Workload-Preserving Differentially Private Synthetic Data for Causal Inference via Maximum-Entropy Calibration

# Workload-Preserving Differentially Private Synthetic Data for Causal Inference via Maximum-Entropy Calibration
Source: [https://arxiv.org/html/2607.08122](https://arxiv.org/html/2607.08122)
[Amir Asiaee](https://arxiv.org/html/2607.08122v1/mailto:[email protected]?Subject=Your%20UAI%202026%20paper)Kaveh AryanDepartment of Informatics King’s College London London, UK

###### Abstract

Workload\-based differentially private \(DP\) synthetic data methods privately measure aggregate queries and post\-process the noisy answers into synthetic records\. Generic workloads can achieve strong distributional fidelity, but causal estimands such as the average treatment effect \(ATE\) depend on treatment\-arm balance and outcome moments that generic marginals need not preserve\. We propose*causal workloads*: DP query sets designed around the orthogonal moments used by doubly robust causal estimators\. The released workload can be used directly by stable moment\-map estimators or reconstructed by maximum\-entropy calibration into reusable synthetic data; our theory decomposes ATE error into sampling, privacy, workload\-approximation, Monte Carlo, and calibration terms\. We also introduceCausal\-AIM, an adaptive workload selector, and a noise\-aware multiple\-imputation \(NA\+MI\) procedure for confidence intervals from DP synthetic data\. Because the workload is released once, the same DP synthetic table can support ATE, ATT, and subgroup analyses without additional privacy spending\. Empirically, causal workloads are most useful at strict privacy budgets and for calibrated uncertainty, while generic workloads often retain an advantage for point RMSE as privacy relaxes\. The broader lesson is a tradeoff: distributional fidelity can help point accuracy, but valid causal inference requires preserving causal moments and propagating DP noise rather than treating synthetic rows as real\.

## 1Introduction

Differential privacy is increasingly used for public release of sensitive tabular datasets in domains such as health, education, and social science\[dwork2014foundations\]\. A common deployment pattern is to release DP synthetic microdata so that downstream analysts can reuse existing statistical and machine learning workflows\[liang\_generating\_2026\]\. State\-of\-the\-art DP synthesis methods often follow a select–measure–reconstruct pipeline: select a set of low\-dimensional queries, measure them privately, and reconstruct a distribution \(often maximum entropy / graphical model\) from the noisy answers\[mckenna2021nist,mckenna2022aim\]\. However,*causal inference*introduces additional structure and fragility\. Even if synthetic data match many marginals, causal estimands can be biased by subtle distortions in overlap, confounding structure, or conditional outcome models\. Recent work emphasizes that DP noise inflates the variance of causal estimators and invalidates naive confidence intervals unless it is explicitly modeled\[farzam2024causal,schroder2025private,dp\-cate2025,ohnishi2024covbal\]\.

This paper asks a concrete question:

> *Which DP queries must a synthetic data mechanism preserve to enable valid causal inference, and what are the fundamental privacy–causal tradeoffs?*

We answer by designing*causal workloads*and giving explicit bounds for stable workload\-based estimators and for the calibrated synthetic\-data route\.

#### Key idea: causal workloads as orthogonal moments\.

Modern causal estimators, including doubly robust \(DR\) estimators\[robins1994aipw\]and double/debiased machine learning\[chernozhukov2018dml\], rely on score functions whose expectation identifies the target estimand\. We propose to choose a workload that directly measures these orthogonal moments, or a basis approximation thereof, under DP\. The released workload can then be used in two ways: the directqq\-route evaluates a stable estimator as a function of the released moments, while the synthetic\-data route reconstructs a maximum\-entropy distribution matched to those moments and samples reusable synthetic records\. Our theory controls the moment\-level error in both routes, with additional calibration and Monte Carlo terms for synthetic\-data analysis\.

#### Why synthetic data at all?

If the goal is ATE only, one could release a DP ATE estimate directly\. Synthetic data is valuable when many analysts ask many questions, or when we want to preserve a whole*class*of causal estimands \(ATE, average treatment effect on the treated \(ATT\)𝔼​\[Y​\(1\)−Y​\(0\)∣T=1\]\\mathbb\{E\}\[Y\(1\)\-Y\(0\)\\mid T\{=\}1\], subgroup effects, balancing diagnostics, etc\.\) without repeated access to the private data\. Causal workloads formalize this “class of analyses” viewpoint; Section[7](https://arxiv.org/html/2607.08122#S7)provides direct evidence that a single release supports ATE, ATT, and subgroup analyses with no additional privacy spending, a reuse property structurally unavailable to direct DP ATE estimators\.

#### Our contributions\.

1. 1\.We define*causal workloads*: DP query sets built from orthogonal\-score moments, with two downstream routes: direct moment plug\-ins and maximum\-entropy synthetic\-data reconstruction\.
2. 2\.We prove finite\-sample error bounds that connect ATE error to released\-workload error, and decompose synthetic\-data error into sampling, privacy, workload\-approximation, Monte Carlo, and calibration terms\. All proofs are collected in Appendix[E](https://arxiv.org/html/2607.08122#A5)\.
3. 3\.We introduceCausal\-AIM, an adaptive causal workload selector, and NA\+MI, a noise\-aware multiple\-imputation procedure for confidence intervals from DP synthetic data\.
4. 4\.Empirically, Causal \+ NA\+MI is the only private method with near\-nominal coverage on all four benchmarks \(99\.8−100%99\.8\{\-\}100\\%atε≤1\\varepsilon\\leq 1\), while naive synthetic\-data analyses remain at or below35\.2%35\.2\\%; the same release also supports ATE, ATT, and subgroup analyses without extra privacy spending\.

## 2Related Work

#### A brief DP\-synthesis primer\.

A mechanism is\(ε,δ\)\(\\varepsilon,\\delta\)\-differentially private if replacing one record changes its output distribution by at most a factoreεe^\{\\varepsilon\}, up to slackδ\\delta\[dwork2014foundations\]\. Workload\-based DP synthetic\-data methods first choose a set of aggregate queries \(the*workload*\), release noisy answers to those queries, and then reconstruct a distribution whose query answers match the noisy measurements\. The final synthetic records are sampled from this reconstructed distribution; reconstruction and sampling are post\-processing, so they spend no additional privacy budget\. This select–measure–reconstruct template underlies MWEM \(Multiplicative Weights Exponential Mechanism;hardt2012mwem\), which iteratively fixes poorly matched queries; Private\-PGM, which fits a graphical model to noisy low\-order marginals\[mckenna2019privatepgm\]; MST \(Maximum\-Spanning\-Tree marginal selection with Private\-PGM reconstruction;mckenna2021nist\), which won the NIST synthetic\-data challenge; and AIM \(Adaptive and Iterative Mechanism;mckenna2022aim\), which adaptively selects marginals against a target workload\.

#### Workload\-based DP synthetic data\.

A large class of methods measures a set of marginals / low\-dimensional queries and reconstructs a distribution by imposing structure such as a graphical\-model factorization or an optimization objective over candidate distributions, including Private\-PGM\[mckenna2019privatepgm\]and its NIST\-MST/MST instantiations\[mckenna2021nist\], the adaptive AIM mechanism\[mckenna2022aim\], and recent optimal\-transport based approaches\[donhauser2024privpgd\]\. Benchmarks suggest these methods can outperform GAN\-style synthesis on tabular data for many metrics\[tao2022benchmark,bowen2020comparative\]\. These methods are designed for general\-purpose fidelity; none explicitly targets causal estimands\.

#### Inference from DP synthetic data\.

Naively treating DP synthetic data as real can produce invalid inference, e\.g\. inflated type\-I error\[perez2024mwutest\]\. Noise\-aware procedures combine Bayesian modeling of the DP noise with multiple imputation to recover calibrated uncertainty\[raisa2023noiseaware\]\. For record\-level synthetic microdata—tables whose rows represent individual units rather than only aggregate summaries—classical multiple\-imputation inference dates back toreiter2003inferenceandraghunathan2003multiple\. Our noise\-aware multiple imputation \(NA\+MI\) procedure builds on this line but specializes it to causal estimands by choosing workloads informed by the orthogonal score structure\.

#### Causal inference under DP\.

Recent papers propose DP causal estimators with point and interval estimation guarantees\. PrivATE\[schroder2025private\]develops doubly robust DP confidence intervals for ATE using output perturbation\.ohnishi2024covbalpropose DP covariate balancing with finite\-sample guarantees\. DP\-CATE\[dp\-cate2025\]extends orthogonal learning to heterogeneous treatment effects under DP\.farzam2024causalstudy causal inference under DP broadly and analyze how DP mechanisms can inflate ATE variance and distort individual\-level effects\. Earlier,niu2022dp\_cateproposed DP estimation of heterogeneous causal effects, and concurrent work bylebeda2025model\_agnosticdevelops a model\-agnostic DP causal framework\. Our work focuses instead on*synthetic data*as the release object: rather than releasing a single DP estimate, we release a DP synthetic dataset that supports a whole class of causal analyses\.

