldberg, and F. Lobo. A survey ofoptimization by building and using probabilisticmodels. Computational optimization and applications,21(1):5–20, 2002.[5] M. Pelikan, M. Pelikan, D. E. Goldberg, and D. E.Goldberg. Hierarchical boa solves ising spin glasses andmaxsat. In In Proc. of the Genetic and EvolutionaryComputation Conference (GECCO 2003), number 2724in LNCS, pages 1271–1282. Springer, 2003.

Full Implementation of an Estimation of Distribution Algorithm on a GPU

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Full Implementation of an Estimation of Distribution Algorithm on a GPU

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