\section{Complex-Corrected SGD}
\label{sec:algorithm}

Let $\theta\in\mathbb R^p$. A fixed response, when present, is suppressed in the notation. We observe $W=Z+U$, where $U\sim N(0,\Omega)$ is independent of $Z$ and the covariance $\Omega\succeq0$ is known. No invertibility or diagonal structure is required. Assume that $Z\in\mathbb R^d$ is sub-Gaussian about its mean $m_Z$: for some $\sigma_Z<\infty$,
\[
\mathbb E e^{s v^\top(Z-m_Z)}\le e^{\sigma_Z^2s^2/2}
\quad(s\in\mathbb R,\ \|v\|=1).
\]
For a loss $\ell(\theta;z)$ differentiable in $\theta$, with
$\mathbb E|\ell(\theta;Z)|<\infty$ for every $\theta$, write
\[
F(\theta)=\mathbb E_Z\ell(\theta;Z),\qquad
g(\theta;z)=\nabla_\theta\ell(\theta;z).
\]
The following assumptions separate input regularity from clean sampling
noise. All norms are Euclidean, with $\|\cdot\|_{\rm op}$ denoting the
induced operator norm.

\begin{assumption}[Holomorphic score]\label{ass:entire}
For every $\theta$, $z\mapsto g(\theta;z)$ admits an entire extension $g^{\mathbb C}(\theta;\cdot)$ on $\mathbb C^d$, real-valued on $\mathbb R^d$.
\end{assumption}

\begin{assumption}[Exponential envelope]\label{ass:envelope}
There are a finite constant $a\ge0$ and a measurable function
$M:\mathbb R^p\to[0,\infty)$ such that
\begin{equation}\label{eq:parameter_envelope}
\|D_qg^{\mathbb C}(\theta;q)\|_{\rm op}
\le M(\theta)e^{a\|q\|},
\qquad \theta\in\mathbb R^p,\ q\in\mathbb C^d.
\end{equation}
The exponent $a$ is independent of $\theta$; the amplitude $M(\theta)$
need not be bounded.
In addition, for finite constants $A,B\ge0$,
\begin{equation}\label{eq:amplitude_control}
M(\theta)^2\le A+B\|\nabla F(\theta)\|^2,
\qquad \theta\in\mathbb R^p.
\end{equation}
\end{assumption}

\begin{assumption}[Clean sampling variance]\label{ass:clean_variance}
Differentiation may be interchanged with the expectation defining $F$,
and for constants $\sigma_0^2,\rho_0\ge0$, independent of $\theta$,
\[
\mathbb E_Z\|g(\theta;Z)-\nabla F(\theta)\|^2
\le\sigma_0^2+\rho_0\|\nabla F(\theta)\|^2.
\]
This condition concerns the clean stochastic score;
no variance bound is assumed on the corrected score.
\end{assumption}

The amplitude control transfers the input growth bound to the standard
SGD relative variance condition. The correction identity itself uses
only the entire extension and the bound~\eqref{eq:parameter_envelope};
it requires neither amplitude control nor convexity.

\begin{algorithm}[H]
\caption{Complex-Corrected SGD}
\label{alg:ccsgd}
\begin{algorithmic}[1]
\Require Initial point $\theta_1$, step sizes $\{\eta_t\}_{t=1}^T$, known noise covariance $\Omega$
\For{$t=1,\ldots,T$}
    \State Observe a fresh $W_t=Z_t+U_t$
    \State Draw $V_t\sim N(0,\Omega)$ independently of the observation and the past
    \State $G_t\gets\operatorname{Re}g^{\mathbb C}(\theta_t;W_t+iV_t)$
    \State $\theta_{t+1}\gets\theta_t-\eta_tG_t$
\EndFor
\State \Return $\theta_{T+1}$
\end{algorithmic}
\end{algorithm}

\paragraph{Optional averaging.}
For the same observation $W_t$, one may draw $m$ independent
$V_{t,j}\sim N(0,\Omega)$ and replace $G_t$ by
\[
G_t^{(m)}=\frac1m\sum_{j=1}^m
\operatorname{Re}g^{\mathbb C}(\theta_t;W_t+iV_{t,j}).
\]
This preserves conditional unbiasedness with respect to the past.
Given $(\mathcal F_{t-1},W_t)$, the covariance is $1/m$ times that of
a single draw, reducing simulation variability and stabilizing the update
at the cost of additional score evaluations. Variability due to the
shared observation remains.

\begin{lemma}[Exact correction]\label{lem:exact_correction}
Under Assumptions~\ref{ass:entire}--\ref{ass:envelope}, for every $(z,\theta)$,
\[
\mathbb E_{U,V}\bigl[\operatorname{Re}g^{\mathbb C}(\theta;z+U+iV)\bigr]
=g(\theta;z).
\]
Consequently, if $\theta_t$ is measurable with respect to the past $\mathcal F_{t-1}$, $(Z_t,U_t,V_t)$ is independent of that past, and differentiation may be interchanged with the expectation defining $F$, then
$\mathbb E[\operatorname{Re}g^{\mathbb C}(\theta_t;W_t+iV_t)\mid\mathcal F_{t-1}]=\nabla F(\theta_t)$.
\end{lemma}


\paragraph{Observed responses.}
For noise-free responses $Y=r(Z)$, keep $Y$ fixed when extending
$z\mapsto g(\theta;z,Y)$ to complex inputs. The same argument gives
\[
\mathbb E_{U,V}[\operatorname{Re}g^{\mathbb C}(\theta;Z+U+iV,Y)
\mid Z,Y]=g(\theta;Z,Y),
\]
provided the score satisfies the entire and growth conditions at each
fixed $(\theta,y)$. For the objective
$F(\theta)=\mathbb E\ell(\theta;Z,Y)$, sampling and moment conditions
must hold for the joint pair $(Z,Y)$.
Appendix~\ref{app:regression} derives these bounds for the regression
experiment.