Parallel random number generation#
There are four main strategies implemented that can be used to produce repeatable pseudorandom numbers across multiple processes (local or distributed).
SeedSequence
spawning#
NumPy allows you to spawn new (with very high probability) independent
BitGenerator
and Generator
instances via their spawn()
method.
This spawning is implemented by the SeedSequence
used for initializing
the bit generators random stream.
SeedSequence
implements an algorithm to process a userprovided seed,
typically as an integer of some size, and to convert it into an initial state for
a BitGenerator
. It uses hashing techniques to ensure that lowquality seeds
are turned into high quality initial states (at least, with very high
probability).
For example, MT19937
has a state consisting of 624 uint32
integers. A naive way to take a 32bit integer seed would be to just set
the last element of the state to the 32bit seed and leave the rest 0s. This is
a valid state for MT19937
, but not a good one. The Mersenne Twister
algorithm suffers if there are too many 0s. Similarly, two adjacent 32bit
integer seeds (i.e. 12345
and 12346
) would produce very similar
streams.
SeedSequence
avoids these problems by using successions of integer hashes
with good avalanche properties to ensure that flipping any bit in the input
has about a 50% chance of flipping any bit in the output. Two input seeds that
are very close to each other will produce initial states that are very far
from each other (with very high probability). It is also constructed in such
a way that you can provide arbitrarysized integers or lists of integers.
SeedSequence
will take all of the bits that you provide and mix them
together to produce however many bits the consuming BitGenerator
needs to
initialize itself.
These properties together mean that we can safely mix together the usual
userprovided seed with simple incrementing counters to get BitGenerator
states that are (to very high probability) independent of each other. We can
wrap this together into an API that is easy to use and difficult to misuse.
Note that while SeedSequence
attempts to solve many of the issues related to
userprovided small seeds, we still recommend
using secrets.randbits
to generate seeds with 128 bits of entropy to
avoid the remaining biases introduced by humanchosen seeds.
from numpy.random import SeedSequence, default_rng
ss = SeedSequence(12345)
# Spawn off 10 child SeedSequences to pass to child processes.
child_seeds = ss.spawn(10)
streams = [default_rng(s) for s in child_seeds]
For convenience the direct use of SeedSequence
is not necessary.
The above streams
can be spawned directly from a parent generator
via spawn
:
parent_rng = default_rng(12345)
streams = parent_rng.spawn(10)
Child objects can also spawn to make grandchildren, and so on.
Each child has a SeedSequence
with its position in the tree of spawned
child objects mixed in with the userprovided seed to generate independent
(with very high probability) streams.
grandchildren = streams[0].spawn(4)
This feature lets you make local decisions about when and how to split up streams without coordination between processes. You do not have to preallocate space to avoid overlapping or request streams from a common global service. This general “treehashing” scheme is not unique to numpy but not yet widespread. Python has increasinglyflexible mechanisms for parallelization available, and this scheme fits in very well with that kind of use.
Using this scheme, an upper bound on the probability of a collision can be
estimated if one knows the number of streams that you derive. SeedSequence
hashes its inputs, both the seed and the spawntreepath, down to a 128bit
pool by default. The probability that there is a collision in
that pool, pessimisticallyestimated ([1]), will be about \(n^2*2^{128}\) where
n is the number of streams spawned. If a program uses an aggressive million
streams, about \(2^{20}\), then the probability that at least one pair of
them are identical is about \(2^{88}\), which is in solidlyignorable
territory ([2]).
Sequence of integer seeds#
As discussed in the previous section, SeedSequence
can not only take an
integer seed, it can also take an arbitrarylength sequence of (nonnegative)
integers. If one exercises a little care, one can use this feature to design
ad hoc schemes for getting safe parallel PRNG streams with similar safety
guarantees as spawning.
For example, one common use case is that a worker process is passed one root seed integer for the whole calculation and also an integer worker ID (or something more granular like a job ID, batch ID, or something similar). If these IDs are created deterministically and uniquely, then one can derive reproducible parallel PRNG streams by combining the ID and the root seed integer in a list.
# default_rng() and each of the BitGenerators use SeedSequence underneath, so
# they all accept sequences of integers as seeds the same way.
from numpy.random import default_rng
def worker(root_seed, worker_id):
rng = default_rng([worker_id, root_seed])
# Do work ...
root_seed = 0x8c3c010cb4754c905776bdac5ee7501
results = [worker(root_seed, worker_id) for worker_id in range(10)]
This can be used to replace a number of unsafe strategies that have been used
in the past which try to combine the root seed and the ID back into a single
integer seed value. For example, it is common to see users add the worker ID to
the root seed, especially with the legacy RandomState
code.
# UNSAFE! Do not do this!
worker_seed = root_seed + worker_id
rng = np.random.RandomState(worker_seed)
It is true that for any one run of a parallel program constructed this way, each worker will have distinct streams. However, it is quite likely that multiple invocations of the program with different seeds will get overlapping sets of worker seeds. It is not uncommon (in the author’s selfexperience) to change the root seed merely by an increment or two when doing these repeat runs. If the worker seeds are also derived by small increments of the worker ID, then subsets of the workers will return identical results, causing a bias in the overall ensemble of results.
Combining the worker ID and the root seed as a list of integers eliminates this risk. Lazy seeding practices will still be fairly safe.
This scheme does require that the extra IDs be unique and deterministically
created. This may require coordination between the worker processes. It is
recommended to place the varying IDs before the unvarying root seed.
spawn
appends integers after the userprovided seed, so if
you might be mixing both this ad hoc mechanism and spawning, or passing your
objects down to library code that might be spawning, then it is a little bit
safer to prepend your worker IDs rather than append them to avoid a collision.
# Good.
worker_seed = [worker_id, root_seed]
# Less good. It will *work*, but it's less flexible.
worker_seed = [root_seed, worker_id]
With those caveats in mind, the safety guarantees against collision are about the same as with spawning, discussed in the previous section. The algorithmic mechanisms are the same.
Independent streams#
Philox
is a counterbased RNG based which generates values by
encrypting an incrementing counter using weak cryptographic primitives. The
seed determines the key that is used for the encryption. Unique keys create
unique, independent streams. Philox
lets you bypass the
seeding algorithm to directly set the 128bit key. Similar, but different, keys
will still create independent streams.
import secrets
from numpy.random import Philox
# 128bit number as a seed
root_seed = secrets.getrandbits(128)
streams = [Philox(key=root_seed + stream_id) for stream_id in range(10)]
This scheme does require that you avoid reusing stream IDs. This may require coordination between the parallel processes.
Jumping the BitGenerator state#
jumped
advances the state of the BitGenerator asif a large number of
random numbers have been drawn, and returns a new instance with this state.
The specific number of draws varies by BitGenerator, and ranges from
\(2^{64}\) to \(2^{128}\). Additionally, the asif draws also depend
on the size of the default random number produced by the specific BitGenerator.
The BitGenerators that support jumped
, along with the period of the
BitGenerator, the size of the jump and the bits in the default unsigned random
are listed below.
BitGenerator 
Period 
Jump Size 
Bits per Draw 

