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 matrix-vector 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.