Parallel Random Number Generators

Posted by Cleve Moler,

This is the second of a multi-part series about the MATLAB random number generators. If you ask for help rng, you will get lots of information, including the fact that there are three modern generators.

'twister'              Mersenne Twister
'combRecursive'        Combined Multiple Recursive
'multFibonacci'        Multiplicative Lagged Fibonacci

My previous post was about twister. Today's post is about the other two, 'combRecursive' and 'multFibonacci', which are designed for parallel computing.

Contents

Parallel computing demo

I frequently use the card game Blackjack to demonstrate parallel computing. At the same time I can demonstrate the random number generators. I regard Blackjack as a financial instrument, not unlike the stock of a publicly traded company. Simulating the size of an investment as a function of time is a typical application of the Monte Carlo technique.

Begin by opening a pool of workers.

   parpool;
Starting parallel pool (parpool) using the 'local' profile ... connected to 2 workers.

Four players each play twenty thousand hands of Blackjack.

   p = 4;
   n = 20000;

Collect the results in this array.

   B = zeros(n,p);

A parallel for loop executes a Blackjack program that knows nothing about parallel computing.

   parfor k = 1:p
      B(:,k) = cumsum(blackjacksim(n));
   end

Plot the results.

   plot(B)
   title(['Blackjack , $ ', int2str(sum(B(end,:)))])
   xlabel('Number of hands')
   ylabel('Stake, ($)')
   axis([0 n -2500 2500])

Parallel streams

The card dealing calls the random number generator. It is essential that the different parallel workers have different, independent streams of random numbers. The default MATLAB generator twister does not offer this feature. Simply starting twister with different seeds on different workers does not provide statistically independent streams. So we turn to the other generators. To see which one is in use here, run a small spmd, "single program, multiple data" block.

   spmd
      r = rng
      s = r.State'
      format short
      x = rand(1,7)
   end
Lab 1: 
  r = 
       Type: 'combRecursive'
       Seed: 0
      State: [12x1 uint32]
  s =
    Columns 1 through 6
    1720035765  2052922678  1637499698  3048064580  1173461082  2391850890
    Columns 7 through 12
    1862757735  2368998908  1385613640  1660833332   146924518  3104031825
  x =
      0.8789    0.6969    0.0409    0.4609    0.7528    0.2871    0.5241
Lab 2: 
  r = 
       Type: 'combRecursive'
       Seed: 0
      State: [12x1 uint32]
  s =
    Columns 1 through 6
     323405913  3817048408  3712601073  1070773748  1552739185  3267875480
    Columns 7 through 12
    1594297407  2533167732  3377045245  3413340742  2651847732  1248925296
  x =
      0.1072    0.3194    0.1048    0.6623    0.0878    0.3692    0.8035

We see that we have two workers, that they are both using the combRecursive generator, that they have the same seed, but different states, so they are generating different random numbers.

combRecursive

Also known as mrg32k3a. A 32-bit combined multiple recursive generator (CMRG), due to Pierre L'Ecuyer, at the Universite de Montreal, and his colleagues, described in the papers referenced below. This generator is similar to the CMRG implemented in the RngStreams package. It has a period of $2^{127}$, and supports up to $2^{63}$ parallel streams, via sequence splitting, and $2^{51}$ substreams each of length $2^{76}$. Here is a link to the C source code. combmrg2.c

The state of the backbone generator is a 2-by-3 array S that evolves at each step according to the linear recurrence expressed succinctly in MATLAB by

m1 = 2^32 - 209;
m2 = 2^32 - 22853;
x1 = mod(1403580*S(1,2)-810728*S(1,3),m1);
x2 = mod(527612*S(2,1)-1370589*S(2,3),m2);
S = [x1 S(1,1) S(1,2))
     x2 S(2,1) S(2,2)];

A single precision random real u is then produced by

z = mod(x1-x2,m1);
if z > 0, u = z/(m1+1); else, u = m1/(m1+1); end

The important feature of this generator is that it is possible to create different initial states for each worker in a parallel pool so that the resulting streams of random numbers are statistically independent.

multFibonacci

Also known as mlfg6331_64. A 64-bit multiplicative lagged Fibonacci generator (MLFG), developed by Michael Mascagni and Ashok Srinivasan at Florida State University. This generator, which has lags $l=63$ and $k=31$, is similar to the MLFG implemented in the SPRNG package. It has a period of approximately $2^{124}$. It supports up to $2^{61}$ parallel streams, via parameterization, and $2^{51}$ substreams each of length $2^{72}$.

The state of this generator is a length 63 vector of 64-bit integers S. The recurrence relation is

$$ x_n = x_{n-k} \times x_{n-l} (mod 2^{64}) $$

Each random double precision value is created using one 64-bit integer from the generator; the possible values are all multiples of $2^{-53}$ strictly within the interval (0,1).

Again, the important feature of this generator is that it is possible to create different initial states for each worker in a parallel pool so that the resulting streams of random numbers are statistically independent.

Which one?

Which one should you use? Most of the time, stick with the default and you'll be OK. You will get 'twister' in serial computations and 'combRecursive' on the workers in a parallel pool. You can use

rng('shuffle')

at the beginning of a session if you want different sequences of random numbers in different sessions. Otherwise, don't worry about setting the generator or the seed.

If you want to experiment, you can use rng to try different generators and different starting seeds on your computation. If you find a problem where it makes a significant difference, please let us know.

Thanks

Thanks again to Peter Perkins.

References

Pierre L'Ecuyer, R. Simard, E. J. Chen, and W. D. Kelton. "An Objected-Oriented Random-Number Package with Many Long Streams and Substreams." Operations Research, 50(6):1073-1075. 2002. <http://www.iro.umontreal.ca/~lecuyer/myftp/papers/streams00.pdf>

Pierre L'Ecuyer. "Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators." Operations Research 47(1):159-164. 1999. <http://dx.doi.org/10.1287/opre.47.1.159>

Michael Mascagni and Ashok Srinivasan. "Parameterizing Parallel Multiplicative Lagged-Fibonacci Generators." Parallel Computing, 30: 899-916. 2004. <http://www.cs.fsu.edu/~asriniva/papers/mlfg.ps>

Michael Mascagni and Ashok Srinivasan. "SPRNG: A Scalable Library for Pseudorandom Number Generation." ACM Transactions on Mathematical Software, Vol 26 436-461. 2000. <http://www.cs.fsu.edu/~asriniva/papers/sprngacm.ps>


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Published with MATLAB® R2014b

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