This paper investigates the problem of online statistical inference of model parameters in stochastic optimization problems via the Kiefer-Wolfowitz algorithm with random search directions. We first present the asymptotic distribution for the Polyak-Ruppert-averaging type Kiefer-Wolfowitz (AKW) estimators, whose asymptotic covariance matrices depend on the distribution of search directions and the function-value query complexity. The distributional result reflects the trade-off between statistical efficiency and function query complexity. We further analyze the choice of random search directions to minimize certain summary statistics of the asymptotic covariance matrix. Based on the asymptotic distribution, we conduct online statistical inference by providing two construction procedures of valid confidence intervals.
翻译:本文件调查了通过使用随机搜索方向的Kiefer-Wolfowitz算法在随机优化问题中模型参数的在线统计推论问题。我们首先为Polyak-Ruppert-avecing type Kiefer-Wolfowitz (AKW) 测算器提供了无症状共变量分布,其无症状共变量矩阵取决于搜索方向的分布和功能值查询复杂性。分布结果反映了统计效率和功能查询复杂性之间的平衡。我们进一步分析了随机搜索方向的选择,以尽量减少无症状共变量矩阵的某些汇总统计数据。根据无症状分布,我们通过提供两个有效信任间隔的构建程序进行在线统计推断。