#### Double/debiased machine learning\.

Our causal workload design draws on the orthogonal moment framework ofchernozhukov2018dml, which shows that Neyman\-orthogonal scores enable root\-nninference for treatment effects when nuisance functions are estimated at slower rates\. We use this structure to choose workloads that preserve the moments needed for orthogonal scores\.

#### Causal prior for synthetic data\.

A related line encodes causal or analysis\-specific prior knowledge*into the synthetic generation process itself*: reward\-guided generation steers a generator toward a target downstream utility such as regression\-coefficient recovery\[jackson2026reward\], and post\-hoc constraint wrappers inject partial causal structure, such as trusted or forbidden edges and monotonicity, into arbitrary tabular generators\[asiaee2026causalwrap\]\. Our objective is different: rather than encoding a prior during generation, we design the DP measurements so that the released dataset preserves a target causal estimand, and we account for the DP noise in downstream inference\.

## 3Problem Setup

We observe i\.i\.d\. dataD=\{\(Xi,Ti,Yi\)\}i=1nD=\\\{\(X\_\{i\},T\_\{i\},Y\_\{i\}\)\\\}\_\{i=1\}^\{n\}, whereT∈\{0,1\}T\\in\\\{0,1\\\}is treatment\. The target is the average treatment effect \(ATE\)τ:=𝔼​\[Y​\(1\)−Y​\(0\)\]\.\\tau:=\\mathbb\{E\}\[Y\(1\)\-Y\(0\)\]\.

###### Assumption 1\(Unconfoundedness and positivity\)\.

\(Y​\(1\),Y​\(0\)\)⟂T∣X\(Y\(1\),Y\(0\)\)\\perp T\\mid Xand the propensity score\[rosenbaum1983propensity\]e​\(x\):=Pr⁡\(T=1∣X=x\)e\(x\):=\\Pr\(T=1\\mid X=x\)satisfiese​\(x\)∈\[η,1−η\]e\(x\)\\in\[\\eta,1\-\\eta\]almost surely for someη\>0\\eta\>0\.

Under Assumption[1](https://arxiv.org/html/2607.08122#Thmassumption1),τ=𝔼​\[m1​\(X\)−m0​\(X\)\]\\tau=\\mathbb\{E\}\[m\_\{1\}\(X\)\-m\_\{0\}\(X\)\], wheremt​\(x\):=𝔼​\[Y∣T=t,X=x\]m\_\{t\}\(x\):=\\mathbb\{E\}\[Y\\mid T=t,X=x\]\.

## 4Causal Workloads and Maximum\-Entropy Calibration

Figure[1](https://arxiv.org/html/2607.08122#S4.F1)gives an overview of the full release\-and\-inference procedure\.

### 4\.1A Workload That Targets Orthogonal Moments

Letϕ:𝒳→ℝp\\phi:\\mathcal\{X\}\\to\\mathbb\{R\}^\{p\}be a feature map \(e\.g\. spline basis, random Fourier features, tree\-based leaves, or indicators for a discretization ofXX\)\. Define the*moment vectors*

qt\(0\)​\(D\)\\displaystyle q\_\{t\}^\{\(0\)\}\(D\):=1n​∑i=1n𝟙​\{Ti=t\}​ϕ​\(Xi\)∈ℝp,\\displaystyle:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\mathbbm\{1\}\\\{T\_\{i\}=t\\\}\\,\\phi\(X\_\{i\}\)\\in\\mathbb\{R\}^\{p\},\(1\)qt\(1\)​\(D\)\\displaystyle q\_\{t\}^\{\(1\)\}\(D\):=1n​∑i=1n𝟙​\{Ti=t\}​Yi​ϕ​\(Xi\)∈ℝp\.\\displaystyle:=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\mathbbm\{1\}\\\{T\_\{i\}=t\\\}\\,Y\_\{i\}\\,\\phi\(X\_\{i\}\)\\in\\mathbb\{R\}^\{p\}\.\(2\)Here,qt\(0\)q\_\{t\}^\{\(0\)\}records treatment\-arm feature masses, andqt\(1\)q\_\{t\}^\{\(1\)\}records outcome\-weighted feature moments\. As a simple bin\-indicator example, each coordinate ofqt\(1\)q\_\{t\}^\{\(1\)\}is the joint arm\-bin mass times the mean outcome in that bin, with the mass scaled by1/n1/n\. These four blocks are the private measurements used in the experimental pipeline\. Second\-order Gram moments are useful for some direct regression plug\-ins, but they are not part of the default release; Appendix[A](https://arxiv.org/html/2607.08122#A1)gives the feature\-encoding example and the extension\.

###### Definition 1\(Causal workload\)\.

Fix a feature mapϕ\\phi\. The*causal workload*𝒬ϕ\\mathcal\{Q\}\_\{\\phi\}used in our main procedure is the ordered collection of query functions whose answers are\{q0\(0\),q1\(0\),q0\(1\),q1\(1\)\}\\\{q\_\{0\}^\{\(0\)\},q\_\{1\}^\{\(0\)\},q\_\{0\}^\{\(1\)\},q\_\{1\}^\{\(1\)\}\\\}\. Its stacked answer is

q​\(D\):=\[q0\(0\)​\(D\),q1\(0\)​\(D\),q0\(1\)​\(D\),q1\(1\)​\(D\)\]∈ℝ4​p\.q\(D\):=\\bigl\[q\_\{0\}^\{\(0\)\}\(D\),q\_\{1\}^\{\(0\)\}\(D\),q\_\{0\}^\{\(1\)\}\(D\),q\_\{1\}^\{\(1\)\}\(D\)\\bigr\]\\in\\mathbb\{R\}^\{4p\}\.Here “workload” means the collection of DP queries, whileq​\(⋅\)q\(\\cdot\)denotes the realized answer vector\.

This base4​p4pworkload is what we release in the experiments\. It is*one principled choice*, not a claimed optimum: it targets the first\-order treatment and outcome moments used by orthogonal\-score inference on nuisances projected onto theϕ\\phibasis \(Proposition[2](https://arxiv.org/html/2607.08122#Thmproposition2), Appendix[B](https://arxiv.org/html/2607.08122#A2)\), without asserting minimax optimality, uniqueness, or a lower bound\.

![Refer to caption](https://arxiv.org/html/2607.08122v1/x1.png)Figure 1:End\-to\-end pipeline\. The data holder chooses the feature mapϕ\\phi, builds the causal workload, and releases Gaussian\-mechanism\-noised moments once\. Downstream analysis is post\-processing: one may use the noisy moments directly in a stable moment\-map estimator, or reconstruct a maximum\-entropy distribution \(via Private\-PGM\), sample synthetic data, and run NA\+MI inference\.
### 4\.2Private Measurement of the Workload

LetW=\(X,T,Y\)W=\(X,T,Y\)and letm=4​pm=4pbe the dimension of the causal workloadq​\(D\)q\(D\)defined above\. Fora∈\[m\]a\\in\[m\], letha​\(Wi\)h\_\{a\}\(W\_\{i\}\)denote recordii’s contribution to coordinateaa, soqa​\(D\)=n−1​∑iha​\(Wi\)q\_\{a\}\(D\)=n^\{\-1\}\\sum\_\{i\}h\_\{a\}\(W\_\{i\}\); for any distributionPPoverWW,qa​\(P\)=𝔼P​ha​\(W\)q\_\{a\}\(P\)=\\mathbb\{E\}\_\{P\}h\_\{a\}\(W\)\. We useq​\(D\)q\(D\)for the exact, non\-released empirical answer on the confidential data;q⋆=q​\(P⋆\)q^\{\\star\}=q\(P^\{\\star\}\)for the population answer under the true data\-generating lawP⋆P^\{\\star\}; andq~\\widetilde\{q\}for the DP noisy release\. The fixed causal mechanism measures allmmcoordinates once with the Gaussian mechanism:

q~=q​\(D\)\+Z,Z∼𝒩​\(0,σ2​Im\)\.\\widetilde\{q\}=q\(D\)\+Z,\\qquad Z\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\_\{m\}\)\.AssumeYYis clipped to\[−B,B\]\[\-B,B\]and‖ϕ​\(X\)‖2≤ϕmax\\left\\lVert\\phi\(X\)\\right\\rVert\_\{2\}\\leq\\phi\_\{\\max\}\. LetΔ2:=supD∼D′‖q​\(D\)−q​\(D′\)‖2\\Delta\_\{2\}:=\\sup\_\{D\\sim D^\{\\prime\}\}\\left\\lVert q\(D\)\-q\(D^\{\\prime\}\)\\right\\rVert\_\{2\}be the replacement\-adjacencyℓ2\\ell\_\{2\}sensitivity\.Δ2≤2​ϕmax​1\+B2n\.\\Delta\_\{2\}\\leq\\frac\{2\\phi\_\{\\max\}\\sqrt\{1\+B^\{2\}\}\}\{n\}\.The standard Gaussian mechanism\[dwork2014foundations\]usesσ=Δ2​2​log⁡\(1\.25/δ\)ε\.\\sigma=\\frac\{\\Delta\_\{2\}\\sqrt\{2\\log\(1\.25/\\delta\)\}\}\{\\varepsilon\}\.For an adaptive mechanism, the same notation is applied to the measured coordinate setS⊂\[m\]S\\subset\[m\]:q~S\\widetilde\{q\}\_\{S\}denotes the stored noisy answers onSS\.

