论文标题
随机圆形方差和概率界限：一种新方法
Stochastic rounding variance and probabilistic bounds: A new approach
论文作者
论文摘要
随机圆形（SR）提供了确定性IEEE-754浮点圆形模式的替代方案。在某些应用（例如PDE，ODE和神经网络）中，SR经验从经验上改善了数值行为和收敛到准确的解决方案，而没有提供合理的理论背景。 Ipsen，Zhou，Higham和Mary的最新作品为基本线性代数内核计算了SR概率误差界。例如，向前误差的内部产品SR概率界限与$ \ sqrt $ nu成正比，而不是nu对于默认的舍入模式。为了计算边界，这些作品表明，计算中累积的错误形成了martingale。本文提出了一个替代框架，以根据方差的计算来表征SR错误。我们在数值算法中指出了常见的误差模式，并提出了一个界限其方差的引理。对于每个概率以及通过bienaym {é} -Chebyshev不平等，这种结合会导致在几种情况下绑定的更好的概率误差。我们的方法具有为所有符合我们模型的算法提供紧密概率绑定的优势。我们展示了如何应用该方法来为内部产品和Horner多项式评估提供SR误差界。
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{é}-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.
