用伽玛随机变量的线性组合的极值概率评价大规模非线性最小二乘问题的随机算法
Assessing stochastic algorithms for large scale nonlinear least squares problems using extremal probabilities of linear combinations of gamma random variables
摘要：
本文研究了一类大规模非线性最小二乘（NLS）问题的随机算法。我们提出了八个变种的一个实用的随机算法中的主要随机步骤的不确定性进行量化。这样的随机步骤涉及使用蒙特卡罗方法近似NLS目标函数，并且这等价于相应的对称半正定（SPSD）矩阵的迹的估计。对于后者，我们证明了严格的充要条件和充分条件的样本量（转换为成本），以满足规定的概率精度。我们表明，这些条件实际上是可计算的，并产生小的样本量。然后将它们纳入我们的随机算法中，以量化每个随机步骤中的不确定性。我们使用的界限是应用程序的极值尾概率的线性组合的伽玛分布随机变量的更一般的结果。我们推导并证明了新的结果有关的最大和最小尾概率的这种线性组合，这可以被认为是独立的本文的其余部分。
英文摘要：
This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where the uncertainties in the major stochastic steps are quantified. Such stochastic steps involve approximating the NLS objective function using Monte-Carlo methods, and this is equivalent to the estimation of the trace of corresponding symmetric positive semi-definite (SPSD) matrices. For the latter, we prove tight necessary and sufficient conditions on the sample size (which translates to cost) to satisfy the prescribed probabilistic accuracy. We show that these conditions are practically computable and yield small sample sizes. They are then incorporated in our stochastic algorithm to quantify the uncertainty in each randomized step. The bounds we use are applications of more general results regarding extremal tail probabilities of linear combinations of gamma distributed random variables. We derive and prove new results concerning the maximal and minimal tail probabilities of such linear combinations, which can be considered independently of the rest of this paper.
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