/* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*- */
/* vim: set ts=8 sts=2 et sw=2 tw=80: */
/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/. */
#ifndef mozilla_FastBernoulliTrial_h
#define mozilla_FastBernoulliTrial_h
#include "mozilla/Assertions.h"
#include "mozilla/XorShift128PlusRNG.h"
#include <cmath>
#include <stdint.h>
namespace mozilla {
/**
* class FastBernoulliTrial: Efficient sampling with uniform probability
*
* When gathering statistics about a program's behavior, we may be observing
* events that occur very frequently (e.g., function calls or memory
* allocations) and we may be gathering information that is somewhat expensive
* to produce (e.g., call stacks). Sampling all the events could have a
* significant impact on the program's performance.
*
* Why not just sample every N'th event? This technique is called "systematic
* sampling"; it's simple and efficient, and it's fine if we imagine a
* patternless stream of events. But what if we're sampling allocations, and the
* program happens to have a loop where each iteration does exactly N
* allocations? You would end up sampling the same allocation every time through
* the loop; the entire rest of the loop becomes invisible to your measurements!
* More generally, if each iteration does M allocations, and M and N have any
* common divisor at all, most allocation sites will never be sampled. If
* they're both even, say, the odd-numbered allocations disappear from your
* results.
*
* Ideally, we'd like each event to have some probability P of being sampled,
* independent of its neighbors and of its position in the sequence. This is
* called "Bernoulli sampling", and it doesn't suffer from any of the problems
* mentioned above.
*
* One disadvantage of Bernoulli sampling is that you can't be sure exactly how
* many samples you'll get: technically, it's possible that you might sample
* none of them, or all of them. But if the number of events N is large, these
* aren't likely outcomes; you can generally expect somewhere around P * N
* events to be sampled.
*
* The other disadvantage of Bernoulli sampling is that you have to generate a
* random number for every event, which can be slow.
*
* [significant pause]
*
* BUT NOT WITH THIS CLASS! FastBernoulliTrial lets you do true Bernoulli
* sampling, while generating a fresh random number only when we do decide to
* sample an event, not on every trial. When it decides not to sample, a call to
* |FastBernoulliTrial::trial| is nothing but decrementing a counter and
* comparing it to zero. So the lower your sampling probability is, the less
* overhead FastBernoulliTrial imposes.
*
* Probabilities of 0 and 1 are handled efficiently. (In neither case need we
* ever generate a random number at all.)
*
* The essential API:
*
* - FastBernoulliTrial(double P)
* Construct an instance that selects events with probability P.
*
* - FastBernoulliTrial::trial()
* Return true with probability P. Call this each time an event occurs, to
* decide whether to sample it or not.
*
* - FastBernoulliTrial::trial(size_t n)
* Equivalent to calling trial() |n| times, and returning true if any of those
* calls do. However, like trial, this runs in fast constant time.
*
* What is this good for? In some applications, some events are "bigger" than
* others. For example, large allocations are more significant than small
* allocations. Perhaps we'd like to imagine that we're drawing allocations
* from a stream of bytes, and performing a separate Bernoulli trial on every
* byte from the stream. We can accomplish this by calling |t.trial(S)| for
* the number of bytes S, and sampling the event if that returns true.
*
* Of course, this style of sampling needs to be paired with analysis and
* presentation that makes the size of the event apparent, lest trials with
* large values for |n| appear to be indistinguishable from those with small
* values for |n|.
*/
class FastBernoulliTrial {
/*
* This comment should just read, "Generate skip counts with a geometric
* distribution", and leave everyone to go look that up and see why it's the
* right thing to do, if they don't know already.
*
* BUT IF YOU'RE CURIOUS, COMMENTS ARE FREE...
*
* Instead of generating a fresh random number for every trial, we can
* randomly generate a count of how many times we should return false before
* the next time we return true. We call this a "skip count". Once we've
* returned true, we generate a fresh skip count, and begin counting down
* again.
*
* Here's an awesome fact: by exercising a little care in the way we generate
* skip counts, we can produce results indistinguishable from those we would
* get "rolling the dice" afresh for every trial.
*
* In short, skip counts in Bernoulli trials of probability P obey a geometric
* distribution. If a random variable X is uniformly distributed from [0..1),
* then std::floor(std::log(X) / std::log(1-P)) has the appropriate geometric
* distribution for the skip counts.
*
* Why that formula?
*
* Suppose we're to return |true| with some probability P, say, 0.3. Spread
* all possible futures along a line segment of length 1. In portion P of
* those cases, we'll return true on the next call to |trial|; the skip count
* is 0. For the remaining portion 1-P of cases, the skip count is 1 or more.