### 4\.3Two Routes from the Released Workload for Causal Estimation

The DP object produced by measurement is a noisy aggregate vector, not yet a synthetic dataset\. It can be used in two ways\. In the*direct moment route*\(orqq\-route\), an estimator that can be written as a stable functionτ^est​\(q\)\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q\)is evaluated directly atq~\\widetilde\{q\}\(or at the measured coordinatesq~S\\widetilde\{q\}\_\{S\}\); Section[5](https://arxiv.org/html/2607.08122#S5)analyzes this route because it exposes how moment error becomes causal\-estimation error\. In the*synthetic\-data route*, we reconstruct a distributionPsynP^\{\\mathrm\{syn\}\}whose workload answers match the noisy measurements, sample synthetic records from that distribution, and then run standard causal estimators such as DR / augmented inverse\-propensity weighting \(AIPW\) and NA\+MI\. The experiments use this synthetic\-data route; the theory controls the part of this route that is expressible through the reconstructed workload answerq​\(Psyn\)q\(P^\{\\mathrm\{syn\}\}\), plus finite synthetic\-sampling error\. The exponential\-family form below is precisely the reconstruction step: it turns noisy moments into a distribution, and sampling from that distribution produces the synthetic data\. We present each route in two versions: a fixed mechanism that measures all queries at once \(S=\[m\]S=\[m\]\), and an adaptive mechanism,Causal\-AIM, that buildsSSover rounds\.

### 4\.4Maximum\-Entropy Reconstruction for Synthetic Data

Let𝒫\\mathcal\{P\}be a class of distributions overW=\(X,T,Y\)W=\(X,T,Y\)\(discrete support or parametric\)\. Given a measured coordinate setS⊂\[m\]S\\subset\[m\]\(withS=\[m\]S=\[m\]for the fixed full workload\), we define the calibrated synthetic distribution as an information projection:

Psyn=arg⁡minP∈𝒫⁡KL​\(P∥P0\)​s\.t\.​∀a∈S,qa​\(P\)=q~a,P^\{\\mathrm\{syn\}\}=\\arg\\min\_\{P\\in\\mathcal\{P\}\}\\mathrm\{KL\}\(P\\ \\\|\\ P\_\{0\}\)\\ \\text\{s\.t\.\}\\ \\forall a\\in S,\\ q\_\{a\}\(P\)=\\widetilde\{q\}\_\{a\},\(3\)whereP0P\_\{0\}is a reference distribution, such as a uniform distribution on the discretized support, an independent product of noisy one\-way marginals, or a public population prior\. In the discrete case, the solution is an exponential family:

psyn​\(w\)∝p0​\(w\)​exp⁡\(∑a∈Sξa​ha​\(w\)\)p^\{\\mathrm\{syn\}\}\(w\)\\propto p\_\{0\}\(w\)\\exp\\Big\(\\sum\_\{a\\in S\}\\xi\_\{a\}h\_\{a\}\(w\)\\Big\)for Lagrange multipliersξ\\xi\. Appendix[C](https://arxiv.org/html/2607.08122#A3)gives the derivation and the relaxed form used when noisy constraints are infeasible\. This is the same mathematical object used in Private\-PGM\[mckenna2019privatepgm,mckenna2021nist\]\. Thus reconstruction means fitting the maximum\-entropy distributionpξp\_\{\\xi\}whose measured workload answers matchq~S\\widetilde\{q\}\_\{S\}up to the DP noise scale; synthetic\-data generation is the subsequent sampling of records frompξp\_\{\\xi\}\. The fixed synthesis procedure is given in Appendix[C](https://arxiv.org/html/2607.08122#A3)\.

### 4\.5Adaptive Query Selection:Causal\-AIM

AIM selects queries adaptively to reduce worst\-case error in a generic workload\[mckenna2022aim\]\. The fixed mechanism measures the whole causal workload once, soS=\[m\]S=\[m\]\.Causal\-AIMis the adaptive variant: it buildsSSover rounds by selecting the next feature group to measure using a private estimate of causal utility\. Here a candidateϕr\\phi\_\{r\}can be a single feature or a group such as all bins of one covariate; letI​\(r\)⊂\[m\]I\(r\)\\subset\[m\]denote the corresponding coordinates of the released workload\. At iterationkk,Causal\-AIM\(i\) starts from the current measured setSk−1S\_\{k\-1\}and its reconstructed modelPk−1P\_\{k\-1\}, \(ii\) evaluates a private upper bound on ATE error using an orthogonal score residualψ​\(W;θ,ζ\)\\psi\(W;\\theta,\\zeta\), whereψ\\psiis the ATE influence function andζ\\zetacollects nuisance functions, and \(iii\) selects the next feature groupϕrk\\phi\_\{r\_\{k\}\}, or its corresponding moment coordinatesI​\(rk\)I\(r\_\{k\}\), before refitting to obtainPkP\_\{k\}; Appendix[D](https://arxiv.org/html/2607.08122#A4)gives details\. Selected groups are removed from the candidate set: onceI​\(r\)I\(r\)has been measured, its noisy answer is reused in later refits rather than measured again\. After any round, the stored noisy vectorq~Sk\\widetilde\{q\}\_\{S\_\{k\}\}can be used directly in a moment\-map estimator or passed through maximum\-entropy reconstruction to generate synthetic data; Algorithm[1](https://arxiv.org/html/2607.08122#alg1)displays the synthetic\-data route used in our experiments\. This yields a causal\-utility\-driven workload that is usually much smaller than “all marginals up to order 2”\.

Algorithm 1Causal\-AIM\(high\-level\)1:Data

DD, privacy budget

\(ε,δ\)\(\\varepsilon,\\delta\), feature groups

ℛ\\mathcal\{R\}, iterations

KK, reference distribution

P0P\_\{0\}
2:Initialize selected coordinates

S0←∅S\_\{0\}\\leftarrow\\emptyset, available set

A0←ℛA\_\{0\}\\leftarrow\\mathcal\{R\}, and current reconstruction

P\(0\)←P0P^\{\(0\)\}\\leftarrow P\_\{0\}
3:for

k=1,…,min⁡\(K,\|ℛ\|\)k=1,\\dots,\\min\(K,\|\\mathcal\{R\}\|\)do

4:Compute an internal utility

uk​\(r\)u\_\{k\}\(r\)for each unmeasured group

r∈Ak−1r\\in A\_\{k\-1\}, estimating the ATE\-error reduction from adding coordinates

I​\(r\)I\(r\)
5:Select

rkr\_\{k\}using the exponential mechanism applied to

\{uk​\(r\):r∈Ak−1\}\\\{u\_\{k\}\(r\):r\\in A\_\{k\-1\}\\\}, then update

Sk←Sk−1∪I​\(rk\)S\_\{k\}\\leftarrow S\_\{k\-1\}\\cup I\(r\_\{k\}\)and

Ak←Ak−1∖\{rk\}A\_\{k\}\\leftarrow A\_\{k\-1\}\\setminus\\\{r\_\{k\}\\\}
6:Measure only the new coordinates

I​\(rk\)I\(r\_\{k\}\)with the Gaussian mechanism and store their noisy answers

7:Refit

PkP\_\{k\}by solving \([3](https://arxiv.org/html/2607.08122#S4.E3)\) on all stored noisy coordinates

SkS\_\{k\}
8:endfor

9:Output stored noisy moments

q~SK\\widetilde\{q\}\_\{S\_\{K\}\}; for synthetic release, output

PKP\_\{K\}and sample

Dsyn∼PKD^\{\\mathrm\{syn\}\}\\sim P\_\{K\}\(post\-processing\)

## 5Theory: Causal Error Bounds and Tradeoffs

This section tracks how error in the released workload affects ATE estimation along the two routes\. We first analyze direct moment plug\-ins, where the estimator is a function of the released workload, and then add the reconstruction\-calibration and finite\-synthetic\-sample terms needed for the synthetic\-data route\. Thus the theoretical object is an estimator with an explicit workload argument,τ^est​\(q\)\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q\); this is narrower than an arbitrary downstream learner run on synthetic rows\. We state the main decomposition for the ATE; the same weighted\-estimand argument gives ATT and subgroup effects when the corresponding weights are represented by the workload \(Appendix[E\.1](https://arxiv.org/html/2607.08122#A5.SS1)\)\.

