Stochastic Rounding Implicitly Regularizes Tall-and-Thin Matrices

Motivated by the popularity of stochastic rounding in the context of machine learning and the training of large-scale deep neural network models, we consider stochastic nearness rounding of real matrices A with many more rows than columns. We provide novel theoretical evidence, supported by extensive experimental evaluation that, with high probability, the smallest singular value of a stochastically rounded matrix is well bounded away from zero -- regardless of how close A is to being rank deficient and even if A is rank-deficient. In other words, stochastic rounding implicitly regularizes tall and skinny matrices A so that the rounded version has full column rank. Our proofs leverage powerful results in random matrix theory, and the idea that stochastic rounding errors do not concentrate in low-dimensional column spaces.