1\section{The price of inexact curvature}\label{sec:inexact}23The exact protocol uploads $p + p(p+1)/2$ numbers per worker in every derivative phase. Three cheaper variants are common in practice: a \emph{stale} Hessian, equal to the average of the local Hessians evaluated at the local MLEs and shipped once together with the pilot, after which every round costs $p$ numbers; a \emph{single-machine} Hessian evaluated at the current iterate on one worker, as in the surrogate-likelihood approach of \citet{jordan2019communication}; and a \emph{quantized} exact Hessian. The following perturbation theorem covers all three and inexact scores as well. Let $\vartheta$ be the current iterate, $e = \vartheta - \thetaN$, and let the coordinator compute4\begin{equation}\label{eq:inexact-step}5\tilde\vartheta = \vartheta - \widehat H^{-1}\,\widehat g, \qquad \widehat g = g_N(\vartheta) + \delta_g .6\end{equation}78\begin{theorem}[Inexact Newton steps]\label{thm:inexact}9Let \cref{ass:lin,ass:bias,ass:newton} hold, $\vartheta \pto \theta_0$, $\norm{\widehat H - H_N(\thetaN)} = \op(\delta_N)$ with $\delta_N \to 0$, and $\norm{\widehat H^{-1}} = \op(1)$. Then10\begin{equation}\label{eq:inexact}11\norm{\tilde\vartheta - \thetaN} = \op\big(\norm{e}^2 + \delta_N\norm{e} + \norm{\delta_g}\big).12\end{equation}13With $q$ such steps from the order-$r$ pilot, each with a Hessian error of order $\delta_N$,14\begin{equation}\label{eq:inexact-q}15\norm{\iter{r}{q} - \thetaN} = \op\big(\norm{e_0}\,\delta_N^{\,q} + \norm{e_0}^{2^q}\big), \qquad e_0 = \pilot{r} - \thetaN,16\end{equation}17and $\sqrt N(\iter{r}{q} - \thetaN) \pto 0$ provided $\sqrt N\,m^{-r}\delta_N^{\,q} \to 0$ together with the conditions of \cref{thm:frontier}. In particular:18\begin{enumerate}[label=(\alph*)]19\item \emph{Stale averaged local curvature.} Under \cref{ass:third}, $\widehat H = K^{-1}\sum_k H_{D_k}(\thetahat_{D_k})$ satisfies $\delta_N = m^{-1} + N^{-1/2}$, and for $q \ge 1$ the sufficient exponent becomes20\begin{equation}\label{eq:alpha-stale}21\alpha_{\mathrm{stale}}(r,q) = 1 - \frac{1}{2\,(r+q)} .22\end{equation}23\item \emph{Single-machine curvature.} $\widehat H = H_{D_1}(\vartheta)$ satisfies $\delta_N = m^{-1/2}$ (no third-derivative condition is needed), and the sufficient exponent becomes24\begin{equation}\label{eq:alpha-single}25\alpha_{\mathrm{single}}(r,q) = 1 - \frac{1}{2r+q} .26\end{equation}27\item \emph{Relative quantization.} If every entry of the transmitted score and Hessian is rounded to $b$ significant bits ($\varepsilon = 2^{-b}$), then \eqref{eq:inexact} holds with $\delta_N = \varepsilon$ and $\norm{\delta_g} = \op(\varepsilon\norm{e})$, so that one quantized step from the order-$r$ pilot is first-order equivalent to $\thetaN$ whenever $\sqrt N\,\varepsilon\, m^{-r} \to 0$ in addition to $K = o(N^{\alphafr{r}{1}})$. When $K \sim N^{\alpha}$ with $1 - 1/(2r) \le \alpha < \alphafr{r}{1}$, it suffices that28\begin{equation}\label{eq:bits}29b \;\ge\; \Big(\tfrac12 - r\,(1-\alpha)\Big)\log_2 N + \omega(1),30\end{equation}31and for $\alpha < 1 - 1/(2r)$ any fixed number of bits suffices, because the pilot alone is already first-order equivalent.32\end{enumerate}33\end{theorem}3435The proof (\cref{app:proofs}) is a matrix-perturbation argument around the integral form of the Newton remainder,36\begin{equation}\label{eq:inexact-identity}37\tilde\vartheta - \thetaN = \widehat H^{-1}\Big[\widehat H - \int_0^1 H_N(\thetaN + te)\,\dd t\Big]\,e \;-\; \widehat H^{-1}\delta_g,38\end{equation}39whose bracket is bounded by $\norm{\widehat H - H_N(\thetaN)} + \tfrac12\Lambda_N\norm{e}$. The theorem turns