### 5\.1Bounding ATE Error by Moment Error

The two routes in Section[4](https://arxiv.org/html/2607.08122#S4)share the same first question: how accurately does DP release the workload moments? For the direct moment route, the second question is how sensitive an ATE estimator is to perturbing those moments\. For the second question, call a direct moment\-map estimatorτ^est​\(q\)\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q\)*LestL\_\{\\mathrm\{est\}\}\-stable*on a workload domain if

\|τ^est​\(q\)−τ^est​\(q′\)\|≤Lest​‖q−q′‖∞∀admissible​q,q′\.\|\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q\)\-\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q^\{\\prime\}\)\|\\leq L\_\{\\mathrm\{est\}\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}\\quad\\forall\\text\{admissible \}q,q^\{\\prime\}\.Hereqqmay be the population answerq⋆=q​\(P⋆\)q^\{\\star\}=q\(P^\{\\star\}\), the empirical answerq​\(D\)q\(D\), the DP releaseq~\\widetilde\{q\}, or the reconstructed answerq​\(Psyn\)q\(P^\{\\mathrm\{syn\}\}\)\. The formal stability examples below analyze the direct route; the later decomposition composes the same stability logic with the calibration and Monte Carlo terms that appear when the released workload is routed through maximum\-entropy reconstruction and synthetic sampling\. This separates the generic moment\-level guarantee from software\-specific choices about the reconstruction solver and downstream nuisance fitting\.

#### Example 1: projected ridge plug\-in\.

Some direct regression plug\-ins require the arm\-specific Gram momentGt​\(D\):=n−1​∑i𝟙​\{Ti=t\}​ϕ​\(Xi\)​ϕ​\(Xi\)⊤G\_\{t\}\(D\):=n^\{\-1\}\\sum\_\{i\}\\mathbbm\{1\}\\\{T\_\{i\}=t\\\}\\phi\(X\_\{i\}\)\\phi\(X\_\{i\}\)^\{\\top\}; writeqt\(2\)​\(D\):=vec⁡\{Gt​\(D\)\}q\_\{t\}^\{\(2\)\}\(D\):=\\operatorname\{vec\}\\\{G\_\{t\}\(D\)\\\}\. For a workload\-answer vector containing the needed blocks, writeGt​\(q\)G\_\{t\}\(q\)for the Gram block,rt​\(q\)=qt\(1\)r\_\{t\}\(q\)=q\_\{t\}^\{\(1\)\}, andϕ¯​\(q\)=q0\(0\)\+q1\(0\)\\bar\{\\phi\}\(q\)=q\_\{0\}^\{\(0\)\}\+q\_\{1\}^\{\(0\)\}\. Here “projection” means the ridge least\-squares approximation of the arm\-specific outcome regressionmt​\(x\)m\_\{t\}\(x\)in the span ofϕ\\phi\. The arm\-ttcoefficient isβ^t,λ​\(q\)=\{Gt​\(q\)\+λ​I\}−1​rt​\(q\)\\widehat\{\\beta\}\_\{t,\\lambda\}\(q\)=\\\{G\_\{t\}\(q\)\+\\lambda I\\\}^\{\-1\}r\_\{t\}\(q\), with corresponding projected conditional average treatment effect \(CATE\) and ATE:

τ^λ​\(x;q\)\\displaystyle\\widehat\{\\tau\}\_\{\\lambda\}\(x;q\):=ϕ​\(x\)⊤​\{β^1,λ​\(q\)−β^0,λ​\(q\)\},\\displaystyle:=\\phi\(x\)^\{\\top\}\\\{\\widehat\{\\beta\}\_\{1,\\lambda\}\(q\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(q\)\\\},\(4\)τ^λ​\(q\)\\displaystyle\\widehat\{\\tau\}\_\{\\lambda\}\(q\):=ϕ¯​\(q\)⊤​\{β^1,λ​\(q\)−β^0,λ​\(q\)\}\.\\displaystyle:=\\bar\{\\phi\}\(q\)^\{\\top\}\\\{\\widehat\{\\beta\}\_\{1,\\lambda\}\(q\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(q\)\\\}\.\(5\)
###### Theorem 1\(Projected ridge moment stability\)\.

AssumeY∈\[−B,B\]Y\\in\[\-B,B\]and‖ϕ​\(X\)‖2≤ϕmax\\left\\lVert\\phi\(X\)\\right\\rVert\_\{2\}\\leq\\phi\_\{\\max\}\. Fixλ\>0\\lambda\>0and useτ^λ​\(q\)\\widehat\{\\tau\}\_\{\\lambda\}\(q\)as defined in \([5](https://arxiv.org/html/2607.08122#S5.E5)\)\. Suppose that for eachq∘∈\{q,q′\}q\_\{\\circ\}\\in\\\{q,q^\{\\prime\}\\\}andt∈\{0,1\}t\\in\\\{0,1\\\},Gt​\(q∘\)⪰κ​IG\_\{t\}\(q\_\{\\circ\}\)\\succeq\\kappa Ifor someκ≥0\\kappa\\geq 0and‖β^t,λ​\(q∘\)‖2≤Rβ\\left\\lVert\\widehat\{\\beta\}\_\{t,\\lambda\}\(q\_\{\\circ\}\)\\right\\rVert\_\{2\}\\leq R\_\{\\beta\}\. Then for two admissible workload\-answer vectorsq,q′q,q^\{\\prime\}of the same type \(same coordinate ordering and same rule forGtG\_\{t\}\),

\|τ^λ​\(q\)−τ^λ​\(q′\)\|≤Lϕ​‖q−q′‖∞,Lϕ≤Cϕ​\(1\+1κ\+λ\)\.\|\\widehat\{\\tau\}\_\{\\lambda\}\(q\)\-\\widehat\{\\tau\}\_\{\\lambda\}\(q^\{\\prime\}\)\|\\;\\leq\\;L\_\{\\phi\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\},\\ L\_\{\\phi\}\\leq C\_\{\\phi\}\\left\(1\+\\frac\{1\}\{\\kappa\+\\lambda\}\\right\)\.The constantCϕC\_\{\\phi\}depends on\(ϕmax,Rβ,p\)\(\\phi\_\{\\max\},R\_\{\\beta\},p\)but not onnn,ε\\varepsilon, or the particular DP noise draw; Appendix[E\.2](https://arxiv.org/html/2607.08122#A5.SS2)gives one explicit choice\.

#### Example 2: clipped IPW/AIPW moment maps\.

Inverse\-propensity weighting \(IPW\) estimates𝔼​\[Y​\(1\)\]\\mathbb\{E\}\[Y\(1\)\]and𝔼​\[Y​\(0\)\]\\mathbb\{E\}\[Y\(0\)\]by reweighting observed treated and control outcomes by inverse treatment probabilities\. With partition\-cell features, those reweighted sums can be written directly as functions of the released moments\. For partition\-cell features, letpt​j​\(q\)=qt​j\(0\)p\_\{tj\}\(q\)=q\_\{tj\}^\{\(0\)\},rt​j​\(q\)=qt​j\(1\)r\_\{tj\}\(q\)=q\_\{tj\}^\{\(1\)\},sj​\(q\)=p0​j​\(q\)\+p1​j​\(q\)s\_\{j\}\(q\)=p\_\{0j\}\(q\)\+p\_\{1j\}\(q\), and define the clipped cell propensitye^j​\(q\)=clip⁡\{p1​j​\(q\)/sj​\(q\),η,1−η\}\\widehat\{e\}\_\{j\}\(q\)=\\operatorname\{clip\}\\\{p\_\{1j\}\(q\)/s\_\{j\}\(q\),\\eta,1\-\\eta\\\}\. The direct clipped IPW moment map is

τ^IPW​\(q\)=∑j\{r1​j​\(q\)e^j​\(q\)−r0​j​\(q\)1−e^j​\(q\)\}\.\\widehat\{\\tau\}\_\{\\mathrm\{IPW\}\}\(q\)=\\sum\_\{j\}\\left\\\{\\frac\{r\_\{1j\}\(q\)\}\{\\widehat\{e\}\_\{j\}\(q\)\}\-\\frac\{r\_\{0j\}\(q\)\}\{1\-\\widehat\{e\}\_\{j\}\(q\)\}\\right\\\}\.AIPW adds the usual outcome\-regression augmentation to the same cell\-level IPW score; Appendix[E\.3](https://arxiv.org/html/2607.08122#A5.SS3)gives the full expression\.

