\section{Further work} \label{sec:future_work} \subsection{Noise generalization} \label{sec:noise_generalization} \paragraph{Symmetric distributions and van Dantzig conjugates.} For the overview below, write \[ \phi_X(t)=\mathbb E e^{itX},\qquad \phi_Y(t)=\frac{1}{\phi_X(it)}. \] Here the van Dantzig (VD) convention permits analytic or meromorphic continuation: the reciprocal is interpreted through that continuation, with removable singularities filled in, and must be a characteristic function on all of $\mathbb R$. Table~\ref{tab:vd_distributions} collects twenty symmetric examples and nonexamples. Classical pairs, Bessel constructions, and the conditional generalized L\'evy--Bessel family are discussed in~\cite{konstantopoulos2024}. \begingroup \small \setlength{\tabcolsep}{3pt} \renewcommand{\arraystretch}{1.18} \begin{longtable}{@{}>{\centering\arraybackslash}p{0.6cm} >{\raggedright\arraybackslash}p{3.1cm} >{\centering\arraybackslash}p{2.65cm} >{\centering\arraybackslash}p{0.7cm} >{\raggedright\arraybackslash}p{4cm} >{\centering\arraybackslash}p{2.65cm}@{}} \caption{Symmetric distributions and VD conjugates. All sums below have independent summands. The parameters $a,b,\sigma$ are positive.} \label{tab:vd_distributions}\\ \toprule No. & Original variable $X$ & $\phi_X(t)$ & VD? & Conjugate variable $Y$ & $\phi_Y(t)$ \\ \midrule \endfirsthead \multicolumn{6}{c}{\tablename\ \thetable\ (continued)}\\ \toprule No. & Original variable $X$ & $\phi_X(t)$ & VD? & Conjugate variable $Y$ & $\phi_Y(t)$ \\ \midrule \endhead \midrule \multicolumn{6}{r}{Continued on next page}\\ \endfoot \bottomrule \endlastfoot 1 & Degenerate, $X=0$ & $1$ & Yes & Itself & $1$ \\ 2 & Normal $N(0,\sigma^2)$ & $e^{-\sigma^2t^2/2}$ & Yes & Same normal law & $e^{-\sigma^2t^2/2}$ \\ 3 & Rademacher $\pm a$ & $\cos(at)$ & Yes & Hyperbolic secant & $\operatorname{sech}(at)$ \\ 4 & Hyperbolic secant & $\operatorname{sech}(at)$ & Yes & Rademacher $\pm a$ & $\cos(at)$ \\ 5 & Uniform $U(-a,a)$ & $\frac{\sin(at)}{at}$ & Yes & Logistic, scale $a/\pi$ & $\frac{at}{\sinh(at)}$ \\ 6 & Logistic, scale $a/\pi$ & $\frac{at}{\sinh(at)}$ & Yes & Uniform $U(-a,a)$ & $\frac{\sin(at)}{at}$ \\ 7 & Symmetric triangular & $\left(\frac{\sin(at)}{at}\right)^2$ & Yes & Sum of two logistic variables, scale $a/\pi$ & $\left(\frac{at}{\sinh(at)}\right)^2$ \\ 8 & Sum of two Rademacher variables $\pm a$ & $\cos^2(at)$ & Yes & Sum of two hyperbolic-secant variables & $\operatorname{sech}^2(at)$ \\ 9 & Symmetric beta-type & $\mathcal J_\nu(t)$ & Yes & Bessel conjugate & $1/\mathcal I_\nu(t)$ \\ 10 & Arcsine on $[-1,1]$ & $J_0(t)$ & Yes & Bessel conjugate, $\nu=0$ & $1/I_0(t)$ \\ 11 & Semicircle on $[-1,1]$ & $2J_1(t)/t$ & Yes & Bessel conjugate, $\nu=1$ & $t/(2I_1(t))$ \\ 12 & Bessel self-reciprocal & $\mathcal J_\nu(t)/\mathcal I_\nu(t)$ & Yes & Itself & $\mathcal J_\nu(t)/\mathcal I_\nu(t)$ \\ 13 & Symmetric two-component Gaussian mixture & $e^{-\sigma^2t^2/2}\cos(\mu t)$ & Yes & Sum of a normal and a hyperbolic-secant variable & $e^{-\sigma^2t^2/2}\operatorname{sech}(\mu t)$ \\ 14 & Generalized L\'evy--Bessel family & $\mathcal J_\Psi(t)$ & Yes$^*$ & L\'evy--Bessel conjugate & $1/\mathcal I_\Psi(t)$ \\ 15 & Student's $t_\nu$, $0<\nu<\infty$ & $T_\nu(t)$ & No & Does not exist & --- \\ 16 & Cauchy & $e^{-a|t|}$ & No & Does not exist & --- \\ 17 & Laplace & $1/(1+b^2t^2)$ & No & Does not exist & --- \\ 18 & Symmetric stable, $0<\alpha<2$ & $e^{-c|t|^\alpha}$ & No & Does not exist & --- \\ 19 & Generalized normal, $p=1$ & $1/(1+t^2)$ & No & Does not exist & --- \\ 20 & Generalized normal, $00$; its density is bimodal only when $\mu>\sigma$. Notation for Table~\ref{tab:vd_distributions} is collected in Appendix~\ref{app:vd_notation}. \paragraph{Interpretation and optimization scope.