Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Andreas Anastasiou, Krishnakumar Balasubramanian, Murat A. Erdogdu
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:115-137, 2019.

Abstract

We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein?s method and Lindeberg?s argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-anastasiou19a, title = {Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT}, author = {Anastasiou, Andreas and Balasubramanian, Krishnakumar and Erdogdu, Murat A.}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {115--137}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/anastasiou19a/anastasiou19a.pdf}, url = {https://proceedings.mlr.press/v99/anastasiou19a.html}, abstract = {We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein?s method and Lindeberg?s argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense} }
Endnote
%0 Conference Paper %T Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT %A Andreas Anastasiou %A Krishnakumar Balasubramanian %A Murat A. Erdogdu %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-anastasiou19a %I PMLR %P 115--137 %U https://proceedings.mlr.press/v99/anastasiou19a.html %V 99 %X We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein?s method and Lindeberg?s argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense
APA
Anastasiou, A., Balasubramanian, K. & Erdogdu, M.A.. (2019). Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:115-137 Available from https://proceedings.mlr.press/v99/anastasiou19a.html.

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