\section{Experiments}
\label{sec:experiments}

\subsection{More accurate gradients, larger PR variance}
\label{sec:experiment_1}

We compare two gradient estimators for the same stochastic problem,
\begin{equation}\label{eq:experiment_objective}
\min_{\theta\in\mathbb R} F(\theta)
=\mathbb E\!\left[\frac12(\theta-X)^2+\log\cosh\theta\right],
\qquad X\sim N(m,1),
\end{equation}
where the algorithms observe samples but do not use the population mean
$m$. Since $F'(\theta)=\theta+\tanh\theta-m$ and
$F''(\theta)=1+\operatorname{sech}^2\theta\in[1,2]$, the objective is
strongly convex and smooth. Moreover, $|F'''(\theta)|\le2$, so its Hessian
is Lipschitz. We set $m=0$, giving $\theta^*=0$ and $H=2$.

At iteration $t$, the fresh-sample method uses
\[
g_t(\theta)=\theta+\tanh\theta-X_t,
\]
whereas the historical-mean method updates
$\widehat m_t=\widehat m_{t-1}+(X_t-\widehat m_{t-1})/t$ from
$\widehat m_0=0$ and uses
\[
\widetilde g_t(\theta)=\theta+\tanh\theta-\widehat m_t.
\]
Each method takes one SGD step using its own current iterate. At any
fixed $\theta$, their gradient estimation MSEs are
\begin{equation}\label{eq:experiment_gradient_mse}
\mathbb E|g_t(\theta)-F'(\theta)|^2=1,
\qquad
\mathbb E|\widetilde g_t(\theta)-F'(\theta)|^2=\frac1t.
\end{equation}
The fresh-sample method has independent additive noise of variance one.
The historical method corresponds to $\xi_{t+1}=0$ and
$b_t=-(\widehat m_t-m)$, whose cumulative covariance is two by Example~1.
Consequently, the predicted limits are
\begin{equation}\label{eq:experiment_pr_predictions}
\sqrt T(\bar\theta_T-\theta^*)\Rightarrow
\begin{cases}
N(0,1/4),&\text{fresh-sample gradient},\\
N(0,1/2),&\text{historical-mean gradient}.
\end{cases}
\end{equation}
These are total variances for two gradient strategies solving the same
objective.

We run $10{,}000$ independent replications with $\theta_1=2$ and
$\eta_t=0.5t^{-0.65}$. Within each replication, both methods receive the
same sample sequence and one new observation per iteration. We average
$\theta_1,\ldots,\theta_T$ without discarding an initial interval, and
record $\sqrt T(\bar\theta_T-\theta^*)$ at $T=200{,}000$.
The random seed is $20261016$.

\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{experiments/results/experiment_1/pr_histograms.pdf}
\caption{PR fluctuations for the same nonlinear stochastic objective.
Each histogram contains $10{,}000$ independent replications at
$T=200{,}000$. The panels share bins and axes; solid curves are the
asymptotic normal densities in \eqref{eq:experiment_pr_predictions}.
Historical averaging reduces single-step gradient MSE but increases the
PR asymptotic variance.}
\label{fig:pr_histograms}
\end{figure}

At $T=200{,}000$, the empirical PR variances are $0.2501$ for fresh
samples and $0.5120$ for the historical mean, with estimated Monte Carlo
standard errors $0.0035$ and $0.0072$, respectively.
Figure~\ref{fig:pr_histograms} displays the wider historical-mean
distribution.
The empirical single-step gradient MSEs at the largest horizon are
$0.9906$ and $4.9532\times10^{-6}$, compared with the exact values $1$
and $5\times10^{-6}$. Thus a substantial improvement in single-step
gradient accuracy accompanies a larger downstream PR variance.



\subsection{Estimator gain and downstream efficiency}
\label{sec:experiment_2}

We retain the objective \eqref{eq:experiment_objective} with $m=0$ and
replace the running mean by the recursive estimator
\begin{equation}\label{eq:experiment_gain_recursion}
\widehat m_t=\left(1-\frac at\right)\widehat m_{t-1}
+\frac atX_t,\qquad \widehat m_0=0,\quad a>1/2.
\end{equation}
The SGD gradient is $\theta_t+\tanh\theta_t-\widehat m_t$.
The two theoretical criteria, from Example~2, are
\begin{equation}\label{eq:experiment_gain_predictions}
T\,\mathbb E(\widehat m_T-m)^2\longrightarrow\frac{a^2}{2a-1},
\qquad
\operatorname{Var}_{\mathrm{asymp}}
\!\left[\sqrt T(\bar\theta_T-\theta^*)\right]
=\frac14\frac{2a}{2a-1}.
\end{equation}
The first criterion is minimized at $a=1$, while the second decreases
with $a$ toward $1/4$.

