Efficiently Generating a Number in a Range
pcg-random.org> Let's move from an over-engineered approach to an under-engineered one.
The article says this to deride C++s implementation as being too complicated because it supports ranges such as [-3,17] and then promptly goes on to discuss how a modulo based implementation is very biased if the upper end of the range is above 2^31. It's not really clear why the former use case is unimportant but the latter isn't.
It just goes to show that one person's niche use case is another person's main use case. I wish people would just avoid the judgemental term "over engineered" and instead focus on matching appropriate algorithms to appropriate use cases.
The comment there is not about [-3, 17] being an obscure output range from a distribution. It is that the distribution must be able to handle a random generator that outputs numbers in that range.
I think there's a small error there in that the output type of UniformRandomBitGenerator must be actually be unsigned. The larger point still stands though. It is possible to write a conforming UniformRandomBitGenerator that has an output range of [3, 17] and it falls on the distribution to handle this.
Ah, good call. I did slightly misinterpret what was being said. I think my overall point still stands, though.
The modulo approach is biased for all ranges which don't divide the full range but only a small amount of bias at small ranges rather than a large amount of bias at large ranges.
I don't disagree - I just very much hate the phrase "over engineered" as I think it never adds anything to a discussion.
return min + (max - min) / 2
Or in O(1):
Alternativelyreturn minreturn max> Or in O(1)
You mean in one instruction? The parent comment is O(1). The methods in the article are all O(1) too.
In TXR Lisp, the algorithm I put in place basically finds the tightest power-of-two bounding box for the modulus, clisp the pseudo-andom number to that power-of-two range, and then rejects values outside of the modulus.
Example: suppose we wanted values in the range 0 to 11. The tightest power of two is 16, so we generate 4 bit pseudo-random numbers in the 0 to 15 range. If we get a value in the 12 to 15 range, we throw it away and choose another one.
The clipping to the power-of-two bounding box ensures that we reject at most 50% of the raw values.
I don't bother optimizing for small cases. That is, under this 4 bit example, each generated value that is trimmed to 4 bits will be the full output of the PRNG, a 32 bit value. The approach pays off for bignums; the PRNG is called enough times to cover the bits, clipped to the power-of-two box, then subject to the rejection test.
This is referred to as “Bitmask” in the article.
Off the top of my head, for a randomly chosen range of size n, you reject a throw with probability 1/4, right?
It's unintuitive, but I believe the probability ends up being 1-ln(2) if you think of n as being uniformly random.
For any power of two m, then for any range size n (with m/2 < n <= m), the probability of rejection is (m-n)/m. If any n is equally probable, then the average rejection is equal to the rejection of the average n (= 3*m/4): (m/4)/m = 1/4. This is true for any power of two m. I stand my case!
Hm, yeah not sure what I was thinking. Maybe expected number of attempts?
n uniformly random from what distribution? Or are you taking a limit somewhere?
This is well-timed. I don't know much about different random number generators but I do know that we recently had an problem where RNG was a serious performance bottleneck.
Note that this article is not really about RNGs themselves, but mostly how to use one to generate an unbiased number within a given range from a "raw" RNG output which typically generates a stream of 32 or 64bit integers.
Regarding the performance of RNGs themselves it's mostly bound by how "random" your want your NG to be. If you don't really care about quality and need very good performance, for instance to procedurally generate assets in a videogame, there are extremely fast and somewhat decent PRNGs out there, such as XorShift. Of course you won't use that to generate PGP keys...
I only skimmed the article, so maybe they said this, but for choosing from a small range, for example 0..51, you can get several of these from a 32 bit random number with this algorithm
https://stackoverflow.com/questions/6046918/how-to-generate-...
You should be able to run a 64 bit PRNG once and pick at least 8 random cards from a deck.
The article's conclusion was that the PRNG generation method used is usually not the bottleneck, but how you take that to get a result is. Don't know if that applies to the algorithm linked, but the author's point was that bottlenecks are more likely to arise in the code that surrounds the PRNG algorithm than in the call to PRNG itself.
Would you mind talking a little bit more about the scenario and bottleneck? If you're concerned about anonymity you can just email me directly. I'm working on a research project to examine real world scenarios when the fastest cryptographic PRNGs are legitimately insufficient; your case might be useful.
I wonder if the "Bitmask with Rejection" method would be more efficient if you sometimes made the mask one bit larger than strictly necessary.
As it is, if you want a number in the range 0..8, you take 4 bits of randomness, giving you a number in 0..15. This is great, but 7/16 (43.75%) of the time you have to try again. This not only means more loop iterations, it also means you discard 4 bits of randomness, which may have been costly to generate.
If instead you took 5 bits of randomness, you'd be able to accept anything in 0..26 and would only have to reject 27..31, which means only rejecting 5/32 (15.625%) of the time.
0..8 is a particularly bad case, though. If you need numbers in the range 0..14, then it's not worth trying to use 5 bits.
It seems crazy to me that there's no way to produce unbiased numbers in an arbitrary range without rejection sampling and a loop. Is there a proof of this?
Trivial proof. Pigeon hole principle.
Double-precision Floats have more values between 0.0 and 1.0, than between 1.0 and 2.0. In fact, roughly half of ALL double-precision floats exist between -1.0 and 1.0, a very small minority of them exist between 1.0 and 2.0.
