Randomized algorithms, such as randomized sketching or projections, are a promising approach to ease the computational burden in analyzing large datasets. However, randomized algorithms also produce non-deterministic outputs, leading to the problem of evaluating their accuracy. In this paper, we develop a statistical inference framework for quantifying the uncertainty of the outputs of randomized algorithms. We develop appropriate statistical methods -- sub-randomization, multi-run plug-in and multi-run aggregation inference -- by using multiple runs of the same randomized algorithm, or by estimating the unknown parameters of the limiting distribution. As an example, we develop methods for statistical inference for least squares parameters via random sketching using matrices with i.i.d.entries, or uniform partial orthogonal matrices. For this, we characterize the limiting distribution of estimators obtained via sketch-and-solve as well as partial sketching methods. The analysis of i.i.d. sketches uses a trigonometric interpolation argument to establish a differential equation for the limiting expected characteristic function and find the dependence on the kurtosis of the entries of the sketching matrix. The results are supported via a broad range of simulations.

翻译：随机化算法（如随机草图或投影）是降低大数据集计算负担的一种有前景的方法。然而，随机化算法也会产生非确定性输出，从而引发对其精度评估的问题。本文开发了一个统计推断框架，用于量化随机化算法输出的不确定性。我们通过多次运行同一随机化算法，或通过估计极限分布的未知参数，提出了适当的统计方法——子随机化、多次运行插值法和多次运行聚合推断法。以最小二乘参数为例，我们利用具有独立同分布元素的随机矩阵或均匀部分正交矩阵，通过随机草图方法开发了统计推断方法。为此，我们刻画了通过草图求解及部分草图方法获得的估计量的极限分布。对独立同分布草图的分析利用三角插值论证，建立了极限期望特征函数的微分方程，并发现了草图矩阵元素峰度的影响。通过广泛的仿真实验验证了这些结果。