###### Proposition 1\(Clipped IPW/AIPW moment stability\)\.

For partition\-cell features, bounded outcomes, lower\-bounded cell masses, and propensities clipped to\[η,1−η\]\[\\eta,1\-\\eta\], the direct moment\-map IPW and AIPW estimators areLIPWL\_\{\\mathrm\{IPW\}\}\- andLAIPWL\_\{\\mathrm\{AIPW\}\}\-stable\. The constants scale polynomially in1/η1/\\etaand the inverse minimum cell mass; Appendix[E\.3](https://arxiv.org/html/2607.08122#A5.SS3)gives explicit bounds\. This is standard clipped\-score stability\[rosenbaum1983propensity,robins1994aipw,chernozhukov2018dml\]; the paper\-specific step is plugging the resultingLestL\_\{\\mathrm\{est\}\}into the DP moment\-release decomposition\.

For either example, settingq=q​\(D\)q=q\(D\)andq′=q~q^\{\\prime\}=\\widetilde\{q\}converts the deterministic stability bound into a privacy\-noise bound\. For ridge, the following corollary also records the conditioning requirement for the noisy Gram blocks\.

###### Corollary 1\(Noise\-calibrated ridge\)\.

Suppose the workload coordinates enteringτ^λ​\(q\)\\widehat\{\\tau\}\_\{\\lambda\}\(q\)have independent Gaussian noise with coordinate scaleσ\\sigma, and setδm=c​σ​log⁡\(m/γ\)\\delta\_\{m\}=c\\sigma\\sqrt\{\\log\(m/\\gamma\)\}\. On the event‖q~−q​\(D\)‖∞≤δm\\left\\lVert\\widetilde\{q\}\-q\(D\)\\right\\rVert\_\{\\infty\}\\leq\\delta\_\{m\}andmaxt⁡‖G~t−Gt‖op≤λ/2\\max\_\{t\}\\left\\lVert\\widetilde\{G\}\_\{t\}\-G\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq\\lambda/2, the noisy ridge matrices remain well conditioned and

\|τ^λ​\(q~\)−τ^λ​\(q​\(D\)\)\|≲Cϕ​\(1\+1κ\+λ\)​δm\.\|\\widehat\{\\tau\}\_\{\\lambda\}\(\\widetilde\{q\}\)\-\\widehat\{\\tau\}\_\{\\lambda\}\(q\(D\)\)\|\\lesssim C\_\{\\phi\}\\left\(1\+\\frac\{1\}\{\\kappa\+\\lambda\}\\right\)\\delta\_\{m\}\.For full joint\-cell bases with diagonal Gram matrices, the operator\-norm condition follows fromλ≳δm\\lambda\\gtrsim\\delta\_\{m\}; for explicitly released dense Gram blocks, standard Gaussian\-matrix bounds give the analogous sufficient choiceλ≳σ​p\+log⁡\(1/γ\)\\lambda\\gtrsim\\sigma\\sqrt\{p\+\\log\(1/\\gamma\)\}\.

Here and below, SNR means the coordinatewise signal\-to\-noise ratio\|q~a\|/σa\|\\widetilde\{q\}\_\{a\}\|/\\sigma\_\{a\}\. Appendix[E\.2](https://arxiv.org/html/2607.08122#A5.SS2.SSS0.Px1)gives the conditioning intuition and explains how ridge complements the SNR calibration filter\.

Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1)and Proposition[1](https://arxiv.org/html/2607.08122#Thmproposition1)are two examples of the same moment\-stability template: ridge covers direct regression plug\-ins that may require Gram moments, while IPW/AIPW covers cell\-based causal scores computable from the base treatment and outcome moments\.

### 5\.2Accuracy of the DP Moment Release

The previous subsection is deterministic once a moment\-error bound is available\. The next result supplies that bound for the Gaussian mechanism: it converts the sensitivity of the released workload into the coordinate error used by the stability examples and the decomposition below\.

###### Theorem 2\(DP moment accuracy\)\.

Letq​\(D\)=n−1​∑ih​\(Wi\)∈ℝmq\(D\)=n^\{\-1\}\\sum\_\{i\}h\(W\_\{i\}\)\\in\\mathbb\{R\}^\{m\}be the released workload answer, and suppose‖h​\(W\)‖2≤Hq\\left\\lVert h\(W\)\\right\\rVert\_\{2\}\\leq H\_\{q\}for every record\. Measuringq​\(D\)q\(D\)with the Gaussian mechanism under\(ε,δ\)\(\\varepsilon,\\delta\)\-DP yields, with probability at least1−γ1\-\\gamma,

‖q~−q​\(D\)‖∞=O​\(Hq​log⁡\(m/γ\)​log⁡\(1/δ\)n​ε\)\.\\left\\lVert\\widetilde\{q\}\-q\(D\)\\right\\rVert\_\{\\infty\}=O\\\!\\left\(\\frac\{H\_\{q\}\\sqrt\{\\log\(m/\\gamma\)\\log\(1/\\delta\)\}\}\{n\\varepsilon\}\\right\)\.For the base causal workload,Hq=ϕmax​1\+B2H\_\{q\}=\\phi\_\{\\max\}\\sqrt\{1\+B^\{2\}\}\. If optional Gram blocks are appended for a direct regression plug\-in, one may instead takeHq=\{ϕmax2​\(1\+B2\)\+ϕmax4\}1/2H\_\{q\}=\\\{\\phi\_\{\\max\}^\{2\}\(1\+B^\{2\}\)\+\\phi\_\{\\max\}^\{4\}\\\}^\{1/2\}\.

*Remark on norms\.*Theorems[1](https://arxiv.org/html/2607.08122#Thmtheorem1)and[2](https://arxiv.org/html/2607.08122#Thmtheorem2)both use‖ϕ​\(X\)‖2≤ϕmax\\left\\lVert\\phi\(X\)\\right\\rVert\_\{2\}\\leq\\phi\_\{\\max\}; indicator features additionally satisfy‖ϕ​\(X\)‖∞≤1\\left\\lVert\\phi\(X\)\\right\\rVert\_\{\\infty\}\\leq 1, and forpp\-dimensional indicator features both hold simultaneously withϕmax≤p\\phi\_\{\\max\}\\leq\\sqrt\{p\}\.

### 5\.3Overall ATE Error Decomposition

This subsection does not introduce a third route; it composes the direct\-route moment\-stability bound with the two extra errors introduced when the released workload is converted into synthetic microdata\. At the population\-reconstruction level, the controlled estimand isτ^est​\{q​\(Psyn\)\}\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\\\{q\(P^\{\\mathrm\{syn\}\}\)\\\}\. When this quantity is estimated fromnsynn\_\{\\mathrm\{syn\}\}sampled synthetic records, write the result asτ^syn\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}and treat finite synthetic sample size as an additional sampling term\. The experiments instantiate this release–reconstruct–analyze path with DR/AIPW on synthetic microdata; the theorem formalizes the workload\-matching error budget that this pipeline is designed to control\.

Because the iterative max\-entropy solver satisfies the noisy constraints only approximately, the bound carries an explicit calibration term\. For a measured or retained coordinate setSS\(withS=\[m\]S=\[m\]for the fixed full workload when no thresholding is used\), letΠS\\Pi\_\{S\}denote coordinate projection:ΠSv=\(va:a∈S\)\\Pi\_\{S\}v=\(v\_\{a\}:a\\in S\)\. In implementation,ΠS\\Pi\_\{S\}is the Boolean mask over retained moment coordinates\. Define

CalGap​\(S,σ\):=Lest​‖ΠS​\{q~−q​\(Psyn\)\}‖2,\\mathrm\{CalGap\}\(S,\\sigma\):=L\_\{\\mathrm\{est\}\}\\,\\bigl\\lVert\\Pi\_\{S\}\\\{\\widetilde\{q\}\-q\(P^\{\\mathrm\{syn\}\}\)\\\}\\bigr\\rVert\_\{2\},whereLestL\_\{\\mathrm\{est\}\}is the appropriate estimator Lipschitz constant \(LϕL\_\{\\phi\}for the ridge plug\-in of Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1), or the IPW/AIPW constants in Proposition[1](https://arxiv.org/html/2607.08122#Thmproposition1)and Appendix[E\.3](https://arxiv.org/html/2607.08122#A5.SS3)\) andq​\(Psyn\)q\(P^\{\\mathrm\{syn\}\}\)the workload moments of the solver output\.

