Stochastic rounding variance and probabilistic bounds: A new approach

TL;DR

This paper introduces a new variance-based framework for analyzing stochastic rounding errors, providing tighter probabilistic bounds for numerical algorithms like inner products and polynomial evaluation.

Contribution

It proposes an alternative approach to error analysis based on variance, improving probabilistic bounds over existing martingale-based methods.

Findings

Variance-based bounds are tighter than previous error bounds.

Applicable to inner product and polynomial evaluation algorithms.

Provides a unified framework for probabilistic error analysis.

Abstract

Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs and neural networks, SR empirically improves the numerical behavior and convergence to accurate solutions while no sound theoretical background has been provided. Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic error bounds for basic linear algebra kernels. For example, the inner product SR probabilistic bound of the forward error is proportional to $\sqrt$ nu instead of nu for the default rounding mode. To compute the bounds, these works show that the errors accumulated in computation form a martingale. This paper proposes an alternative framework to characterize SR errors based on the computation of the variance. We pinpoint common error patterns in numerical algorithms and propose a lemma that bounds their variance. For each probability and through Bienaym{\'e}-Chebyshev inequality, this bound leads to better probabilistic error bound in several situations. Our method has the advantage of providing a tight probabilistic bound for all algorithms fitting our model. We show how the method can be applied to give SR error bounds for the inner product and Horner polynomial evaluation.