Holger Dammertz September 15, Massively Parallel Random Number Generators
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1 Holger Dammertz September 15, 2010 Massively Parallel Random Number Generators
2 Page 2 Massively Parallel Random Number Generators Overview September 15, 2010 Overview Applications for Pseudo-Random Numbers Monte Carlo Simulation, Integration Test and Content Generation These applications are often easily parallelizable MRIP paradigm: multiple replications in parallel Generating Pseudo-Random Numbers Generate i.i.d. uniform random numbers (for example 32bit) Transform into (0, 1) Additional transformation to target distribution (for example, normal distributed)
3 Page 3 Massively Parallel Random Number Generators Overview September 15, 2010 Structure of a RNG Formal Definition [L E06] (S, µ,f,u,g) S state space µ prob. distr. on S to select initial state (seed) s 0 S f : S S transition function g : S U output function U = (0,1) output space s i = f (s i 1 ),i 1 and u i = g(s i )
4 Page 4 Massively Parallel Random Number Generators Overview September 15, 2010 Structure of a RNG In a parallel Implementation (S, µ,f,u,g) S needed per generating stream (usually per thread) S if possible hold in fast memory S store in global memory after finishing for multiple calls f and g are device functions (usually in a single function) output space often unsigned int transform needed Example: Linear Congruence Generator LCG [Knu81] s i S is an integer (for example 32bit), U = S,g = id f : s i = (as i 1 + c) mod m Needs well chosen a, c and m
5 Page 5 Massively Parallel Random Number Generators Overview September 15, 2010 Structure of a RNG Required properties of a RNG Speed Repeatability Minimal statistical bias Additional properties Random access on u i Independent number streams Long period
6 Page 6 Massively Parallel Random Number Generators Overview September 15, 2010 Structure of a RNG Why is a long period important? From [SPM05]: for a cycle length of n a single simulation should use at most 16 3 n random numbers (to trust the results of statistical simulation). Assume period of 2 48 and simple parallel cuda app with 4096 threads: random numbers per thread
7 Page 7 Massively Parallel Random Number Generators Overview September 15, 2010 Choice of RNG parameters are important Simple LCG, points
8 Page 8 Massively Parallel Random Number Generators Parallelization Techniques September 15, 2010 Two different views on parallel random numbers 1) Single Stream for all Threads Each thread computes parts of one problem Result should not depend on number of threads and should be repeatable Ideally RNG update is also parallel ( speed)
9 Page 9 Massively Parallel Random Number Generators Parallelization Techniques September 15, 2010 Two different views on parallel random numbers 2) Each Thread uses it s own RNG Each thread computes individual solutions Need to guarantee independence of streams If you could have a true RNG both methods would behave exactly the same (but repeatability would be lost). Using Pseudo RNGs this needs to be explicitly designed in the program
10 Page 10 Massively Parallel Random Number Generators Parallelization Techniques September 15, 2010 Parallelizing RNGs Random Seeding Easy to implement Generally very bad parallelization method Need to seed valid states No guarantee of independence Can work for generators with a long period Better alternative: well chosen seeds (per thread) For example: Mersenne Twister dcmt library New GPU Mersenne Twister (MTGP) provides seed tables Parametrization n different RNG configurations for n threads Needs to be especially developed and tested Often restricted to specific n
11 Page 11 Massively Parallel Random Number Generators Parallelization Techniques September 15, 2010 Parallelizing RNGs Block Splitting Can guarantee non overlapping sequences of length m m needs to be known in advance Starting states need to be known or computed
12 Page 12 Massively Parallel Random Number Generators Parallelization Techniques September 15, 2010 Parallelizing RNGs Leap-Frogging Needs RNG to be able to skip n numbers (or else quite inefficient)
