Github user mengxr commented on a diff in the pull request:
https://github.com/apache/spark/pull/1025#discussion_r15534415
--- Diff:
core/src/main/scala/org/apache/spark/util/random/SamplingUtils.scala ---
@@ -88,14 +91,73 @@ private[spark] object SamplingUtils {
*/
def computeFractionForSampleSize(sampleSizeLowerBound: Int, total: Long,
withReplacement: Boolean): Double = {
- val fraction = sampleSizeLowerBound.toDouble / total
if (withReplacement) {
- val numStDev = if (sampleSizeLowerBound < 12) 9 else 5
- fraction + numStDev * math.sqrt(fraction / total)
+ PoissonBounds.getUpperBound(sampleSizeLowerBound) / total
} else {
- val delta = 1e-4
- val gamma = - math.log(delta) / total
- math.min(1, fraction + gamma + math.sqrt(gamma * gamma + 2 * gamma *
fraction))
+ val fraction = sampleSizeLowerBound.toDouble / total
+ BinomialBounds.getUpperBound(1e-4, total, fraction)
}
}
}
+
+/**
+ * Utility functions that help us determine bounds on adjusted sampling
rate to guarantee exact
+ * sample sizes with high confidence when sampling with replacement.
+ */
+private[spark] object PoissonBounds {
+
+ /**
+ * Returns a lambda such that Pr[X > s] is very small, where X ~
Pois(lambda).
+ */
+ def getLowerBound(s: Double): Double = {
+ math.max(s - numStd(s) * math.sqrt(s), 1e-15)
+ }
+
+ /**
+ * Returns a lambda such that Pr[X < s] is very small, where X ~
Pois(lambda).
+ *
+ * @param s sample size
+ */
+ def getUpperBound(s: Double): Double = {
+ math.max(s + numStd(s) * math.sqrt(s), 1e-10)
+ }
+
+ private def numStd(s: Double): Double = {
+ // TODO: Make it tighter.
+ if (s < 6.0) {
+ 12.0
+ } else if (s < 16.0) {
+ 9.0
+ } else {
+ 6.0
+ }
+ }
+}
+
+/**
+ * Utility functions that help us determine bounds on adjusted sampling
rate to guarantee exact
+ * sample size with high confidence when sampling without replacement.
+ */
+private[spark] object BinomialBounds {
+
+ val minSamplingRate = 1e-10
+
+ /**
+ * Returns a threshold `p` such that if we conduct n Bernoulli trials
with success rate = `p`,
+ * it is very unlikely to have more than `fraction * n` successes.
+ */
+ def getLowerBound(delta: Double, n: Long, fraction: Double): Double = {
+ val gamma = - math.log(delta) / n * (2.0 / 3.0)
+ math.max(minSamplingRate,
--- End diff --
It should be okay to return `0` here. The `minSamplingRate` should be
applied to `getUpperBound`.
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