Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regression
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 va...
| Authors | Yijin Ni, Xiaoming Huo |
|---|---|
| Primary category | cs.LG |
| Published | 12 April 2026 |
| Last updated at source | 12 April 2026 |
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