13 Page 13 Massively Parallel Random Number Generators Application Scenarios September 15, 2010 Application Design Considerations On the fly Computation Compute RN when needed in kernel State needs to be stored per stream RNG uses additional resources (registers and memory) Needs properly parallelized implementation Pre-computation Store RN in main memory Only memory access per RN Easily parallelizable (same access as leap frog) Memory requirements can be huge Bandwidth need to be considered
14 Page 14 Massively Parallel Random Number Generators Application Scenarios September 15, 2010 Upload vs. Computation Example Intel(R) Xeon(R) 2.50GHz, GTX 480, 256MB random numbers, Mersenne Twister: MTGP/SFMT, 2 14 threads, blocksize 256 Precomputation Upload CPU-computed RN Precompute on CPU (SFMT): 180ms Upload: 44ms Precompute on GPU Precompute on GPU (MTGP): 9.18ms Consume (Asian Options): 783.5ms Produce and consume (MTGP) Single Kernel: ms
15 Page 15 Massively Parallel Random Number Generators Overview of some RNGs September 15, 2010 Overview of some RNGs LCGs s i = (as i 1 + c) mod m Combining multiple LCGs can give longer period Independent streams: Wichmann-Hill (273 threads) Multiple Recursive Generator s i = ( k ξ =1 Larger period (for k=1 equal to LCG) Blocking: MRG32k3a [LSCK02] a ξ s i ξ ) mod m
16 Page 16 Massively Parallel Random Number Generators Overview of some RNGs September 15, 2010 Overview of some RNGs RNGs based on Cryptographic functions Creates white noise from input MD5 [TW08]: hash function Tiny Encryption Algorithm (TEA) [ZOC10] Different configuration per thread (parametrization) Transform counter and thread id Random Access in the sequence Mersenne Twister New GPU version [Sai10] Long period: 32bit version provides , , Good seeding strategies (see MTGPDC)
17 Page 17 Massively Parallel Random Number Generators Testing of RNGs September 15, 2010 Testing RNGs Why use tests Implementation of RNGs is very sensitive Before using any RNG implementation it should be tested Failed tests: very likely bad sequence Passed tests: guarantees nothing Test Suits Use statistical tests to find flaws DIEHARD + NIST (both integrated in DIEHARDER [Bro09]) TestU01 [LS07]
18 Page 18 Massively Parallel Random Number Generators Our Collection September 15, 2010 Collection of Parallel Random Number Generators Selecting suitable RNGs Domain specific problem Depends on current compiler and hardware Our Collection CUDA implementation of several different RNGs Easy to use Currently tested on Linux Most of the code: MIT License Copy-paste ready code Available at
19 Page 19 Massively Parallel Random Number Generators Our Collection September 15, 2010 Collection of Parallel Random Number Generators Integrated Benchmark Benchmarks compiles and runs on your machine Configure threads and blocks according to your target application Integrated Test Suits DIEHARDER and TestU01 Automatic report generation Easy to extend Integrate your own RNG Integrate your own tests
20 Page 20 Massively Parallel Random Number Generators Some Results September 15, 2010 Comparisons: TestU01 SmallCrush Battery BirthdaySpacings Collision Gap SimpPoker CouponCollector MaxOft RNG Name combinedlcgtaus P P P P P P P P P P P P P P P drand48gpu P P P P P F F P P P F F F F P kiss07 P P P P P P P P P P P P P P P lfsr113 P P P P P P P P P P P P P P P md5 P P P P P P P P P P P P P P P mtgp P P P P P P P P P P P P P P P park miller F F P P P F F P P P F F F F F ranecu P F P P P F F P P P F F F F F tea P P P P P P P P P P P P P P P tt800 P P P P P P P P P P P P P P P MaxOft AD WeightDistrib MatrixRank HammingIndep RandomWalk1 H RandomWalk1 M RandomWalk1 J RandomWalk1 R RandomWalk1 C
21 Page 21 Massively Parallel Random Number Generators Some Results September 15, 2010 Raw Performance Test Raw Performance, GeForce GTX 480, Random Numbers seconds combinedlcgtaus drand48gpu kiss07 lfsr113 mtgp park_miller ranecu tt800
22 Page 21 Massively Parallel Random Number Generators Some Results September 15, 2010 Raw Performance Test 0.3 Raw Performance, GeForce GTX 480, Random Numbers seconds drand48gpu combinedlcgtaus kiss07 lfsr113 md5 mtgp park_miller ranecu tea tt800
