Stochastic rounding: implementation, error analysis and applications

Croci, Matteo; Fasi, Massimiliano; Higham, Nicholas J.; Mary, Theo; Mikaitis, Mantas

Abstract

Stochastic rounding (SR) randomly maps a real number x 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 x. 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 n in floating-point arithmetic, it yields an error bound with constant n−−√u with high probability, where u 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 nu. 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.

Citation

Croci, M., Fasi, M., Higham, N. J., Mary, T., & Mikaitis, M. (2022). Stochastic rounding: implementation, error analysis and applications. Royal Society Open Science, 9(3), Article 211631. https://doi.org/10.1098/rsos.211631

• Journal Article Type: Article
• Acceptance Date: Feb 4, 2022
• Online Publication Date: Mar 9, 2022
• Publication Date: 2022-03
• Deposit Date: Mar 14, 2022
• Publicly Available Date: Jun 30, 2022
• Journal: Royal Society Open Science
• Electronic ISSN: 2054-5703
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 9
• Issue: 3
• Article Number: 211631
• DOI: https://doi.org/10.1098/rsos.211631
• Public URL: https://durham-repository.worktribe.com/output/1212255
• Related Public URLs: http://eprints.maths.manchester.ac.uk/2843/