Stochastic gradient descent (SGD) is a scalable and memory-efficient optimization algorithm for large datasets and stream data, which has drawn a great deal of attention and popularity. The applications of SGD-based estimators to statistical inference such as interval estimation have also achieved great success. However, most of the related works are based on i.i.d. observations or Markov chains. When the observations come from a mixing time series, how to conduct valid statistical inference remains unexplored. As a matter of fact, the general correlation among observations imposes a challenge on interval estimation. Most existing methods may ignore this correlation and lead to invalid confidence intervals. In this paper, we propose a mini-batch SGD estimator for statistical inference when the data is $\phi$-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, we show that the proposed estimator is asymptotically normal, and its limiting distribution can be effectively approximated by the bootstrap procedure. The proposed method is memory-efficient and easy to implement in practice. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.

翻译：随机梯度下降（SGD）是一种适用于大规模数据集和流数据的高可扩展性与内存高效优化算法，近年来受到广泛关注与应用。基于SGD估计量的统计推断方法（如区间估计）也取得了显著成功。然而，现有研究大多基于独立同分布观测或马尔可夫链。当观测数据来自混合时间序列时，如何进行有效的统计推断仍属未探索领域。事实上，观测值间的普遍相关性给区间估计带来了挑战：多数现有方法可能忽略这种相关性，导致置信区间失效。本文针对$\phi$-混合数据场景，提出了一种用于统计推断的小批量SGD估计量，并采用关联的小批量自助法SGD流程构建置信区间。借助《Yu (1994)》提出的"独立块"技巧，我们证明了所提估计量具有渐近正态性，且其极限分布可通过自助法有效近似。该方法具有内存高效、易于实际部署的特点。基于合成数据的仿真实验及真实世界数据集的应用验证了我们的理论。