###### Theorem 3\(ATE error decomposition\)\.

Under Assumption[1](https://arxiv.org/html/2607.08122#Thmassumption1), the boundedness conditions above, and an estimator Lipschitz condition\|τ^est​\(q\)−τ^est​\(q′\)\|≤Lest​‖q−q′‖∞\|\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q\)\-\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\(q^\{\\prime\}\)\|\\leq L\_\{\\mathrm\{est\}\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}for the retained workload coordinates, letτ^syn\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}be the empirical estimate ofτ^est​\{q​\(Psyn\)\}\\widehat\{\\tau\}\_\{\\mathrm\{est\}\}\\\{q\(P^\{\\mathrm\{syn\}\}\)\\\}formed fromnsynn\_\{\\mathrm\{syn\}\}i\.i\.d\. draws from the solver outputPsynP^\{\\mathrm\{syn\}\}\. This condition holds for the ridge plug\-in by Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1)and for partition\-feature clipped IPW/AIPW by Proposition[1](https://arxiv.org/html/2607.08122#Thmproposition1)\. Then

\|τ^syn−τ\|\\displaystyle\|\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}\-\\tau\|≤Op​\(Lest​n−1/2\)⏟sampling\+Op​\(Lest​Hq​log⁡m​log⁡\(1δ\)n​ε\)⏟privacy\\displaystyle\\leq\\underbrace\{O\_\{p\}\\\!\\bigl\(L\_\{\\mathrm\{est\}\}n^\{\-1/2\}\\bigr\)\}\_\{\\text\{sampling\}\}\+\\underbrace\{O\_\{p\}\\\!\\\!\\left\(\\frac\{L\_\{\\mathrm\{est\}\}H\_\{q\}\\sqrt\{\\log m\\,\\log\(\\frac\{1\}\{\\delta\}\)\}\}\{n\\varepsilon\}\\right\)\}\_\{\\text\{privacy\}\}\+Approx​\(ϕ;S\)⏟workload\+Op​\(nsyn−1/2\)⏟Monte Carlo\+CalGap​\(S,σ\)⏟calibration,\\displaystyle\+\\underbrace\{\\mathrm\{Approx\}\(\\phi;S\)\}\_\{\\text\{workload\}\}\\;\+\\;\\underbrace\{O\_\{p\}\\\!\\bigl\(n\_\{\\mathrm\{syn\}\}^\{\-1/2\}\\bigr\)\}\_\{\\text\{Monte Carlo\}\}\+\\underbrace\{\\mathrm\{CalGap\}\(S,\\sigma\)\}\_\{\\text\{calibration\}\},whereApprox​\(ϕ;S\)\\mathrm\{Approx\}\(\\phi;S\)is the approximation error of representingmt​\(x\)m\_\{t\}\(x\)ande​\(x\)e\(x\)using the retained workload coordinates\. WhenPsynP^\{\\mathrm\{syn\}\}satisfies the constraints exactly, the fifth term vanishes and the bound reduces to the four\-term decomposition\.

###### Corollary 2\(Design rule for causal utility\)\.

To keep DP distortion below sampling noise, it suffices \(up to logs\) that

Hq​log⁡mn​ε≪1n⟺ε≫Hq​log⁡mn\.\\displaystyle\\frac\{H\_\{q\}\\sqrt\{\\log m\}\}\{n\\varepsilon\}\\ll\\frac\{1\}\{\\sqrt\{n\}\}\\Longleftrightarrow\\varepsilon\\gg H\_\{q\}\\sqrt\{\\frac\{\\log m\}\{n\}\}\.This recovers a natural “ε\\varepsilonvs\.nn” threshold analogous to parametric DP inference\[smith2011privacy\], with the outcome boundBBcontrolling the difficulty\.

## 6Uncertainty Quantification from DP Synthetic Data

Recent work showsthat synthetic\-data inference must account for both sampling variability and DP noise\[raisa2023noiseaware,perez2024mwutest\]\. We propose a causal version of NA\+MI:

1. 1\.Treat the measured DP momentsq~S\\widetilde\{q\}\_\{S\}as noisy observations of latent true momentsqSq\_\{S\}, withS=\[m\]S=\[m\]for the fixed full workload and adaptiveSSforCausal\-AIM\.
2. 2\.SampleMMposterior drawsqS\(ℓ\)∼𝒩​\(q~S,ΣS\)q\_\{S\}^\{\(\\ell\)\}\\sim\\mathcal\{N\}\(\\widetilde\{q\}\_\{S\},\\Sigma\_\{S\}\),ℓ=1,…,M\\ell=1,\\dots,M: the asymptotic Gaussian posterior forqSq\_\{S\}givenq~S\\widetilde\{q\}\_\{S\}under a flat prior, with covariance given by the known Gaussian\-mechanism noise on the measured coordinates\.
3. 3\.For each draw, constructPsyn,\(ℓ\)P^\{\\mathrm\{syn\},\(\\ell\)\}via \([3](https://arxiv.org/html/2607.08122#S4.E3)\) on the measured coordinates and sample a synthetic datasetDsyn,\(ℓ\)D^\{\\mathrm\{syn\},\(\\ell\)\}\.
4. 4\.Computeτ^\(ℓ\)\\widehat\{\\tau\}^\{\(\\ell\)\}on each synthetic dataset and combine using Rubin’s MI rules\[rubin1987mi\]\.

Appendix[F](https://arxiv.org/html/2607.08122#A6)gives a bias\-aware variant thatinflatesthe MI variance by an estimated workload\-approximation term, with a coverage guarantee; Appendix[G](https://arxiv.org/html/2607.08122#A7)states the full procedure as pseudocode\. The result is a confidence interval that widens asε\\varepsilondecreases \(stronger privacy\), similar in spirit toraisa2023noiseaware\.

## 7Experiments

We evaluate three questions: whether causal workloads improve ATE accuracy over generic workloads, whether NA\+MI yields calibrated confidence intervals, and when adaptiveCausal\-AIMimproves over a fixed workload\.

### 7\.1Setup

Benchmarks\.We use five benchmark settings where the target ATE is known, so both point error and coverage can be evaluated directly\.

- •IHDP\[hill2011bayesian\]:n=747n=747children from the Infant Health and Development Program benchmark, with 25 child and family covariates\. Treatment indicates the program intervention, the observed outcome is a continuous cognitive score, and the standard semi\-simulated release supplies potential\-outcome means for the ground\-truth ATE\.
- •Twins\[louizos2017causal\]:n=11,400n=11\{,\}400same\-sex twin\-pair records with 30 birth and maternal covariates\. The benchmark constructs a confounded treatment indicator by sampling heavier\-versus\-lighter twin status from a covariate\-dependent propensity; the outcome is one\-year mortality, with both potential mortality outcomes observed within each pair\.
- •ACIC 2016\[dorie2019acic\]:n=4,802n=4\{,\}802units from the competition covariate matrix, with anonymized real covariates and three categorical fields one\-hot encoded into 82 numeric columns\. We use DGP 7, which simulates confounded treatment assignment and continuous potential outcomes with heterogeneous treatment effects; the ATE is computed from the exported potential outcomes\.
- •LaLonde/NSW\[lalonde1986evaluating\]: the National Supported Work experimental sample withn=445n=445subjects \(185 treated and 260 controls\)\. Covariates include age, education, race, marital status, and pre\-program earnings; the outcome is 1978 earnings, and the experimental difference\-in\-means is the reference effect\.
- •ACS: California 2018 American Community Survey person microdata fromfolktables\[ding2021retiring\]\. We use 20 demographic covariates and simulate confounded treatment and continuous potential outcomes atn∈\{1000,5000,20000\}n\\in\\\{1000,5000,20000\\\}, so the ATE is known by construction\.

Baselines\.All private methods release synthetic data and then estimate the ATE on synthetic records using doubly robust analysis\.

- •Non\-private DR: an oracle doubly robust estimator with cross\-fitting on the confidential data\.
- •MST \+ naive DR: generic MST\-style synthesis\[mckenna2021nist\], followed by standard DR inference that ignores DP noise\.
- •AIM \+ naive DR: generic AIM synthesis\[mckenna2022aim\], followed by the same naive DR analysis\.
- •Causal workload \+ naive DR: our fixed causal workload mechanism, analyzed without the NA\+MI uncertainty correction\.
- •Causal workload \+ NA\+MI: the fixed causal workload mechanism with noise\-aware multiple imputation\.
- •Causal\-AIM\+ NA\+MI: adaptive causal workload selection with the same NA\+MI analysis\.

Defaults and metrics\.Default runs useL=5L=5quantile bins,M=20M=20MI draws,nsyn=10​nn\_\{\\mathrm\{syn\}\}=10n, clipped standardized outcomes, DR analysis on synthetic data, and no SNR thresholding or calibration ridge in the main pipeline; we keepnsyn=10​nn\_\{\\mathrm\{syn\}\}=10nfor comparability across methods, although the ablation showsnsyn=nn\_\{\\mathrm\{syn\}\}=nalready suffices\. We report ATE RMSE and empirical 95% coverage overε∈\{0\.5,1,2,5\}\\varepsilon\\in\\\{0\.5,1,2,5\\\}withδ=1/n2\\delta=1/n^\{2\}, and run ablations over workload dimension, MI draws, synthetic sample size, overlap strength, and adaptive roundsKK\. Appendix[H](https://arxiv.org/html/2607.08122#A8)gives feature\-construction details, metric definitions, and ablation grids; additional figures appear in Appendix[J](https://arxiv.org/html/2607.08122#A10)\.

