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https://issues.apache.org/jira/browse/RNG-123?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16976619#comment-16976619
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Alex Herbert commented on RNG-123:
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I did note about the two constants for the two different generators having to
have the same oddness for this correlation to be observed. Thus we have this to
add to the javadoc:
{noformat}
Due to the use of an underlying linear congruential generator (LCG) alterations
to the 128 bit seed have the following effect: the first 64-bits alter the
generator state; the second 64 bits choose between one of two alternative LCGs
where the output of the chosen LCG is the same sequence except for an additive
constant determined by the seed bits. The result is that seeds that differ
only in the last 64-bits will have a 50% chance of producing highly correlated
output sequences.
Consider using the fixed increment variant where the 64-bit seed sets the
generator state.
@see https://ieeexplore.ieee.org/document/718715
Section 3.1: Different additive constants in a maximum potency congruential
generator
[Reference as provided]{noformat}
We can then add a simpler variant with a fixed increment to the library.
As for a section in the user guide for working with parallel computations based
on jumps, random seedings and reparameterization (since the library has
generators that that can use each of these strategies), is there a reference
where these strategies are discussed? I see it as a major unknown for any of
these strategies as to whether there will be correlations between the parallel
output. My current strategy is to ensure that the parallel work is done on
different data, thus even if the same generator was used the results are
credible; or to use a different generator class for each set of work (this does
not scale well).
> PCG generators may exhibit massive stream correlation
> -----------------------------------------------------
>
> Key: RNG-123
> URL: https://issues.apache.org/jira/browse/RNG-123
> Project: Commons RNG
> Issue Type: Bug
> Components: core
> Affects Versions: 1.3
> Reporter: Sebastiano Vigna
> Priority: Major
>
> [This is based on an issue I posted on the Rust development mailing list.]
> The documentation of the PCG generators does not state explicit that
> different seeds generate independent sequences, but the existence of a
> 128-bit seed implies somehow the idea that the whole seed is meaningful.
> The user should be made aware that the second parameter (the constant of the
> underlying LCG) is almost useless from a mathematical and statistical
> viewpoint.
> Changing constant to an LCG with power-of-two modulus and a constant like the
> one used in PCG simply adds a constant to the sequence (more precisely, there
> are two equivalence classes of constants, and in each equivalence class the
> sequences are identical, except for an additive constant).
> The minimal scrambling done by the generator usually cannot cancel this fact,
> and as a result changing the constant is equivalent to changing the initial
> state (modulo an additive constant). You can try to run this program:
>
> {noformat}
> import org.apache.commons.rng.core.source32.PcgXshRr32;
> import com.google.common.primitives.Ints;
> public class TestPCG {
> public static void main(final String[] args) {
> final long state = Long.parseLong(args[0]);
> final long c = Long.parseLong(args[1]);
> final long d = Long.parseLong(args[2]);
> if (c % 2 != d % 2) throw new IllegalArgumentException();
> final long C = c << 1 | 1;
> final long D = d << 1 | 1;
> final long r = 1314878037273365987L * ((d - c) >>> 1);
> final PcgXshRr32 rng0 = new PcgXshRr32(new long[] { state, c });
> final PcgXshRr32 rng1 = new PcgXshRr32(new long[] {
> 0xc097ef87329e28a5L *(6364136223846793005L * (state + C) + C - r
> - D) - D, d });
> for(;;) {
> final int a = rng0.nextInt();
> System.out.write(Ints.toByteArray(a), 0, 4);
> final int b = rng1.nextInt();
> System.out.write(Ints.toByteArray(b), 0, 4);
> }
> }
> }{noformat}
> You can pass any state as first argument, and any two constants as the
> following two arguments, as long as they are either both even or both odd .
> The program will set up a second initial state so that the sequences
> generated by the PRNGs using the two constants as seed are based on almost
> identical underlying LCG sequences, in spite of having arbitrary, different
> constants and different initial states. The two streams should be
> independent, but if you pipe the output in PractRand you'll get immediately
> {noformat}
> rng=RNG_stdin32, seed=unknown
> length= 4 megabytes (2^22 bytes), time= 2.1 seconds
> Test Name Raw Processed Evaluation
> BCFN(0+0,13-5,T) R=+263.2 p = 3.4e-103 FAIL !!!!!
> BCFN(0+1,13-5,T) R=+128.6 p = 1.5e-50 FAIL !!!!
> BCFN(0+2,13-6,T) R= +65.2 p = 9.2e-23 FAIL !!
> BCFN(0+3,13-6,T) R= +15.4 p = 1.0e-5 mildly suspicious
> DC6-9x1Bytes-1 R= +59.1 p = 4.2e-33 FAIL !!!
> DC6-6x2Bytes-1 R= +34.1 p = 9.0e-19 FAIL !
> DC6-5x4Bytes-1 R= +15.2 p = 7.7e-8 very suspicious
> [Low4/16]BCFN(0+1,13-6,T) R= +12.0 p = 1.5e-4 unusual
> [Low4/16]FPF-14+6/64:(4,14-8) R= +9.2 p = 1.1e-6 unusual
> [...]
> [Low8/32]FPF-14+6/4:(9,14-9) R= +27.4 p = 1.6e-17 FAIL
> [Low8/32]FPF-14+6/4:(10,14-10) R= +16.4 p = 2.5e-9 suspicious
> [Low8/32]FPF-14+6/4:all R=+283.4 p = 8.4e-255 FAIL !!!!!!
> [Low8/32]Gap-16:A R=+414.8 p = 2.4e-336 FAIL !!!!!!!
> [Low8/32]Gap-16:B R= +1736 p = 5e-1320 FAIL
> !!!!!!!!{noformat}
> You can also offset one of generator by hundred of iterations, but the
> sequences are so correlated that the result won't change. If you peek at the
> state of the two generators you'll see that their difference is constant.
> I think the reader should be made aware of the danger. If you start several
> generators of this kind the state is too small to guarantee that there will
> be no overlap. Once you get overlap, since there are in practice just two
> sequences, you will get a lot of unwanted correlation.
> There's a reason why nobody in the last decades ever considered creating
> "streams" using the constant part of an LCG, and it's that people realized
> very early that it doesn't work (see, e.g.,
> [https://ieeexplore.ieee.org/document/718715]).
>
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