To generate unbiased random numbers between 0.0 and 2.0, it therefore requires you to either reject a significant amount of numbers in the 0.0 to 1.0 range, or perform some kind of many-to-few mapping in the 1.0 to 2.0 range.
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With regards to arbitrary INTEGER ranges, the proof is even easier. A random bitstream has 2^number-of-bits possible random values. Which does NOT divide evenly into an arbitrary integer range.
For example, 5-random bits will represent 32-different values. There's no way to map 32-values and divide them evenly into 0-9 (10 numbers).
I'd expect it's possible by changing the generator at the lowest level, but it makes sense to me that you need a loop if you don't control the underlying generator.
Imagine you want to turn a random number in 1..4 into a random number in 1..3. The original is your only source of randomness, so the rest should be deterministic. Then each outcome in 1..4 has to map to exactly one number in 1..3, but there's no mapping that accepts all of 1..4 while still giving each of 1..3 an equal probability.
What if we allow the mapping function to be stateful?
I guess you could save up a few bits over repeated calls, but it can't help you always execute the first call with a single round of generation.
Could it cap the maximum number of generator calls though? Rejection sampling is technically O(infinity) because you could reject an unbounded number of times. This isn't a problem in practice but it sure is theoretically annoying. With a cap on the maximum number of calls, it would be O(1).
I don't think so.
If you want a number in 1..3, and the generator provides numbers from 1..4, and you want a cap n, then you could model it as a generator that provides numbers in 1..4^n. There's never a way to split that space into three equal parts.
You always end up with unwanted extra possibilities that you still need to handle somehow.
You don't need rejection sampling, but you do need a loop. It is easier to see that a finite number of samples is not sufficient:
If you have only a probability distribution defined by a product space where all distinguishable events have probability p^i, for some finite i, then any subset of the distinguishable events accumulate to a probability r * p^i for some integral r. If your goal is a probability that is not an integral multiple of p^i, you are out of luck with a finite number of samples.
Proposition: There is no way, given some integer r and a sequence of iid random integer samples X1,X2,... uniformly distributed in [0,n) for some n>1, to always construct some finite k and function f : [0,n)^k -> [0,r) such that f(X1,...,Xk) is uniformly distributed over [0,r).
Proof: Suppose such a k and f exist. The distribution of U[0,n)^k is isomorphic to that of U[0,n^k) (just treat it like writing down a k-digit number in base n). And so f must be a function from a set of n^k things to a set of r things. By the pigeonhole principle there must be some integers x,y in [0,r) such that the preimage of x has size at least ceiling(n^k/ r) and the preimage of y has size at most floor(n^k/ r). By the fundamental theorem of arithmetic there exists (for any n,k) some r such that n^k/r is not an integer and so the probabilities of x,y (being proportional to the sizes of their fibres under f) are not equal.
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The gist of this is that you always might need to loop (repeatedly sample) for some ranges and you might need to repeat arbitrarily many times.
One way is by rejection.
Another nice thing is if you want to generate a Bernoulli r.v. for some probability p a computable number, you lazily compute the bits of p simultaneously to a random sequence of bits (distributed as Bernoulli(1/2)) and compare the two. If your random sequence is definitely less than p then generate 0. If definitely greater then generate 1. If not sure then generate some more bits.
In this way any Bernoulli random variable may be generated from an infinite sequence of iid Bernoulli(1/2), and basically any probability space can be modelled in this way too. In this sense, all of probability can be built out of tosses of a fair coin)
Personally I'm a fan of the xoshiro[1] generator I have found it to be faster and give more equiprobable outputs.
The article discusses the xoshiro RNG, and some better alternatives: “One other concern we might have is that some generators have weak low-order bits. For example, the Xoroshiro+ and Xoshiro+ families of PRNGs have low-order bits that fail statistical tests.“
The xoshiro page mentions this too, and says it won’t matter if you are generating random floats, but the article is generating random ints.
The RNG isn’t the point of the article though. It’s discussing the speed and correctness of what happens after the RNG but before you use the results.
Note that this article is not about the random number generating scheme, but about post-processing to get a random number within a certain range. That said, the author specifically points out xoshiro as a random number generating scheme to watch out for:
One other concern we might have is that some generators have weak low-order bits. For example, the Xoroshiro+ and Xoshiro+ families of PRNGs have low-order bits that fail statistical tests. When we perform % 52 (because 52 is even) we pass the lowest bit straight through into the output.
xoshiro has flaws: http://www.pcg-random.org/posts/a-quick-look-at-xoshiro256.h...
xoshiro's response: http://pcg.di.unimi.it/pcg.php
Melissa O'Neill's response back: http://www.pcg-random.org/posts/on-vignas-pcg-critique.html
Thanks. This completely allays my concerns about the PCG's I would actually use.
Edit: And I like that MCG with 64-bit multiplicand that she shows. I might switch to that for Monte Carlo applications.
Interesting read, thanks! The deflection about xoshiro is not particularly convincing, though. It's much more likely that you'll want to multiply your random stream by a multiple of 57 than you'll want to xor it with a 43-bit-shifted version of itself. He also doesn't appear to counter the complaint about the generator getting stuck around 0: http://www.pcg-random.org/posts/xoshiro-repeat-flaws.html
The other parts of the post definitely concern me about using the PCG's described, though. Also, it's interesting to see how PCG can be predicted. I would not have known how to attack it.
His "attack" is literally a brute force of 32 bits of state. It would not scale to PCG with 128 bits of internal state.