} Rows 10 and 11 specialize row 9; row 16 specializes row 18 at $\alpha=1$; and row 19 specializes row 17 at $b=1$. A formal reciprocal alone does not establish a VD conjugate. For example, the Laplace law gives \[ \frac1{\phi_X(it)}=1-b^2t^2, \] which is not a characteristic function, since it is unbounded on $\mathbb R$. The negative entries for Student, Cauchy, stable, and $00$, \[ M_U(t)=\frac{\sinh(bt)}{bt},\qquad \varphi_V(t)=\frac{bt}{\sinh(bt)}, \] with both ratios interpreted as $1$ at $t=0$. The conjugate $V$ has a centered logistic distribution with scale $b/\pi$: \[ f_V(v)=\frac{\pi}{4b\cosh^2(\pi v/(2b))},\qquad F_V(v)=\frac{1}{1+e^{-\pi v/b}}. \] It can be sampled directly from $R\sim\operatorname{Unif}(0,1)$ via \begin{equation}\label{eq:uniform_conjugate_sampler} V=\frac b\pi\log\frac{R}{1-R}. \end{equation} Thus a uniform measurement error is corrected by an independent logistic imaginary perturbation. \paragraph{Symmetric two-point measurement error.} For $\mathbb P(U=b)=\mathbb P(U=-b)=1/2$, \[ M_U(t)=\cosh(bt),\qquad \varphi_V(t)=\frac1{\cosh(bt)}. \] Here $V$ has a scaled hyperbolic-secant distribution: \[ f_V(v)=\frac{1}{2b\cosh(\pi v/(2b))},\qquad F_V(v)=\frac2\pi\arctan\!\left(e^{\pi v/(2b)}\right). \] An exact sampler is \begin{equation}\label{eq:binary_conjugate_sampler} V=\frac{2b}\pi\log\tan\!\left(\frac{\pi R}{2}\right), \qquad R\sim\operatorname{Unif}(0,1). \end{equation} The imaginary perturbation is continuous even though the measurement error is discrete. \paragraph{Exact correction and implementation.} Set $D=U+iV$. Equation~\eqref{eq:conjugate_transform} implies $\mathbb E D^k=0$ for every integer $k\ge1$ \cite{conjugatecriterion2026}. Hence, for fixed $(\theta,z)$, \begin{equation}\label{eq:nongaussian_correction} \mathbb E_{U,V}\operatorname{Re}g^{\mathbb C}(\theta;z+U+iV) =g(\theta;z) \end{equation} for polynomial scores. The identity also holds for entire scores when \begin{equation}\label{eq:nongaussian_taylor} \sum_{k=0}^\infty \frac{\|\partial_z^k g^{\mathbb C}(\theta;z)\|}{k!} \mathbb E|D|^k<\infty. \end{equation} Indeed, this condition permits termwise expectation of the Taylor series; all nonconstant terms vanish. Thus correction is exact, with no finite-order truncation. Algorithm~\ref{alg:nongaussian_ccsgd} implements it using the explicit inverse-distribution samplers above. It requires the error law and its scale, but not the latent input distribution. \begin{algorithm}[H] \caption{CCSGD for uniform or symmetric two-point measurement error} \label{alg:nongaussian_ccsgd} \begin{algorithmic}[1] \Require Initial point $\theta_1$, step sizes $\{\eta_t\}_{t=1}^T$, known error type and scale $b>0$ \For{$t=1,\ldots,T$} \State Observe a fresh $W_t=Z_t+U_t$ \State Draw $R_t\sim\operatorname{Unif}(0,1)$ independently of the observation and the past \If{$U_t\sim\operatorname{Unif}[-b,b]$} \State $V_t\gets(b/\pi)\log\{R_t/(1-R_t)\}$ \Else\Comment{$\mathbb P(U_t=\pm b)=1/2$} \State $V_t\gets(2b/\pi)\log\tan(\pi R_t/2)$ \EndIf \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} For noise-free responses, the observed response is held fixed during the complex evaluation, as in Section~\ref{sec:algorithm}. Independent replicates of $V_t$ may be averaged using the same observation. For independent input-error coordinates, the construction can also be applied coordinatewise with independent conjugate perturbations, subject to the corresponding multivariate integrability condition. \paragraph{Moment conditions for optimization.} Unlike a Gaussian conjugate, these conjugates have exponential moments only below a finite threshold. Specifically, \[ \mathbb E e^{s|V|}<\infty\quad\Longleftrightarrow\quad 0\le s<\lambda, \qquad \lambda= \begin{cases} \pi/b, & U\sim\operatorname{Unif}[-b,b],\\ \pi/(2b), & \mathbb P(U=\pm b)=1/2. \end{cases} \] Under the derivative envelope~\eqref{eq:parameter_envelope}, the strict condition $a<\lambda$ suffices for \eqref{eq:nongaussian_taylor}: the score has exponential growth of any rate slightly larger than $a$, and the conjugate has an exponential moment at a still larger rate. If $2a<\lambda$, boundedness of $U$ gives \[ K_a=\mathbb E e^{2a|Z|}\, \mathbb E[|D|^2e^{2a|D|}]<\infty. \] Together with the clean sampling and amplitude bounds, the same argument as in Section~\ref{sec:main_results} yields the relative variance condition~\eqref{eq:relative_variance}. For the CLTs, choose $\delta>0$ with $(2+\delta)a<\lambda$; the higher-moment proof then applies as well. Thus the optimization analysis transfers when these moment conditions and the corresponding objective and covariance assumptions are verified. Polynomial scores satisfy the required input moment conditions for both pairs. Arbitrary entire scores still require growth control, and random responses require the corresponding joint moment bounds. \paragraph{Closure under independent linear combinations.