# Statistical Inference with Stochastic Gradient Methods under φ-mixing Data

arXiv: https://arxiv.org/abs/2302.12717 · HTML: https://mlsanity.com/paper/2302.12717/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 48 references that are in the map.
- First posted on arXiv 24 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Taming the Noise in Reinforcement Learning via Soft Updates](https://mlsanity.com/paper/1512.08562/) (2015)
- [Finite-Time Analysis of Asynchronous Stochastic Approximation and Q-Learning](https://mlsanity.com/paper/2002.00260/) (2020)
- [Statistical Inference for Model Parameters in Stochastic Gradient Descent](https://mlsanity.com/paper/1610.08637/) (2016)
- [Statistical Inference of the Value Function for Reinforcement Learning in Infinite Horizon Settings](https://mlsanity.com/paper/2001.04515/) (2020)
- [Two Time-scale Off-Policy TD Learning: Non-asymptotic Analysis over Markovian Samples](https://mlsanity.com/paper/1909.11907/) (2019)
- [Off-Policy Confidence Interval Estimation with Confounded Markov Decision Process](https://mlsanity.com/paper/2202.10589/) (2022)
- [Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scaling](https://mlsanity.com/paper/2106.03156/) (2021)
- [A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions](https://mlsanity.com/paper/2010.09800/) (2020)
- [Online Bootstrap Inference For Policy Evaluation in Reinforcement Learning](https://mlsanity.com/paper/2108.03706/) (2021)
- [Statistical Inference for Online Decision Making via Stochastic Gradient Descent](https://mlsanity.com/paper/2010.07341/) (2020)
- [Reinforcement Learning with Heterogeneous Data: Estimation and Inference](https://mlsanity.com/paper/2202.00088/) (2022)
- [Data Sampling Affects the Complexity of Online SGD over Dependent Data](https://mlsanity.com/paper/2204.00006/) (2022)
- [Online Statistical Inference for Parameters Estimation with Linear-Equality Constraints](https://mlsanity.com/paper/2105.10315/) (2021)

## Built on by (most cited first, 7 of 7)

- [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (2023)
- [Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference](https://mlsanity.com/paper/2312.10894/) (2023)
- [Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximation](https://mlsanity.com/paper/2510.12375/) (2025)
- [Asymptotic Analysis of Sample-averaged Q-learning](https://mlsanity.com/paper/2410.10737/) (2024)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)
- [An Online Bootstrap for Time Series](https://mlsanity.com/paper/2310.19683/) (2023)
- [Limit Theory for Online Expectile Estimation under Short and Long Range Dependence](https://mlsanity.com/paper/2610.16030/) (2026)

## Abstract (arXiv)

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 φ-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from citeyu1994rates, 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.