We scan $a\in\{0.6,0.75,1,1.25,1.5,2,3,5,8\}$ using $5{,}000$
independent replications per gain. The initial SGD iterate, step size,
and averaging convention are the same as in Section~\ref{sec:experiment_1}.
Within each replication, all gains receive the same sample sequence.
We record both criteria at $T=50{,}000$ and $200{,}000$, using seed
$20261017$. Equation~\eqref{eq:experiment_gain_recursion} is applied
throughout the finite initial interval as well.

\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{experiments/results/experiment_2/gain_sweep.pdf}
\caption{Estimator gain and the two efficiency criteria. The left panel
shows the scaled single-step estimation MSE; the right panel shows the
PR empirical variance. Solid black curves are the limits in
\eqref{eq:experiment_gain_predictions}; error bars are $1.96$ Monte Carlo
standard errors. The vertical reference marks $a=1$, and the horizontal
reference marks the downstream asymptotic lower bound $1/4$.}
\label{fig:gain_sweep}
\end{figure}

Figure~\ref{fig:gain_sweep} exhibits the predicted difference in parameter
choices. At $T=200{,}000$, the scaled single-step MSE is $0.9785$ at
$a=1$ and increases to $4.3006$ at $a=8$, while the PR empirical variance
decreases from $0.5062$ to $0.2621$. The corresponding theoretical values
are $1$ and $4.2667$ for the scaled MSE, and $0.5$ and $0.2667$ for the
PR variance. Across the sampled gains, the single-step criterion is
smallest at $a=1$, whereas the downstream variance is smallest at the
largest sampled gain. At $a=0.6$, the empirical PR variance increases
from $1.3213$ to $1.3628$ between the two horizons, approaching its
asymptotic value $1.5$ more slowly near the threshold $a=1/2$.

\subsection{Attaining the downstream variance bound}
\label{sec:experiment_3}

We first state the theoretical predictions for the objective
\eqref{eq:experiment_objective} with $m=0$ and $H=2$. The three auxiliary
gain schedules yield
\begin{equation}\label{eq:experiment_attainment_predictions}
\Sigma_{\mathrm{PR}}=
\begin{cases}
1/2,&\gamma_t=1/t,\\
4/15,&\gamma_t=8/t,\\
1/4,&\gamma_t=t^{-0.6}.
\end{cases}
\end{equation}
The last value is the sharp downstream bound of
Proposition~\ref{prop:lower_bound}, attained by the polynomial-gain
construction of Proposition~\ref{prop:attainability}. Its single-step
estimation MSE has the slower rate $t^{-0.6}$, compared with $t^{-1}$ for
the other two schedules. This predicts the counterintuitive combination
of slower auxiliary estimation and optimal downstream PR variance.

We simulate all three estimators using
\[
\widehat m_t=(1-\gamma_t)\widehat m_{t-1}+\gamma_tX_t,
\qquad \widehat m_0=0,
\]
and the same downstream SGD update as before. The polynomial schedule
has $\gamma_1=1$, so its historical weights are exactly normalized.
Its exponent $\rho=0.6$ satisfies $\rho>2\alpha-1$ with $\alpha=0.65$.
We use $10{,}000$ independent replications, paired across the three
methods, $T=1{,}000{,}000$, and seed $20261018$. All remaining SGD settings
match Section~\ref{sec:experiment_1}.

\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{experiments/results/experiment_3/attainment_histograms.pdf}
\caption{Downstream PR distributions for three auxiliary gain schedules.
Each panel contains $10{,}000$ independent replications at $T=1{,}000{,}000$,
with shared bins and axes. Solid curves are the theoretical normal
densities with variances $1/2$, $4/15$, and $1/4$, respectively.
The polynomial-gain estimator has slower single-step MSE decay but
attains the smallest downstream asymptotic variance.}
\label{fig:attainment_histograms}
\end{figure}

The empirical PR variances are $0.5101$, $0.2682$, and $0.2524$,
respectively, with Monte Carlo standard errors $0.0073$, $0.0038$, and
$0.0036$. Figure~\ref{fig:attainment_histograms} compares the empirical
distributions with the predictions in
\eqref{eq:experiment_attainment_predictions}.
At the same horizon, the auxiliary-estimation MSEs are
$1.0031\times10^{-6}$, $4.3386\times10^{-6}$, and
$1.2750\times10^{-4}$. Thus the polynomial estimator has the largest
single-step error among the three methods, yet the smallest empirical
PR variance, illustrating its theoretically optimal downstream limit.