Annals of Statistics

Statistical inference for model parameters in stochastic gradient descent

Xi Chen, Jason D. Lee, Xin T. Tong, and Yichen Zhang

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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.

Article information

Ann. Statist., Volume 48, Number 1 (2020), 251-273.

Received: October 2017
Revised: July 2018
First available in Project Euclid: 17 February 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62J10: Analysis of variance and covariance 62M02: Markov processes: hypothesis testing
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stochastic gradient descent asymptotic variance batch-means estimator high-dimensional inference time-inhomogeneous Markov chain


Chen, Xi; Lee, Jason D.; Tong, Xin T.; Zhang, Yichen. Statistical inference for model parameters in stochastic gradient descent. Ann. Statist. 48 (2020), no. 1, 251--273. doi:10.1214/18-AOS1801.

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Supplemental materials

  • Supplement to “Statistical inference for model parameters in stochastic gradient descent”. We provide the proofs of all the theorectial results as well as additional simulation studies.