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Stochastic rounding: implementation, error analysis and applications.

Matteo Croci1, Massimiliano Fasi2, Nicholas J Higham3

  • 1Oden Institute, University of Texas at Austin, Austin, TX, 78712, USA.

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|March 16, 2022
PubMed
Summary
This summary is machine-generated.

Stochastic rounding (SR) offers improved accuracy in floating-point computations by randomly selecting the nearest representable number. This method enhances numerical stability, particularly in machine learning and differential equation solving.

Keywords:
IEEE 754bfloat16binary16floating-point arithmeticmachine learningrounding error analysis

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Area of Science:

  • Computer Science
  • Numerical Analysis
  • Applied Mathematics

Background:

  • Finite precision arithmetic poses challenges in scientific computing.
  • Traditional rounding methods like round-to-nearest (RN) can introduce significant errors and suffer from stagnation.
  • Stochastic rounding (SR), a probabilistic rounding technique, is gaining renewed interest for its potential to mitigate these issues.

Purpose of the Study:

  • To survey the mathematical properties, probabilistic error analysis, and implementation of stochastic rounding.
  • To highlight the advantages of SR over traditional rounding methods, particularly in computational accuracy.
  • To explore the applications of SR in machine learning and the numerical solution of differential equations.

Main Methods:

  • Probabilistic error analysis of stochastic rounding.
  • Comparison of error bounds between SR and round-to-nearest (RN).
  • Implementation strategies for SR in computer arithmetic.

Main Results:

  • SR provides a high-probability error bound with a constant factor of for inner product computations, outperforming RN's worst-case bound of .
  • SR is immune to stagnation, a critical issue where small updates are lost in RN.
  • SR demonstrates practical utility in machine learning and numerical differential equation solving.

Conclusions:

  • Stochastic rounding offers a robust alternative to traditional rounding methods in floating-point arithmetic.
  • Its probabilistic nature and immunity to stagnation lead to improved numerical stability and accuracy.
  • SR is a valuable technique for enhancing computations in demanding applications like machine learning and scientific simulations.