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）是一种针对大规模数据集和流数据具有可扩展性和内存效率的优化算法，近年来受到广泛关注与青睐。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，现有相关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，如何开展有效的统计推断仍属未探索领域。事实上，观测数据间的普遍相关性对区间估计构成挑战，多数现有方法可能忽略该相关性而导致置信区间失效。本文针对φ-混合数据提出一种小批量SGD估计量用于统计推断，并利用关联的小批量自助法SGD流程构建置信区间。通过引用\cite{yu1994rates}中的"独立块"技巧，我们证明所提估计量具有渐近正态性，且其极限分布可被自助法流程有效逼近。该方法具有内存效率高、易于实现的优点。合成数据的模拟实验及真实数据集的案例分析均验证了我们的理论。