\(2^{19937}1\) 
\(2^{128}\) 
32 

\(2^{128}\) 
\(~2^{127}\) ([3]) 
64 

\(2^{128}\) 
\(~2^{127}\) ([3]) 
64 

\(2^{256}\) 
\(2^{128}\) 
64 
The jump size is \((\phi1)*2^{128}\) where \(\phi\) is the golden ratio. As the jumps wrap around the period, the actual distances between neighboring streams will slowly grow smaller than the jump size, but using the golden ratio this way is a classic method of constructing a lowdiscrepancy sequence that spreads out the states around the period optimally. You will not be able to jump enough to make those distances small enough to overlap in your lifetime.
jumped
can be used to produce long blocks which should be long enough to not
overlap.
import secrets
from numpy.random import PCG64
seed = secrets.getrandbits(128)
blocked_rng = []
rng = PCG64(seed)
for i in range(10):
blocked_rng.append(rng.jumped(i))
When using jumped
, one does have to take care not to jump to a stream that
was already used. In the above example, one could not later use
blocked_rng[0].jumped()
as it would overlap with blocked_rng[1]
. Like
with the independent streams, if the main process here wants to split off 10
more streams by jumping, then it needs to start with range(10, 20)
,
otherwise it would recreate the same streams. On the other hand, if you
carefully construct the streams, then you are guaranteed to have streams that
do not overlap.