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.

翻译：随机舍入（SR）提供了确定性IEEE-754浮点舍入模式的替代方案。在偏微分方程、常微分方程和神经网络等应用中，尽管缺乏可靠的理论背景，SR凭经验改善了数值行为并提高了对精确解的收敛性。Ipsen、Zhou、Higham和Mary近期的工作已计算出基本线性代数内核的SR概率误差边界。例如，内积前向误差的SR概率边界与$\sqrt\nu$成正比，而默认舍入模式则与$\nu$成正比。为计算这些边界，研究发现计算过程中累积的误差构成鞅。本文提出一种基于方差计算的替代框架来表征SR误差。我们识别数值算法中的常见误差模式，并提出一个限定其方差的引理。通过Bienaymé-切比雪夫不等式，该边界在多种情况下能导出更优的概率误差界。我们的方法能为所有符合模型的算法提供紧凑的概率边界，并展示了如何将其应用于内积和Horner多项式求值的SR误差边界计算。