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：本文研究了通过具有随机搜索方向的Kiefer-Wolfowitz算法，对随机优化问题中模型参数进行在线统计推断的问题。我们首先给出了Polyak-Ruppert平均型Kiefer-Wolfowitz（AKW）估计量的渐近分布，其渐近协方差矩阵依赖于搜索方向的分布以及函数值查询复杂度。该分布结果反映了统计效率与函数查询复杂度之间的权衡。我们进一步分析了随机搜索方向的选择，以最小化渐近协方差矩阵的某些汇总统计量。基于渐近分布，我们通过提供两种有效的置信区间构造程序来进行在线统计推断。