tor-browser

FastBernoulliTrial.h (16995B)

---

/* -*- 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 > double(SIZE_MAX))
    *       mSkipCount = SIZE_MAX;
    *    else
    *       mSkipCount = skipCount;
    *
    * that leads to undefined behavior 64-bit machines: SIZE_MAX coerced to
    * double can equal 2^64, so if skipCount equaled 2^64 converting it to
    * size_t would induce undefined behavior.
    *
    * Jakob Olesen cleverly suggested flipping the sense of the comparison to
    * skipCount < double(SIZE_MAX).  The conversion will evaluate to 2^64 or
    * the double just below it: either way, skipCount is guaranteed to have a
    * value that's safely convertible to 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 < double(SIZE_MAX))
     mSkipCount = skipCount;
   else
     mSkipCount = SIZE_MAX;

   return true;
 }
};

} /* namespace mozilla */

#endif /* mozilla_FastBernoulliTrial_h */