Academic Journal
STOCHASTIC ROUNDING VARIANCE AND PROBABILISTIC BOUNDS: A NEW APPROACH
| العنوان: | STOCHASTIC ROUNDING VARIANCE AND PROBABILISTIC BOUNDS: A NEW APPROACH |
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| المؤلفون: | El Arar, El-Mehdi, Sohier, Devan, de Oliveira Castro, Pablo, Petit, Eric |
| المساهمون: | Laboratoire d'Informatique Parallélisme Réseaux Algorithmes Distribués (LI-PaRAD), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), Université Paris-Saclay, Intel Corporation Hillsboro, Intel Corporation USA, ANR-20-CE46-0009,INTERFLOP,Plateforme d'analyse pour l'arithmétique flottante(2020) |
| المصدر: | ISSN: 1064-8275 ; SIAM Journal on Scientific Computing ; https://hal.science/hal-03722888 ; SIAM Journal on Scientific Computing, In press, 45 (5), pp.C255-C275. ⟨10.1137/22M1510819⟩. |
| بيانات النشر: | HAL CCSD Society for Industrial and Applied Mathematics |
| سنة النشر: | 2023 |
| المجموعة: | Université de Versailles Saint-Quentin-en-Yvelines: HAL-UVSQ |
| مصطلحات موضوعية: | Stochastic rounding, Floating-point arithmetic, Concentration inequality, Inner product, Polynomial evaluation, Horner algorithm, 65G50, 65F05, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA] |
| الوصف: | 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 √ 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 Bienaymé-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. |
| نوع الوثيقة: | article in journal/newspaper |
| اللغة: | English |
| Relation: | info:eu-repo/semantics/altIdentifier/arxiv/2207.10321; hal-03722888; https://hal.science/hal-03722888; https://hal.science/hal-03722888v3/document; https://hal.science/hal-03722888v3/file/main.pdf; ARXIV: 2207.10321 |
| DOI: | 10.1137/22M1510819 |
| الاتاحة: | https://hal.science/hal-03722888 https://hal.science/hal-03722888v3/document https://hal.science/hal-03722888v3/file/main.pdf https://doi.org/10.1137/22M1510819 |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.92F2E7E0 |
| قاعدة البيانات: | BASE |
| DOI: | 10.1137/22M1510819 |
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