### 7\.2Results

Table[1](https://arxiv.org/html/2607.08122#S7.T1)summarizes all methods atε=1\\varepsilon\{=\}1, Figures[2](https://arxiv.org/html/2607.08122#S7.F2)and[3](https://arxiv.org/html/2607.08122#S7.F3)show RMSE and coverage across privacy budgets, and Figure[5](https://arxiv.org/html/2607.08122#S7.F5)shows the adaptive fixed\-workload comparison on ACIC\.

Table 1:ATE RMSE and 95% CI Coverage atε=1\.0\\varepsilon=1\.0,δ=1/n2\\delta=1/n^\{2\}\. Each cell is RMSE / Coverage; among private methods, bold marks the best value and underlining marks the second\-best value in each dataset column\.Generic workloads win on RMSE at loose budgets, but lose on coverage\.MST \+ naive DR achieves the lowest private RMSE on all four benchmarks atε≥2\\varepsilon\\geq 2and on three of four atε=1\\varepsilon=1\. At strict budgets, causal workload \+ NA\+MI becomes more competitive, matching or beating MST on two of four benchmarks atε=0\.5\\varepsilon=0\.5and on IHDP atε=1\\varepsilon=1\. However, MST’s point accuracy comes with invalid uncertainty quantification: atε≤1\\varepsilon\\leq 1, all naive methods have coverage at most35\.2%35\.2\\%\(Figure[3](https://arxiv.org/html/2607.08122#S7.F3)\)\.

NA\+MI restores calibrated inference\.Causal workload \+ NA\+MI is the only private method with near\-nominal coverage atε≤1\\varepsilon\\leq 1on all four benchmarks \(99\.8−100%99\.8\{\-\}100\\%; Table[1](https://arxiv.org/html/2607.08122#S7.T1)\)\. The price is wider intervals: NA\+MI intervals are3434–300×300\\timeswider than naive intervals at strict budgets \(Figure[4](https://arxiv.org/html/2607.08122#S7.F4)\) because they account for DP measurement noise\. This is the intended behavior when privacy noise dominates the sampling error; the narrow naive intervals are overconfident rather than genuinely informative\.

Adaptive workloads are operating\-point choices\.On ACIC atε≥1\\varepsilon\\geq 1,Causal\-AIMsubstantially reduces RMSE relative to the fixed causal workload, especially at smallKK\(Figure[5](https://arxiv.org/html/2607.08122#S7.F5)\)\. Atε=0\.5\\varepsilon=0\.5, additional adaptive rounds can hurt because each round consumes privacy budget that may exceed the gain from better feature targeting\. On IHDP, no testedKKrecovers the fixed workload’s calibration \(Appendix[K](https://arxiv.org/html/2607.08122#A11)\), so fixed causal workload \+ NA\+MI remains the conservative default when coverage is the priority\.

Robustness and reuse\.On the ACS semi\-synthetic study, NA\+MI again provides the coverage advantage: at\(n=1000,ε=0\.5\)\(n=1000,\\varepsilon=0\.5\), Causal \+ NA\+MI has coverage1\.001\.00while MST \+ naive DR has coverage0\.070\.07, and NA\+MI also wins RMSE in that cell \(0\.880\.88vs\.1\.171\.17\)\. Ablations show thatM=20M=20MI draws is conservative, synthetic sample sizes abovensyn=nn\_\{\\mathrm\{syn\}\}=nadd little, and SNR thresholding is mainly useful as a safeguard when moments approach the noise floor \(Figures[9](https://arxiv.org/html/2607.08122#A10.F9),[10](https://arxiv.org/html/2607.08122#A10.F10),[11](https://arxiv.org/html/2607.08122#A10.F11), and[12](https://arxiv.org/html/2607.08122#A10.F12)\)\. Finally, one DP synthetic release supports multiple estimands: ATE, ATT, and a subgroup effect all achieve nominal coverage in the reuse experiment \(Appendix[K](https://arxiv.org/html/2607.08122#A11)\)\.

Fidelity is not causal utility\.Generic workloads dominate marginal\-fidelity metrics such as average marginal total\-variation distance \(TVD\) in nearly every configuration \(Figure[13](https://arxiv.org/html/2607.08122#A10.F13)\), yet those metrics do not rank methods by ATE error or coverage\. This is the empirical counterpart of the workload argument: preserving generic low\-dimensional marginals is not enough when the downstream target depends on treatment\-arm outcome and balance moments\.

Practical takeaway\.Use Causal \+ NA\+MI when valid uncertainty quantification is the primary goal, especially at strict privacy budgets\. UseCausal\-AIMwhen point RMSE is prioritized and its operating point is favorable for the dataset\. Use generic workloads when the release is intended for broad marginal fidelity or exploratory analysis and calibrated causal intervals are not required\.

![Refer to caption](https://arxiv.org/html/2607.08122v1/x2.png)Figure 2:ATE RMSE versus privacy budget \(ε\\varepsilon\) on four benchmarks \(nrep=500n\_\{\\mathrm\{rep\}\}=500,δ=1/n2\\delta=1/n^\{2\}\)\. MST \+ naive DR leads on RMSE atε≥2\\varepsilon\\geq 2; atε=0\.5\\varepsilon=0\.5, Causal \+ NA\+MI matches or beats it on half the benchmarks, and on IHDP atε=1\\varepsilon=1\. The value of causal workload methods lies chiefly in enabling calibrated coverage \(Figure[3](https://arxiv.org/html/2607.08122#S7.F3)\)\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x3.png)Figure 3:Empirical 95% CI coverage across privacy budgets and four datasets \(nrep=500n\_\{\\mathrm\{rep\}\}=500\)\. The dashed line marks nominal 0\.95\. All naive methods are at or below35\.2%35\.2\\%atε≤1\\varepsilon\\leq 1\. Causal workload \+ NA\+MI achieves99\.8−100%99\.8\{\-\}100\\%coverage atε≤1\\varepsilon\\leq 1on all four benchmarks; Coverage can decline at highε\\varepsilondue to approximation bias \(Appendix[L](https://arxiv.org/html/2607.08122#A12)\)\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x4.png)Figure 4:Average 95% CI length across methods, datasets, and privacy budgets \(nrep=500n\_\{\\mathrm\{rep\}\}=500\)\. Noise\-aware MI intervals are3434–300×300\\timeswider than naive intervals atε≤1\\varepsilon\\leq 1\(depending on dataset and comparator\) because they account for both DP noise and sampling variability\. Naive intervals are narrow but far below nominal coverage \(Figure[3](https://arxiv.org/html/2607.08122#S7.F3)\), so their apparent precision is spurious\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x5.png)Figure 5:ATE RMSE ofCausal\-AIM\(adaptive\) versus a fixed causal workload on ACIC DGP 7 \(nrep=100n\_\{\\mathrm\{rep\}\}=100\)\. The left panel overlays all privacy budgets: horizontal lines mark the fixed workload and curves showCausal\-AIMas a function of adaptive roundsKK; the right panel shows the most frequently selected features\. Atε≥1\\varepsilon\\geq 1,Causal\-AIMreduces RMSE by78−96%78\{\-\}96\\%over the fixed workload atK≤3K\\leq 3\(53−91%53\{\-\}91\\%atK=5K\{=\}5\)\. Atε=0\.5\\varepsilon=0\.5, additional rounds degrade performance because each round consumes budget that exceeds the benefit of better feature targeting; few rounds \(K≤3K\\leq 3\) are recommended at lowε\\varepsilon\.

## 8Conclusion

This paper argues that DP synthetic data for causal inference should preserve the moments used by the causal estimand, not only generic distributional fidelity\. The proposed causal workload releases treatment\-arm feature masses and outcome\-feature moments, reconstructs a maximum\-entropy synthetic distribution, and supports noise\-aware inference through NA\+MI\. The theory tracks how DP moment error, workload approximation, calibration mismatch, and synthetic Monte Carlo error enter ATE estimation; the experiments show the corresponding tradeoff: generic workloads often lead on RMSE at loose budgets, but their naive intervals undercover severely, while causal workload \+ NA\+MI provides calibrated inference at strict privacy budgets\. The distinction between the directqq\-route and the synthetic\-data route clarifies what the theory controls and what the release enables: one measured workload can be used directly by stable moment maps or converted into reusable synthetic records for ATE, ATT, subgroup effects, and model checks under the same privacy cost\. The main limitations are the need to choose a feature mapϕ\\phi, approximation bias at highε\\varepsilon, wider intervals when DP noise dominates, and our focus on the AIM/Private\-PGM reconstruction family; Appendix[L](https://arxiv.org/html/2607.08122#A12)discusses these limitations, the coverage–privacy diagnostic, and future directions such as continuous feature maps, richer subgroup/CATE workloads, and end\-to\-end DP\-aware feature selection\.

###### Acknowledgements\.

AA was partially supported by NIH grant R01MH139379 and by Patient\-Centered Outcomes Research Institute \(PCORI\) award ME\-2023C1\-32148\.

