论文标题
随机舍入数值算法的积极影响
The Positive Effects of Stochastic Rounding in Numerical Algorithms
论文作者
论文摘要
最近，随机舍入（SR）已在专业硬件中实现，但是当前大多数计算节点尚未支持此舍入模式。几项作品从经验上说明了在神经网络和普通微分方程等各个领域中随机舍入的好处。对于某些算法，例如总和，内部产品或矩阵向量乘法，已证明SR比传统的确定性界限更好地提供了概率误差界限。在本文中，我们扩展了在计算机架构中更广泛采用SR的理论基础。首先，我们分析了两种SR模式的偏见：SR-NEARNESS和SR-UP或DOWN。我们在欧拉（Euler）的前进方法的一个案例研究中证明了IEEE-754默认舍入模式和sr-up或down的默认舍入式舍入错误，而sr-nearness却没有偏见。其次，我们证明了与SR的霍纳多项式评估方法的正向误差有关的o（$ \ sqrt $ n）概率，从而改善了已知的确定性O（n）结合。
Recently, stochastic rounding (SR) has been implemented in specialized hardware but most current computing nodes do not yet support this rounding mode. Several works empirically illustrate the benefit of stochastic rounding in various fields such as neural networks and ordinary differential equations. For some algorithms, such as summation, inner product or matrixvector multiplication, it has been proved that SR provides probabilistic error bounds better than the traditional deterministic bounds. In this paper, we extend this theoretical ground for a wider adoption of SR in computer architecture. First, we analyze the biases of the two SR modes: SR-nearness and SR-up-or-down. We demonstrate on a case-study of Euler's forward method that IEEE-754 default rounding modes and SR-up-or-down accumulate rounding errors across iterations and that SR-nearness, being unbiased, does not. Second, we prove a O($\sqrt$ n) probabilistic bound on the forward error of Horner's polynomial evaluation method with SR, improving on the known deterministic O(n) bound.
