arXiv subjects

El-Mehdi El Arar

Publications and source records attributed to El-Mehdi El Arar.

8 recordsLinked to original sources

arXiv2026

What is New in Stochastic Rounding: a Survey on Theory, Hardware, and Applications

Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length $n$ summation, the worst-case error of SR grows as $\sqrt{n}$ with high probability, unlike for standard modes, like round-to-nearest (RN), which grows as $n$. For this reason, the former is increasingly employed in large-scale, low-precision computations as an RN alternative. Additionally, SR alleviates stagnation, whereby relatively small summands are completely rounded off and do not contribute to the sum. We provide an update to [Croci et al., Roy. Soc. Open Sci. 9.3 (2022), pp. 1-25], a survey which discusses the development and use of SR between 1949 and 2022, citing over 100 references. Since then, there has been a surge of new research, and this update covers almost four years of further progress in applying, analysing, and implementing SR. Our main focus is limited-precision stochastic rounding, a new variant that fixes the precision of the random numbers used. We provide insights into industrial and numerical analysis activities surrounding SR, highlighting the next possible steps in making this rounding mode more widely available in hardware.

math.NA↗

arXiv2026

Probabilistic Error Analysis of Limited-Precision Stochastic Rounding: Horner's Algorithm and Pairwise Summation

Stochastic rounding (SR) is a probabilistic rounding mode that mitigates errors in large-scale numerical computations, especially when prone to stagnation effects. Beyond numerical analysis, SR has shown significant benefits in practical applications such as deep learning and climate modelling. The definition of classical SR requires that results of arithmetic operations are known with infinite precision. This is often not possible, and when it is, the resulting hardware implementation can become prohibitively expensive in terms of energy, area, and latency. A more practical alternative is limited-precision SR, which only requires that the outputs of arithmetic operations are available in higher, finite, precision. We extend previous work on limited-precision SR presented in [El Arar et al., SIAM J. Sci. Comput. 47(5) (2025), B1227-B1249], which developed a framework to evaluate the trade-off between accuracy and hardware resource cost in SR implementations. Within this framework, we study the Horner algorithm and pairwise summation, providing both theoretical insights and practical experiments in these settings when using limited-precision SR.

math.NA↗

arXiv2025

Mixed precision accumulation for neural network inference guided by componentwise forward error analysis

This work proposes a mathematically founded mixed precision accumulation strategy for the inference of neural networks. Our strategy is based on a new componentwise forward error analysis that explains the propagation of errors in the forward pass of neural networks. Specifically, our analysis shows that the error in each component of the output of a linear layer is proportional to the condition number of the inner product between the weights and the input, multiplied by the condition number of the activation function. These condition numbers can vary widely from one component to the other, thus creating a significant opportunity to introduce mixed precision: each component should be accumulated in a precision inversely proportional to the product of these condition numbers. We propose a numerical algorithm that exploits this observation: it first computes all components in low precision, uses this output to estimate the condition numbers, and recomputes in higher precision only the components associated with large condition numbers. We test our algorithm on various networks and datasets and confirm experimentally that it can significantly improve the cost--accuracy tradeoff compared with uniform precision accumulation baselines.

cs.LG↗

arXiv2025

Error Analysis of Sum-Product Algorithms under Stochastic Rounding

The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been shown to be an effective alternative for obtaining tight error bounds. This analysis considers the distribution of errors and evaluates the algorithm's performance on average. Using martingales and the Azuma-Hoeffding inequality, it provides error bounds that are valid with a certain probability and in O($\sqrt$nu) instead of deterministic worst-case bounds in O(nu), where n is the number of operations and u is the unit roundoff. In this paper, we present a general method that automatically constructs a martingale for any computation scheme with multi-linear errors based on additions, subtractions, and multiplications. We apply this generalization to algorithms previously studied with SR, such as pairwise summation and the Horner algorithm, and prove equivalent results. We also analyze a previously unstudied algorithm, Karatsuba polynomial multiplication, which illustrates that the method can handle reused intermediate computations.

stat.CO↗

arXiv2025

Probabilistic error analysis of limited-precision stochastic rounding

Classical probabilistic rounding error analysis is particularly well suited to stochastic rounding (SR), and it yields strong results when dealing with floating-point algorithms that rely heavily on summation. For many numerical linear algebra algorithms, one can prove probabilistic error bounds that grow as O(nu), where n is the problem size and u is the unit roundoff. These probabilistic bounds are asymptotically tighter than the worst-case ones, which grow as O(nu). For certain classes of algorithms, SR has been shown to be unbiased. However, all these results were derived under the assumption that SR is implemented exactly, which typically requires a number of random bits that is too large to be suitable for practical implementations. We investigate the effect of the number of random bits on the probabilistic rounding error analysis of SR. To this end, we introduce a new rounding mode, limited-precision SR. By taking into account the number r of random bits used, this new rounding mode matches hardware implementations accurately, unlike the ideal SR operator generally used in the literature. We show that this new rounding mode is biased and that the bias is a function of r. As r approaches infinity, however, the bias disappears, and limited-precision SR converges to the ideal, unbiased SR operator. We develop a novel model for probabilistic error analysis of algorithms employing SR. Several numerical examples corroborate our theoretical findings.

math.NA↗

arXiv2024

Bounds on non-linear errors for variance computation with stochastic rounding

The main objective of this work is to investigate non-linear errors and pairwise summation using stochastic rounding (SR) in variance computation algorithms. We estimate the forward error of computations under SR through two methods: the first is based on a bound of the variance and Bienaym{é}-Chebyshev inequality, while the second is based on martingales and Azuma-Hoeffding inequality. The study shows that for pairwise summation, using SR results in a probabilistic bound of the forward error proportional to log(n)u rather than the deterministic bound in O(log(n)u) when using the default rounding mode. We examine two algorithms that compute the variance, called ''textbook'' and ''two-pass'', which both exhibit non-linear errors. Using the two methods mentioned above, we show that these algorithms' forward errors have probabilistic bounds under SR in O($\sqrt$ nu) instead of nu for the deterministic bounds. We show that this advantage holds using pairwise summation for both textbook and two-pass, with probabilistic bounds of the forward error proportional to log(n)u.

math.NA↗

arXiv2023

Stochastic rounding variance and probabilistic bounds: A new approach

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.

math.NA↗

arXiv2022

The Positive Effects of Stochastic Rounding in Numerical Algorithms

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.

math.NA↗