\section{Conclusion}
In this work, we develop a novel statistical inference framework for streaming PCA using Oja’s algorithm. We derive finite-sample and high-probability deviation bounds for the coordinates of the estimated eigenvector, establish a Bernstein-type concentration bound on the residual of the Oja vector, establish a Central Limit Theorem for suitable subsets of entries, and devise an efficient subsampling-based variance estimation algorithm. By leveraging the structure of the Oja updates, we provide entrywise confidence intervals, bypassing expensive resampling techniques such as bootstrapping. Our theoretical results are supported by extensive numerical experiments, indicating that our proposed estimator achieves accuracy similar to the multiplier bootstrap method while requiring significantly less time. 

We believe that our subsampling algorithm can be adapted to any SGD problem where the covariance matrix of the estimator $\hat{\theta}_n$ scales as $c_n$ times some scale-free matrix $\V$, where $c_n$ is known. This structure aligns with subsampling and m-out-of-n bootstrap methods, where the variance estimated from a subsample of size $m$ is scaled by $m/n$ to approximate the variance of the full sample estimator. Our findings also highlight the potential for improved uncertainty quantification techniques in streaming non-convex optimization problems beyond PCA, since Oja-type updates can be found in many important non-convex optimization algorithms such as matrix sensing, matrix completion, and subspace estimation. Further directions include deflation-based methods to apply our method to variance estimation for top $k$ eigenvectors.
