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 $\mathbf{A}$ 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 $\mathbf{A}$ is to being rank deficient and even if $\mathbf{A}$ is rank-deficient. In other words, stochastic rounding \textit{implicitly regularizes} tall and skinny matrices $\mathbf{A}$ 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.

翻译：受机器学习和大规模深度神经网络模型训练中随机舍入广泛应用的启发，我们考虑对行数远大于列数的实矩阵 $\mathbf{A}$ 进行随机邻近舍入。我们通过大量实验验证，提供了新的理论证据表明：无论 $\mathbf{A}$ 接近秩亏损的程度如何，甚至当 $\mathbf{A}$ 本身秩亏损时，随机舍入后的矩阵的最小奇异值以高概率远离零。换言之，随机舍入对高瘦矩阵 $\mathbf{A}$ 具有隐式正则化效应，使得舍入后的版本具有满列秩。我们的证明利用了随机矩阵理论中的强大结果，其核心思想在于随机舍入误差不会集中在低维列空间内。