El-Mehdi El Arar

El-Mehdi El Arar, Massimiliano Fasi, Silviu-Ioan Filip et al.

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.

El-Mehdi El Arar, Silviu-Ioan Filip, Theo Mary et al.

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 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 practical 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.