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Stochastic gradient descent (SGD) is a scalable and memory-efficientoptimization algorithm for large datasets and stream data, which has drawn agreat deal of attention and popularity. The applications of SGD-basedestimators to statistical inference such as interval estimation have alsoachieved 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 timeseries, how to conduct valid statistical inference remains unexplored. As amatter of fact, the general correlation among observations imposes a challengeon interval estimation. Most existing methods may ignore this correlation andlead to invalid confidence intervals. In this paper, we propose a mini-batchSGD estimator for statistical inference when the data is $\phi$-mixing. Theconfidence intervals are constructed using an associated mini-batch bootstrapSGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, weshow that the proposed estimator is asymptotically normal, and its limitingdistribution can be effectively approximated by the bootstrap procedure. Theproposed method is memory-efficient and easy to implement in practice.Simulation studies on synthetic data and an application to a real-world datasetconfirm our theory.

Statistical Inference with Stochastic Gradient Methods under $\phi$-mixing Data