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This article considers stochastic algorithms for efficiently solving a classof large scale non-linear least squares (NLS) problems which frequently arisein applications. We propose eight variants of a practical randomized algorithmwhere the uncertainties in the major stochastic steps are quantified. Suchstochastic steps involve approximating the NLS objective function usingMonte-Carlo methods, and this is equivalent to the estimation of the trace ofcorresponding symmetric positive semi-definite (SPSD) matrices. For the latter,we prove tight necessary and sufficient conditions on the sample size (whichtranslates to cost) to satisfy the prescribed probabilistic accuracy. We showthat these conditions are practically computable and yield small sample sizes.They are then incorporated in our stochastic algorithm to quantify theuncertainty in each randomized step. The bounds we use are applications of moregeneral results regarding extremal tail probabilities of linear combinations ofgamma distributed random variables. We derive and prove new results concerningthe maximal and minimal tail probabilities of such linear combinations, whichcan be considered independently of the rest of this paper.

siam/asa journal on uncertainty quantification

Assessing stochastic algorithms for large scale nonlinear least squares  problems using extremal probabilities of linear combinations of gamma random  variables