} The admissible noise class is closed under finite independent linear combinations. Let $U_1,\ldots,U_r$ be independent symmetric errors with moment generating functions finite on $\mathbb R$, and let $V_1,\ldots,V_r$ be mutually independent conjugates, independent of all the errors, such that $M_{U_j}(t)\varphi_{V_j}(t)=1$. For real coefficients $\beta_1,\ldots,\beta_r$, set \[ U=\sum_{j=1}^r\beta_jU_j,\qquad V=\sum_{j=1}^r\beta_jV_j. \] Independence gives \[ M_U(t)\varphi_V(t) =\prod_{j=1}^r \bigl[M_{U_j}(\beta_jt)\varphi_{V_j}(\beta_jt)\bigr]=1. \] Thus $V$ is a conjugate of $U$ and can be sampled by combining the component samplers. Moreover, for $D_j=U_j+iV_j$, independence and the multinomial expansion give $\mathbb E(\sum_j\beta_jD_j)^k=0$ for every integer $k\ge1$: every term contains a positive-order moment of some $D_j$, which vanishes. Consequently the same exact correction applies to polynomial scores, and to entire scores satisfying~\eqref{eq:nongaussian_taylor} for the combined perturbation. For example, the sum of two independent $\operatorname{Unif}[-b,b]$ errors has a symmetric triangular distribution on $[-2b,2b]$; its conjugate is the sum of two independent logistic variables of scale $b/\pi$. Similarly, the sum of $n$ independent errors taking values $\pm b$ with equal probabilities has law $b(2B-n)$, where $B\sim\operatorname{Bin}(n,1/2)$; its conjugate is the sum of $n$ independent hyperbolic-secant variables with the density above. A Gaussian error plus an independent uniform error is corrected by the sum of the corresponding independent Gaussian and logistic perturbations. These constructions concern sums of independent variables; arbitrary distributional mixtures need not preserve the conjugate property. The moment requirements must also be retained. Suppose each component conjugate satisfies $\mathbb E e^{s|V_j|}<\infty$ for $0\le s<\lambda_j$, taking $\lambda_j=\infty$ for Gaussian components. Then the triangle inequality and independence imply \[ \mathbb E e^{s|V|} \le\prod_{j=1}^r\mathbb E e^{s|\beta_j||V_j|}<\infty, \qquad 0\le s<\lambda_{\rm eff},\qquad \lambda_{\rm eff} =\min_{j:\beta_j\ne0}\frac{\lambda_j}{|\beta_j|}. \] The minimum is interpreted as $\infty$ if all coefficients vanish. Since $M_U$ is finite on $\mathbb R$, the combined real error also has absolute exponential moments of every linear order. Hence, with sub-Gaussian latent inputs, $a<\lambda_{\rm eff}$ suffices for the entire-score identity, while $2a<\lambda_{\rm eff}$ supplies the correction variance and some $(2+\delta)$-moment. The optimization arguments then transfer subject to the same clean sampling, amplitude, objective, and covariance conditions. \subsection{Function generalization} Exact Gaussian correction restricts the admissible score class. In the scalar Gaussian location model with positive noise variance, Stefanski \cite{stefanski1989} shows that a target admitting an integrable unbiased estimator for every real mean must extend to an entire function. Applied componentwise at fixed $\theta$, this rules out globally exact unbiased correction of non-entire scores in that setting. Entire analyticity alone does not ensure the integrability required by CCSGD; growth control remains essential. For non-entire scores, a natural direction is approximate or regularized correction with explicit bias, variance, and optimization-error bounds. With singular $\Omega$, the analyticity restriction concerns the directions affected by noise.