*
* skip: 0 1 or more
* |------------------^-----------------------------------------|
* portion: 0.3 0.7
* P 1-P
*
* But the "1 or more" section of the line is subdivided the same way: *within
* that section*, in portion P the second call to |trial()| returns true, and
* in portion 1-P it returns false a second time; the skip count is two or
* more. So we return true on the second call in proportion 0.7 * 0.3, and
* skip at least the first two in proportion 0.7 * 0.7.
*
* skip: 0 1 2 or more
* |------------------^------------^----------------------------|
* portion: 0.3 0.7 * 0.3 0.7 * 0.7
* P (1-P)*P (1-P)^2
*
* We can continue to subdivide:
*
* skip >= 0: |------------------------------------------------- (1-P)^0 --|
* skip >= 1: | ------------------------------- (1-P)^1 --|
* skip >= 2: | ------------------ (1-P)^2 --|
* skip >= 3: | ^ ---------- (1-P)^3 --|
* skip >= 4: | . --- (1-P)^4 --|
* .
* ^X, see below
*
* In other words, the likelihood of the next n calls to |trial| returning
* false is (1-P)^n. The longer a run we require, the more the likelihood
* drops. Further calls may return false too, but this is the probability
* we'll skip at least n.
*
* This is interesting, because we can pick a point along this line segment
* and see which skip count's range it falls within; the point X above, for
* example, is within the ">= 2" range, but not within the ">= 3" range, so it
* designates a skip count of 2. So if we pick points on the line at random
* and use the skip counts they fall under, that will be indistinguishable
* from generating a fresh random number between 0 and 1 for each trial and
* comparing it to P.
*
* So to find the skip count for a point X, we must ask: To what whole power
* must we raise 1-P such that we include X, but the next power would exclude
* it? This is exactly std::floor(std::log(X) / std::log(1-P)).
*
* Our algorithm is then, simply: When constructed, compute an initial skip
* count. Return false from |trial| that many times, and then compute a new
* skip count.
*
* For a call to |trial(n)|, if the skip count is greater than n, return false
* and subtract n from the skip count. If the skip count is less than n,
* return true and compute a new skip count. Since each trial is independent,
* it doesn't matter by how much n overshoots the skip count; we can actually
* compute a new skip count at *any* time without affecting the distribution.
* This is really beautiful.
*/
public:
/**
* Construct a fast Bernoulli trial generator. Calls to |trial()| return true
* with probability |aProbability|. Use |aState0| and |aState1| to seed the
* random number generator; both may not be zero.
*/
FastBernoulliTrial(double aProbability, uint64_t aState0, uint64_t aState1)
: mProbability(0),
mInvLogNotProbability(0),
mGenerator(aState0, aState1),
mSkipCount(0) {
setProbability(aProbability);
}
/**
* Return true with probability |mProbability|. Call this each time an event
* occurs, to decide whether to sample it or not. The lower |mProbability| is,
* the faster this function runs.
*/
bool trial() {
if (mSkipCount) {
mSkipCount--;
return false;
}
return chooseSkipCount();
}
/**
* Equivalent to calling trial() |n| times, and returning true if any of those
* calls do. However, like trial, this runs in fast constant time.
*
* What is this good for? In some applications, some events are "bigger" than
* others. For example, large allocations are more significant than small
* allocations. Perhaps we'd like to imagine that we're drawing allocations
* from a stream of bytes, and performing a separate Bernoulli trial on every
* byte from the stream. We can accomplish this by calling |t.trial(S)| for
* the number of bytes S, and sampling the event if that returns true.
*
* Of course, this style of sampling needs to be paired with analysis and
* presentation that makes the "size" of the event apparent, lest trials with
* large values for |n| appear to be indistinguishable from those with small
* values for |n|, despite being potentially much more likely to be sampled.
*/
bool trial(size_t aCount) {
if (mSkipCount > aCount) {
mSkipCount -= aCount;
return false;
}
return chooseSkipCount();
}
void setRandomState(uint64_t aState0, uint64_t aState1) {
mGenerator.setState(aState0, aState1);
}
void setProbability(double aProbability) {
MOZ_ASSERT(0 <= aProbability && aProbability <= 1);
mProbability = aProbability;
if (0 < mProbability && mProbability < 1) {
/*
* Let's look carefully at how this calculation plays out in floating-
* point arithmetic. We'll assume IEEE, but the final C++ code we arrive
* at would still be fine if our numbers were mathematically perfect. So,
* while we've considered IEEE's edge cases, we haven't done anything that
* should be actively bad when using other representations.
*
* (In the below, read comparisons as exact mathematical comparisons: when
* we say something "equals 1", that means it's exactly equal to 1. We
* treat approximation using intervals with open boundaries: saying a
* value is in (0,1) doesn't specify how close to 0 or 1 the value gets.
