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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.10814 |
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| _version_ | 1866914467253583872 |
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| author | Ni, Yijin Huo, Xiaoming |
| author_facet | Ni, Yijin Huo, Xiaoming |
| contents | We study online covariance matrix estimation for Polyak--Ruppert averaged stochastic gradient descent (SGD). The online batch-means estimator of Zhu, Chen and Wu (2023) achieves an operator-norm convergence rate of $O(n^{-(1-α)/4})$, which yields $O(n^{-1/8})$ at the optimal learning-rate exponent $α\rightarrow 1/2^+$. A rigorous per-block bias analysis reveals that re-tuning the block-growth parameter improves the batch-means rate to $O(n^{-(1-α)/3})$, achieving $O(n^{-1/6})$. The modified estimator requires no Hessian access and preserves $O(d^2)$ memory. We provide a complete error decomposition into variance, stationarity bias, and nonlinearity bias components. A weighted-averaging variant that avoids hard truncation is also discussed. We establish the minimax rate $Θ(n^{-(1-α)/2})$ for Hessian-free covariance estimation from the SGD trajectory: a Le Cam lower bound gives $Ω(n^{-(1-α)/2})$, and a trajectory-regression estimator--which estimates the Hessian by regressing SGD increments on iterates--achieves $O(n^{-(1-α)/2})$, matching the lower bound. The construction reveals that the bottleneck is the sublinear accumulation of information about the Hessian from the SGD drift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10814 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regression Ni, Yijin Huo, Xiaoming Machine Learning Statistics Theory We study online covariance matrix estimation for Polyak--Ruppert averaged stochastic gradient descent (SGD). The online batch-means estimator of Zhu, Chen and Wu (2023) achieves an operator-norm convergence rate of $O(n^{-(1-α)/4})$, which yields $O(n^{-1/8})$ at the optimal learning-rate exponent $α\rightarrow 1/2^+$. A rigorous per-block bias analysis reveals that re-tuning the block-growth parameter improves the batch-means rate to $O(n^{-(1-α)/3})$, achieving $O(n^{-1/6})$. The modified estimator requires no Hessian access and preserves $O(d^2)$ memory. We provide a complete error decomposition into variance, stationarity bias, and nonlinearity bias components. A weighted-averaging variant that avoids hard truncation is also discussed. We establish the minimax rate $Θ(n^{-(1-α)/2})$ for Hessian-free covariance estimation from the SGD trajectory: a Le Cam lower bound gives $Ω(n^{-(1-α)/2})$, and a trajectory-regression estimator--which estimates the Hessian by regressing SGD increments on iterates--achieves $O(n^{-(1-α)/2})$, matching the lower bound. The construction reveals that the bottleneck is the sublinear accumulation of information about the Hessian from the SGD drift. |
| title | Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regression |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2604.10814 |