23 Page 22 Massively Parallel Random Number Generators Some Results September 15, 2010 Performance Test: Asian Option Example from [HT07] 0.5 Asian Options, GeForce GTX 480, 96 runs, simulations seconds combinedlcgtaus drand48gpu kiss07 lfsr113 mtgp park_miller ranecu tt800
24 Page 22 Massively Parallel Random Number Generators Some Results September 15, 2010 Performance Test: Asian Option Example from [HT07] 1.6 Asian Options, GeForce GTX 480, 96 runs, simulations seconds combinedlcgtaus drand48gpu kiss07 lfsr113 md5 mtgp park_miller ranecu tea tt800
25 Page 23 Massively Parallel Random Number Generators Some Results September 15, 2010 Quality vs. Speed Reduced rounds of MD5/TEA RNG For some applications speed more important than quality MD5 and TEA allow to reduce number of iterations Quality of random numbers may degrade Avalanche effect MD5: passes most of the DIEHARDER tests after 16 rounds
26 Page 24 Massively Parallel Random Number Generators Some Results September 15, 2010 Quality vs. Speed, TEA, DIEHARDER tests Dieharder tests Tiny Encryption Algorithm Number of rounds weak passed
27 Page 25 Massively Parallel Random Number Generators Some Results September 15, 2010 Quality vs. Speed, TEA performance 0.35 Raw Performance, Random Numbers GTX480, Tiny Encryption Algorithm Runtime in ms Number of rounds
28 Page 26 Massively Parallel Random Number Generators Some Results September 15, 2010 Quality vs. Speed, MD5 performance 0.35 Raw Performance, Random Numbers GTX480, MD Runtime in ms Number of rounds
29 Page 27 Massively Parallel Random Number Generators Conclusion September 15, 2010 Conclusion Many good (parallel) RNGs exist Several different properties Choice of fitting RNG application dependent Some Picks KISS + Simple code and state management Random seeding: may be ok for non-critical applications MTGPU + Very good quality and sophisticated seeding + Long period Relatively complex code Fixed block/thread layout MD5, TEA + Random access + No Seeding Slow
30 Page 28 Massively Parallel Random Number Generators Conclusion September 15, 2010 Conclusion Precomputing on GPU May be an alternative to in kernel computation RNG Collection Always evaluate your RNG choice and implementation Our framework provides an easy platform for testing
31 Page 28 Massively Parallel Random Number Generators Conclusion September 15, 2010 Conclusion Precomputing on GPU May be an alternative to in kernel computation RNG Collection Always evaluate your RNG choice and implementation Our framework provides an easy platform for testing Questions? Acknowledgements Framework: Christoph Schied This work was supported by a NVIDIA Professor Partnership Award with Hendrik P.A. Lensch
32 Page 28 Massively Parallel Random Number Generators References September 15, 2010 Robert G. Brown. Dieharder: A random number test suite Lee Howes and David Thomas. Efficient Random Number Generation and Application Using CUDA. In GPU Gems 3, pages , D.E. Knuth. The art of computer programming: Seminumerical algorithms, volume 2, P. L Ecuyer. Uniform random number generation. Handbooks in Operations Research and Management Science, 13:55 81, P. L Ecuyer and R. Simard. TestU01: AC library for empirical testing of random number generators. ACM Transactions on Mathematical Software (TOMS), 33(4):22, Pierre L Ecuyer, Richard Simard, E. Jack Chen, and W. David Kelton. An object-oriented random-number package with many long streams and substreams.
33 Page 28 Massively Parallel Random Number Generators References September 15, 2010 Oper. Res., 50(6): , Mutsuo Saito. A variant of mersenne twister suitable for graphic processors. CoRR, abs/ , Marcus Schoo, Krzysztof Pawlikowski, and Donald C. Mcnickle. A survey and empirical comparison of modern pseudo-random number generators for distributed stochastic simulations, S. Tzeng and L.Y. Wei. Parallel white noise generation on a GPU via cryptographic hash. In Proceedings of the 2008 symposium on Interactive 3D graphics and games, pages ACM, F. Zafar, M. Olano, and A. Curtis. GPU Random Numbers via the Tiny Encryption Algorithm
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