Stochastic rounding variance and probabilistic bounds: A new approach

E. E. Arar, D. Sohier, P. D. O. Castro, E. Petit
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引用次数: 1

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.
随机舍入方差和概率界:一种新方法
随机舍入(SR)为确定性的IEEE-754浮点舍入模式提供了另一种选择。在偏微分方程(PDEs)、偏微分方程(ode)和神经网络等一些应用中,SR经验地改善了数值行为和收敛到精确解的能力,但没有提供可靠的理论背景。Ipsen, Zhou, Higham和Mary最近的工作已经计算了基本线性代数核的SR概率误差界限。例如,前向误差的内积SR概率界与$\sqrt$ nu成正比,而不是默认舍入模式下的nu。为了计算边界,这些工作表明,计算中累积的误差形成一个鞅。本文提出了一种基于方差计算来表征SR误差的替代框架。我们指出了数值算法中常见的错误模式,并提出了一个引理来限制它们的方差。对于每个概率,并通过Bienaym{\'e}-Chebyshev不等式,该界在几种情况下导致更好的概率误差界。我们的方法的优点是为所有拟合我们模型的算法提供了一个紧密的概率界。我们展示了如何将该方法应用于内积和Horner多项式评估的SR误差界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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