* When we use closed boundaries like [2**-53, 1], we're careful to ensure
* the boundary values are actually representable.)
*
* - After the comparison above, we know mProbability is in (0,1).
*
* - The gaps below 1 are 2**-53, so that interval is (0, 1-2**-53].
*
* - Because the floating-point gaps near 1 are wider than those near
* zero, there are many small positive doubles ε such that 1-ε rounds to
* exactly 1. However, 2**-53 can be represented exactly. So
* 1-mProbability is in [2**-53, 1].
*
* - log(1 - mProbability) is thus in (-37, 0].
*
* That range includes zero, but when we use mInvLogNotProbability, it
* would be helpful if we could trust that it's negative. So when log(1
* - mProbability) is 0, we'll just set mProbability to 0, so that
* mInvLogNotProbability is not used in chooseSkipCount.
*
* - How much of the range of mProbability does this cause us to ignore?
* The only value for which log returns 0 is exactly 1; the slope of log
* at 1 is 1, so for small ε such that 1 - ε != 1, log(1 - ε) is -ε,
* never 0. The gaps near one are larger than the gaps near zero, so if
* 1 - ε wasn't 1, then -ε is representable. So if log(1 - mProbability)
* isn't 0, then 1 - mProbability isn't 1, which means that mProbability
* is at least 2**-53, as discussed earlier. This is a sampling
* likelihood of roughly one in ten trillion, which is unlikely to be
* distinguishable from zero in practice.
*
* So by forbidding zero, we've tightened our range to (-37, -2**-53].
*
* - Finally, 1 / log(1 - mProbability) is in [-2**53, -1/37). This all
* falls readily within the range of an IEEE double.
*
* ALL THAT HAVING BEEN SAID: here are the five lines of actual code:
*/
double logNotProbability = std::log(1 - mProbability);
if (logNotProbability == 0.0)
mProbability = 0.0;
else
mInvLogNotProbability = 1 / logNotProbability;
}
chooseSkipCount();
}
private:
/* The likelihood that any given call to |trial| should return true. */
double mProbability;
/*
* The value of 1/std::log(1 - mProbability), cached for repeated use.
*
* If mProbability is exactly 0 or exactly 1, we don't use this value.
* Otherwise, we guarantee this value is in the range [-2**53, -1/37), i.e.
* definitely negative, as required by chooseSkipCount. See setProbability for
* the details.
*/
double mInvLogNotProbability;
/* Our random number generator. */
non_crypto::XorShift128PlusRNG mGenerator;
/* The number of times |trial| should return false before next returning true.
*/
size_t mSkipCount;
/*
* Choose the next skip count. This also returns the value that |trial| should
* return, since we have to check for the extreme values for mProbability
* anyway, and |trial| should never return true at all when mProbability is 0.
*/
bool chooseSkipCount() {
/*
* If the probability is 1.0, every call to |trial| returns true. Make sure
* mSkipCount is 0.
*/
if (mProbability == 1.0) {
mSkipCount = 0;
return true;
}
/*
* If the probabilility is zero, |trial| never returns true. Don't bother us
* for a while.
*/
if (mProbability == 0.0) {
mSkipCount = SIZE_MAX;
return false;
}
/*
* What sorts of values can this call to std::floor produce?
*
* Since mGenerator.nextDouble returns a value in [0, 1-2**-53], std::log
* returns a value in the range [-infinity, -2**-53], all negative. Since
* mInvLogNotProbability is negative (see its comments), the product is
* positive and possibly infinite. std::floor returns +infinity unchanged.
* So the result will always be positive.
*
* Converting a double to an integer that is out of range for that integer
* is undefined behavior, so we must clamp our result to SIZE_MAX, to ensure
* we get an acceptable value for mSkipCount.
*
* The clamp is written carefully. Note that if we had said:
*
* if (skipCount > SIZE_MAX)
* skipCount = SIZE_MAX;
*
* that leads to undefined behavior 64-bit machines: SIZE_MAX coerced to
* double is 2^64, not 2^64-1, so this doesn't actually set skipCount to a
* value that can be safely assigned to mSkipCount.
*
* Jakob Olesen cleverly suggested flipping the sense of the comparison: if
* we require that skipCount < SIZE_MAX, then because of the gaps (2048)
* between doubles at that magnitude, the highest double less than 2^64 is
* 2^64 - 2048, which is fine to store in a size_t.
*
* (On 32-bit machines, all size_t values can be represented exactly in
* double, so all is well.)
*/
double skipCount =
std::floor(std::log(mGenerator.nextDouble()) * mInvLogNotProbability);
if (skipCount < SIZE_MAX)
mSkipCount = skipCount;
else
mSkipCount = SIZE_MAX;
return true;
}
};
} /* namespace mozilla */
#endif /* mozilla_FastBernoulliTrial_h */