### Abstract

The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions. Our main contributions are twofold. First, in the fixed dimension setup, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator, which is computationally more efficient and only uses the iterates from SGD. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests. Second, for high-dimensional linear regression, using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficients and confidence intervals, which is computationally attractive and applicable to online data.

Original language | English (US) |
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Pages (from-to) | 251-273 |

Number of pages | 23 |

Journal | Annals of Statistics |

Volume | 48 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Asymptotic variance
- Batch-means estimator
- High-dimensional inference
- Stochastic gradient descent
- Time-inhomogeneous Markov chain

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## Cite this

*Annals of Statistics*,

*48*(1), 251-273. https://doi.org/10.1214/18-AOS1801