BOUNDS ON NON-LINEAR ERRORS FOR VARIANCE COMPUTATION WITH STOCHASTIC ROUNDING

El-Mehdi El Arar (1, 2, 3) , Devan Sohier (1, 2, 3) , Pablo de Oliveira Castro (1, 2, 3) , Eric Petit (4)

1 LI-PaRAD - Laboratoire d'Informatique Parallélisme Réseaux Algorithmes Distribués
2 UVSQ - Université de Versailles Saint-Quentin-en-Yvelines
3 Université Paris-Saclay
4 Intel Corporation [Hillsboro]

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 Bienaymé-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(√ 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.

Mots clés: Stochastic rounding; Floating-point arithmetic; Variance computation; Non-linear error; Doob-Meyer decomposition; Pairwise summation

Domaines: Analyse numérique [math.NA]; Analyse numérique [cs.NA]; Analyse numérique [math.NA]

El-Mehdi El Arar, Devan Sohier, Pablo de Oliveira Castro, Eric Petit. BOUNDS ON NON-LINEAR ERRORS FOR VARIANCE COMPUTATION WITH STOCHASTIC ROUNDING. 2023. ⟨hal-04056057⟩