\begin{abstract}
We study stochastic optimization with Gaussian measurement error in the
inputs. For scores admitting an entire extension, complex-corrected
stochastic gradient descent evaluates the score at a noisy observation
plus an independent imaginary Gaussian perturbation. Building on the
classical complex corrected-score identity, this gives a directly
implementable learning rule whose gradient is exactly unbiased at every
iteration, rather than corrected only to finite order. Under an exponential
derivative bound with parameter-dependent amplitude, a gradient-based
amplitude control, and sub-Gaussian latent inputs, we derive the conditional
moment estimates that place this recursion within classical stochastic
approximation. We obtain convex objective bounds, almost-sure convergence
to the clean optimizer $\theta^*$ and mean-square rates under strong
convexity, and nonconvex stationarity bounds. We also establish
last-iterate and averaged central limit laws for step sizes
$c/(t+\gamma)^\alpha$, $1/2<\alpha\le1$, including the distinct averaging
covariance at $\alpha=1$. The construction allows any known positive
semidefinite noise covariance. We discuss exact correction for admissible
non-Gaussian errors, including uniform and symmetric two-point noise,
under suitable integrability conditions. Explicit examples and numerical
experiments illustrate parameter convergence and improved prediction
over naive training.
\end{abstract}