the qualitative statements in the literature---the $m^{-1/2}$ factor of \citet[Theorem~3.1]{jordan2019communication}, the $(\beta/\lambda)^2 \asymp m^{-1}$ contraction of averaged local Hessians in \citet[Theorem~2]{shamir2014communication} and \citet{fan2023communication}---into exponents. \Cref{tab:protocols} tabulates them with the communication they require. Case~(b) with $r = 1$ gives $\alpha = (q+1)/(q+2)$, exactly the rounds frontier that \citet{fan2023communication} state for CEASE up to logarithms, and $q = 1$ gives the $2/3$ of \citet{jordan2019communication}; case~(a) with $(r,q) = (1,1)$ gives $3/4$, the regime of \citet{zhu2021least}. The stale variant has effective depth $r + q$, linear in $q$, against $2^q r$ for exact curvature: with stale curvature a second round is worth one more local level; with exact curvature it is worth doubling the local order.4041Case~(c) is, to our knowledge, new in this form. The precision needed for the derivative uplink is governed by how far the pilot already is from $\thetaN$: a pilot of order $r$ with $K \sim N^\alpha$ has error $m^{-r} = N^{-r(1-\alpha)}$, and a relative error $\varepsilon$ in the step costs $\varepsilon N^{-r(1-\alpha)}$, which is negligible at the $N^{-1/2}$ scale if and only if \eqref{eq:bits} holds. For the half-block jackknife at $\alpha = 0.8$ the requirement is $0.1\log_2 N$ bits, two bits of mantissa at $N = 2^{20}$, whereas the plain average at the same $\alpha$ would need $0.3\log_2N = 6$ bits and is outside its own frontier ($3/4$) in any case. The mantissa bits multiply the $p(p+1)/2$ Hessian entries per worker and round, so the saving is $(64 - b)/64$ of the Hessian payload. The statement is asymptotic in the same sense as \cref{thm:frontier}; the ablation of \cref{sec:mc} shows at which $b$ the quantization error becomes visible at finite $N$.4243\begin{table}[t]44\centering\scriptsize45\begin{threeparttable}46\caption{Protocol variants: local fits, synchronized phases, uplink per worker, and sufficient exponents.}\label{tab:protocols}47\begin{tabular}{@{}p{3.0cm}p{4.2cm}cp{3.3cm}p{3.6cm}@{}}48\toprule49Variant & Curvature used by the coordinator & Local fits & Uplink per worker (numbers) & Sufficient exponent $\alpha$ \\50\midrule51$(r,0)$ & none & $\sum_j s_j$ & $p$ & $1 - 1/(2r)$ \\52$(r,q)$ exact & aggregated $H_N(\theta)$ at each iterate & $\sum_j s_j$ & $p + q\,(p + p(p+1)/2)$ & $1 - 1/(2^{q+1}r)$ \\53$(r,q)$ stale & $K^{-1}\sum_k H_{D_k}(\thetahat_{D_k})$, shipped once & $\sum_j s_j$ & $p + p(p+1)/2 + q\,p$ & $1 - 1/(2(r+q))$ \\54$(r,q)$ single & $H_{D_1}(\theta)$ on worker 1 & $\sum_j s_j$ & $p + q\,p$ & $1 - 1/(2r+q)$ \\55$(r,1)$ quantized & aggregated, $b$ mantissa bits & $\sum_j s_j$ & $p + (p + p(p+1)/2)\,b/64$ & $1 - 1/(4r)$ if \eqref{eq:bits} \\56AEE \citep{lin2011aggregated} & local Jacobians at local estimates & 1 & $p + p(p+1)/2$ & $< 1/4$ (their Thm 4.2) \\57CSL \citep{jordan2019communication}, $t$ rounds & machine 1, full surrogate & 1 & $t\,p$ & $1 - 1/(2 + t)$ from the average \\58exact pooling (exponential families) & sufficient statistic & 1 & $\dim(T)$ & any $K \le N$ \\59\bottomrule60\end{tabular}61\begin{tablenotes}\footnotesize62\item Downlink per worker: $qp$ numbers for every variant with $q \ge 1$. Headers, serialization and the pilot's validity flag are excluded; measured bytes appear in \cref{sec:bench}. Exponents are sufficient conditions from \cref{thm:frontier,thm:inexact}; the CSL and AEE rows are the cited authors' results translated to our notation; the quantized row is in units of 64-bit numbers.63\end{tablenotes}64\end{threeparttable}65\end{table}66