Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914108269395968 |
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| author | Agrawalla, Bhavya Balasubramanian, Krishnakumar Ghosal, Promit |
| author_facet | Agrawalla, Bhavya Balasubramanian, Krishnakumar Ghosal, Promit |
| contents | Stochastic gradient descent (SGD) has emerged as the quintessential method in a data scientist's toolbox. Using SGD for high-stakes applications requires, however, careful quantification of the associated uncertainty. Towards that end, in this work, we establish a high-dimensional Central Limit Theorem (CLT) for linear functionals of online SGD iterates for overparametrized least-squares regression with non-isotropic Gaussian inputs. We first show that a bias-corrected CLT holds when the number of iterations of the online SGD, $t$, grows sub-linearly in the dimensionality, $d$. In order to use the developed result in practice, we further develop an online approach for estimating the variance term appearing in the CLT, and establish high-probability bounds for the developed online estimator. Together with the CLT result, this provides a fully online and data-driven way to numerically construct confidence intervals. This enables practical high-dimensional algorithmic inference with SGD and to the best of our knowledge, is the first such result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_09727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression Agrawalla, Bhavya Balasubramanian, Krishnakumar Ghosal, Promit Statistics Theory Optimization and Control Probability Machine Learning Stochastic gradient descent (SGD) has emerged as the quintessential method in a data scientist's toolbox. Using SGD for high-stakes applications requires, however, careful quantification of the associated uncertainty. Towards that end, in this work, we establish a high-dimensional Central Limit Theorem (CLT) for linear functionals of online SGD iterates for overparametrized least-squares regression with non-isotropic Gaussian inputs. We first show that a bias-corrected CLT holds when the number of iterations of the online SGD, $t$, grows sub-linearly in the dimensionality, $d$. In order to use the developed result in practice, we further develop an online approach for estimating the variance term appearing in the CLT, and establish high-probability bounds for the developed online estimator. Together with the CLT result, this provides a fully online and data-driven way to numerically construct confidence intervals. This enables practical high-dimensional algorithmic inference with SGD and to the best of our knowledge, is the first such result. |
| title | Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression |
| topic | Statistics Theory Optimization and Control Probability Machine Learning |
| url | https://arxiv.org/abs/2302.09727 |