Uncertainty quantification for estimation through stochastic optimization solutions in an online setting has gained popularity recently. This paper introduces a novel inference method focused on constructing confidence intervals with efficient computation and fast convergence to the nominal level. Specifically, we propose to use a small number of independent multi-runs to acquire distribution information and construct a t-based confidence interval. Our method requires minimal additional computation and memory beyond the standard updating of estimates, making the inference process almost cost-free. We provide a rigorous theoretical guarantee for the confidence interval, demonstrating that the coverage is approximately exact with an explicit convergence rate and allowing for high confidence level inference. In particular, a new Gaussian approximation result is developed for the online estimators to characterize the coverage properties of our confidence intervals in terms of relative errors. Additionally, our method also allows for leveraging parallel computing to further accelerate calculations using multiple cores. It is easy to implement and can be integrated with existing stochastic algorithms without the need for complicated modifications.

翻译：在线环境下通过随机优化解进行估计的不确定性量化近年来备受关注。本文提出一种新的推断方法，旨在构建具有高效计算性能且快速收敛到名义水平的置信区间。具体而言，我们建议使用少量独立多次运行获取分布信息，并构建基于t分布的置信区间。该方法在标准估计更新基础上几乎不增加额外计算与内存开销，使得推断过程近乎零成本。我们为置信区间提供了严格的理论保障，证明其覆盖概率以显式收敛速度逼近精确值，并支持高置信水平推断。特别地，本文为在线估计量建立了新的高斯逼近结果，以相对误差形式刻画置信区间的覆盖性质。此外，该方法还可利用并行计算借助多核处理器进一步加速计算。该方法易于实现，无需复杂修改即可与现有随机算法集成。