Probabilistic Rounding Error Analysis From A Statistical Perspective

The conventional probabilistic rounding error analysis in numerical linear algebra provides worst-case bounds with an associated failure probability, which can still be pessimistic. In this paper, we develop a new probabilistic rounding error analysis from a statistical perspective. By assuming both the data and the relative error are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computational kernels. Our analytical expressions have three notable characteristics: they are statistical and do not involve a failure probability; they are sharper than other deterministic and probabilistic bounds, using mean square error as the metric; they are correct to all orders of unit roundoff and valid for any dimension. Furthermore, numerical experiments validate the accuracy of our derivations and demonstrate that our analytical expressions are generally at least two orders of magnitude tighter than alternative worst-case bounds, exemplified through the inner products. We also discuss a scenario involving inner products where the underlying assumptions are invalid, i.e., input data are dependent, rendering the analytical expressions inapplicable.