Stochastic Rounding and its Probabilistic Backward Error Analysis

Connolly, Michael P. and Higham, Nicholas J. and Mary, Theo (2020) Stochastic Rounding and its Probabilistic Backward Error Analysis. [MIMS Preprint]

Stochastic rounding rounds a real number to the next larger or smaller floating-point number with probabilities $1$ minus the relative distances to those numbers. % It has a larger worst-case error than round to nearest % but has useful statistical properties. It is gaining attention in deep learning because it can improve the accuracy of the computations. We compare basic properties of stochastic rounding with those for round to nearest, finding properties in common as well as significant differences. We prove that for stochastic rounding the rounding errors are mean independent random variables with zero mean. We derive a new version of our probabilistic error analysis theorem from [{\em SIAM J. Sci. Comput.}, 41 (2019), pp.\ A2815--A2835], weakening the assumption of independence of the random variables to mean independence. These results imply that for a wide range of linear algebra computations the backward error for stochastic rounding is unconditionally bounded by a multiple of $\sqrt{n}u$ to first order, with a certain probability, where $n$ is the problem size and $u$ is the unit roundoff. This is the first scenario where the rule of thumb that one can replace $nu$ by $\sqrt{n}u$ in a rounding error bound has been shown to hold without any additional assumptions on the rounding errors. We also explain how stochastic rounding avoids the phenomenon of stagnation in sums, whereby small addends are obliterated by round to nearest when they are too small relative to the sum.