  In this paper, we revisit \textsf{ROOT-SGD}, an innovative method for
stochastic optimization to bridge the gap between stochastic optimization and
statistical efficiency. The proposed method enhances the performance and
reliability of \textsf{ROOT-SGD} by integrating a carefully designed
\emph{diminishing stepsize strategy}. This approach addresses key challenges in
optimization, providing robust theoretical guarantees and practical benefits.
Our analysis demonstrates that \textsf{ROOT-SGD} with diminishing achieves
optimal convergence rates while maintaining computational efficiency. By
dynamically adjusting the learning rate, \textsf{ROOT-SGD} ensures improved
stability and precision throughout the optimization process. The findings of
this study offer valuable insights for developing advanced optimization
algorithms that are both efficient and statistically robust.
