Editing Rounding (section)

===Randomized rounding to an integer===

====Alternating tie-breaking====
One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second occurrence, round down, and so on. Alternatively, the first 0.5 fractional part rounding can be determined by a [[random seed]].  "Up" and "down" can be any two rounding methods that oppose each other - toward and away from positive infinity or toward and away from zero.

If occurrences of 0.5 fractional parts occur significantly more than a restart of the occurrence "counting", then it is effectively bias free. With guaranteed zero bias, it is useful if the numbers are to be summed or averaged.

====Random tie-breaking====
If the fractional part of {{mvar|x}} is 0.5, choose {{mvar|y}} randomly between {{math|''x'' + 0.5}} and {{math|''x'' − 0.5}}, with equal probability. All others are rounded to the closest integer.

Like round-half-to-even and round-half-to-odd, this rule is essentially free of overall bias, but it is also fair among even and odd {{mvar|y}} values. An advantage over alternate tie-breaking is that the last direction of rounding on the 0.5 fractional part does not have to be "remembered".

====Stochastic rounding====
Rounding as follows to one of the closest integer toward negative infinity and the closest integer toward positive infinity, with a probability dependent on the proximity is called [[stochastic]] rounding and will give an unbiased result on average.<ref name="stochastic">{{cite arXiv |title=Deep Learning with Limited Numerical Precision |eprint=1502.02551 |first1=Suyog |last1=Gupta |first2=Ankur |last2=Angrawl |first3=Kailash |last3=Gopalakrishnan |first4=Pritish |last4=Narayanan |page=3 |date=9 February 2016 |class=cs.LG}}</ref>
:<math>\operatorname {Round} (x) = \begin{cases}
\lfloor x \rfloor & \text { with probability } 1 - (x - \lfloor x \rfloor) = \lfloor x \rfloor - x + 1 \\[5mu]
\lfloor x \rfloor + 1 & \text { with probability } {x - \lfloor x \rfloor}
\end{cases}
</math>

For example, 1.6 would be rounded to 1 with probability 0.4 and to 2 with probability 0.6.

Stochastic rounding can be accurate in a way that a rounding [[function (mathematics)|function]] can never be. For example, suppose one started with 0 and added 0.3 to that one hundred times while rounding the running total between every addition. The result would be 0 with regular rounding, but with stochastic rounding, the expected result would be 30, which is the same value obtained without rounding. This can be useful in [[machine learning]] where the training may use low precision arithmetic iteratively.<ref name="stochastic"/> Stochastic rounding is also a way to achieve 1-dimensional [[dithering]].