Stochastic rounding: implementation, error analysis and applications.

Stochastic rounding (SR) randomly maps a real number to one of the two nearest values in a finite precision number system. The probability of choosing either of these two numbers is 1 minus their relative distance to . This rounding mode was first proposed for use in computer arithmetic in the 1950s and it is currently experiencing a resurgence of interest. If used to compute the inner product of two vectors of length in floating-point arithmetic, it yields an error bound with constant with high probability, where is the unit round-off. This is not necessarily the case for round to nearest (RN), for which the worst-case error bound has constant . A particular attraction of SR is that, unlike RN, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity is lost. We survey SR by discussing its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, with a focus on machine learning and the numerical solution of differential equations.