\section{Introduction}
\label{sec:introduction}

Consider a nonlinear stochastic optimization problem,
\[
\min_{\theta\in\mathbb R}F(\theta)
=\mathbb E\!\left[\frac12(\theta-X)^2+\log\cosh\theta\right],
\qquad F'(\theta)=\theta+\tanh\theta-m,
\]
where $m=\mathbb EX$ is unknown. A stochastic gradient can use either a
fresh observation $X_t$ or the running estimate
$\widehat m_t=t^{-1}\sum_{s=1}^tX_s$ in place of $m$. The latter uses
more observations and becomes progressively more accurate, suggesting
that it should also improve the final averaged estimate.

For $X\sim N(0,1)$, however, replacing $X_t$ by $\widehat m_t$ reduces
the gradient estimation MSE from $1$ to $1/t$ while increasing the
Polyak--Ruppert (PR) asymptotic variance from $1/4$ to $1/2$.
How can a progressively more accurate gradient estimate produce a less
precise averaged iterate? Successive running estimates reuse observations,
so their errors are temporally correlated and accumulate coherently.
Their single-step MSE alone does not determine the variance of the
downstream average.

Related historical dependence arises from online moment estimation,
streaming feature normalization, and online subspace estimation, as well
as recursive inner-expectation estimation in compositional
optimization~\cite{wang2017} and nuisance-dependent gradients~\cite{yu2025}.

This note studies these gradient errors through a unified weighted
historical representation.
We characterize their contribution to PR fluctuations through cumulative
weights and derive explicit covariance formulas. The examples and a
sharp variance bound within the normalized historical linear estimator
class clarify why optimizing a single-step estimator and optimizing its
downstream PR average can favor different choices.

The same distinction yields a second counterintuitive conclusion. A recursive
auxiliary estimator with single-step MSE of order $t^{-\rho}$,
$1/2<\rho<1$, can attain the smallest downstream PR asymptotic covariance
within the normalized historical linear estimator class. A running mean
has the faster $t^{-1}$ single-step rate but a larger downstream covariance.
Thus improving the auxiliary estimator's single-step convergence rate can
conflict with optimizing the final averaged SGD estimate. Proposition~\ref{prop:attainability}
and Section~\ref{sec:experiment_3} establish and illustrate this distinction.