#### Reproducibility\.

All experiments, tables, and figures in this paper can be reproduced with the code, data\-processing scripts, and notebooks available at[https://github\.com/AsiaeeLab/causal\-aim](https://github.com/AsiaeeLab/causal-aim)\.

## References

Workload\-Preserving Differentially Private Synthetic Data for Causal Inference via Maximum\-Entropy Calibration \(Supplementary Material\)

## Appendix AFeature Encodings and Gram Moments

This appendix clarifies the feature\-encoding point used in Section[5](https://arxiv.org/html/2607.08122#S5)\. The experiments use*concatenated one\-hot*features: each covariate is discretized separately, one\-hot encoded separately, and then the groups are stacked\. For example, with age in\{young,old\}\\\{\\mathrm\{young\},\\mathrm\{old\}\\\}and sex in\{female,male\}\\\{\\mathrm\{female\},\\mathrm\{male\}\\\}, the experimental\-style feature vector is

ϕcat​\(X\)=\(1,1​\{young\},1​\{old\},1​\{female\},1​\{male\}\)\.\\phi\_\{\\mathrm\{cat\}\}\(X\)=\\bigl\(1,\\ 1\\\{\\mathrm\{young\}\\\},\\ 1\\\{\\mathrm\{old\}\\\},\\ 1\\\{\\mathrm\{female\}\\\},\\ 1\\\{\\mathrm\{male\}\\\}\\bigr\)\.The base workload then releases, for each arm, noisy versions of the age\-bin masses, sex\-bin masses, and the corresponding outcome\-weighted moments\.

A*full joint\-cell*encoding instead uses indicators for the Cartesian\-product cells, e\.g\.

ϕjoint​\(X\)=\(1​\{young,female\},1​\{young,male\},1​\{old,female\},1​\{old,male\}\)\.\\phi\_\{\\mathrm\{joint\}\}\(X\)=\\bigl\(1\\\{\\mathrm\{young,female\}\\\},\\ 1\\\{\\mathrm\{young,male\}\\\},\\ 1\\\{\\mathrm\{old,female\}\\\},\\ 1\\\{\\mathrm\{old,male\}\\\}\\bigr\)\.Exactly one coordinate ofϕjoint​\(X\)\\phi\_\{\\mathrm\{joint\}\}\(X\)is active for each record\. Therefore

ϕjoint​\(X\)​ϕjoint​\(X\)⊤​is diagonal, and​Gt​\(D\)=diag⁡\{qt\(0\)​\(D\)\}\.\\phi\_\{\\mathrm\{joint\}\}\(X\)\\phi\_\{\\mathrm\{joint\}\}\(X\)^\{\\top\}\\text\{ is diagonal, and \}G\_\{t\}\(D\)=\\operatorname\{diag\}\\\{q\_\{t\}^\{\(0\)\}\(D\)\\\}\.For concatenated one\-hot features, off\-diagonal entries are real co\-occurrence moments; for instance,

1n​∑i1​\{Ti=t\}​1​\{oldi\}​1​\{malei\}\\frac\{1\}\{n\}\\sum\_\{i\}1\\\{T\_\{i\}=t\\\}1\\\{\\mathrm\{old\}\_\{i\}\\\}1\\\{\\mathrm\{male\}\_\{i\}\\\}is not determined by the separate old and male counts\. Thus a direct ridge plug\-in under concatenated or overlapping features would need to appendqt\(2\)=vec⁡\(Gt\)q\_\{t\}^\{\(2\)\}=\\operatorname\{vec\}\(G\_\{t\}\)as optional queries\. Our experiments do not use this augmented Gram release; they release the base4​p4pworkload, reconstruct synthetic data, and then run DR/AIPW on the synthetic records\.

## Appendix BFirst\-Order Sufficiency of the Orthogonal\-Score Workload

This sectionclarifies whythe causal workload of Section[4](https://arxiv.org/html/2607.08122#S4)is*one principled choice*rather than a claimed optimum\. Letζ=\(m0,m1,e\)\\zeta=\(m\_\{0\},m\_\{1\},e\)denote the nuisances \(η\\etaremains the positivity constant\),ψ​\(W;θ,ζ\)\\psi\(W;\\theta,\\zeta\)a Neyman\-orthogonal score for a smooth targetθ\\theta, andℱϕ\\mathcal\{F\}\_\{\\phi\}the linear nuisance class spanned byϕ\\phi\.

###### Proposition 2\(First\-order sufficiency\)\.

Restrict the nuisance functions to the feature class spanned byϕ\\phi\. For smooth causal targets identified by a Neyman\-orthogonal score, the workload𝒬ϕ=\{q0\(0\),q1\(0\),q0\(1\),q1\(1\)\}\\mathcal\{Q\}\_\{\\phi\}=\\\{q\_\{0\}^\{\(0\)\},q\_\{1\}^\{\(0\)\},q\_\{0\}^\{\(1\)\},q\_\{1\}^\{\(1\)\}\\\}identifies the arm\-specific feature masses, outcome\-feature moments, and treatment\-balance moments used by projected outcome, propensity, IPW, and AIPW moment maps\. Direct ridge plug\-ins may additionally require Gram moments, as discussed in Appendix[A](https://arxiv.org/html/2607.08122#A1); these are optional extensions rather than part of the default experimental release\.

###### Proof\.

The outcome\-feature momentsqt\(1\)=𝔼​\[1​\{T=t\}​Y​ϕ​\(X\)\]q\_\{t\}^\{\(1\)\}=\\mathbb\{E\}\[1\\\{T=t\\\}Y\\phi\(X\)\]encode arm\-specific outcome structure in the chosen feature class, whileq1\(0\)=𝔼​\[T​ϕ​\(X\)\]q\_\{1\}^\{\(0\)\}=\\mathbb\{E\}\[T\\phi\(X\)\]andq0\(0\)=𝔼​\[\(1−T\)​ϕ​\(X\)\]q\_\{0\}^\{\(0\)\}=\\mathbb\{E\}\[\(1\-T\)\\phi\(X\)\]encode treatment\-feature balance\. Once projected nuisances or direct cell scores are identified from these moments, orthogonality makes remaining nuisance errors second order, so a synthetic distribution matching the causal workload supports inference for the projected target subject to the explicit privacy, approximation, Monte Carlo, and calibration terms\. ∎

#### Necessity intuition and scope\.

If two local alternatives inℱϕ\\mathcal\{F\}\_\{\\phi\}agree on a release but differ in an arm\-specific outcome\-feature or treatment\-feature moment, some orthogonal\-score target has different first\-order pathwise derivatives under the two alternatives, so a release that cannot distinguish those directions cannot support uniformly regular root\-nninference over the projected class\. This is a first\-order identification argument only: it is*not*a minimax lower bound, does*not*prove uniqueness of the workload, and does*not*show that the base query count is optimal among all DP mechanisms; those questions are outside the present scope \(Appendix[L](https://arxiv.org/html/2607.08122#A12)\)\.

## Appendix CMaximum\-Entropy Reconstruction Details

This section provides details of the reconstruction step used in Algorithm[2](https://arxiv.org/html/2607.08122#alg2)\. Assume first thatW=\(X,T,Y\)W=\(X,T,Y\)has finite support𝒲\\mathcal\{W\}, as in a discretized synthetic\-data model\. Letha:𝒲→ℝh\_\{a\}:\\mathcal\{W\}\\to\\mathbb\{R\},a=1,…,ma=1,\\ldots,m, be the ordered workload query functions, withm=4​pm=4pfor the base causal workload and largermmif optional moments are appended\. For a measured coordinate setS⊂\[m\]S\\subset\[m\], writehS\(w\)=\(ha\(w\):a∈S\)⊤h\_\{S\}\(w\)=\(h\_\{a\}\(w\):a\\in S\)^\{\\top\}andq~S=\(q~a:a∈S\)⊤\\widetilde\{q\}\_\{S\}=\(\\widetilde\{q\}\_\{a\}:a\\in S\)^\{\\top\}\. Letp0​\(w\)\>0p\_\{0\}\(w\)\>0be a base distribution\. The role ofp0p\_\{0\}is to say what distribution we prefer among all distributions that match the released moments: it may be uniform over the discretized support, an independent product distribution, a product distribution fit to already measured one\-way marginals, or a public prior from an external population\.

If the noisy constraints are feasible exactly, reconstruction is the information projection

minp∈Δ​\(𝒲\)​∑w∈𝒲p​\(w\)​log⁡p​\(w\)p0​\(w\)s\.t\.∑w∈𝒲p​\(w\)​ha​\(w\)=q~a,a∈S\.\\min\_\{p\\in\\Delta\(\\mathcal\{W\}\)\}\\sum\_\{w\\in\\mathcal\{W\}\}p\(w\)\\log\\frac\{p\(w\)\}\{p\_\{0\}\(w\)\}\\quad\\text\{s\.t\.\}\\quad\\sum\_\{w\\in\\mathcal\{W\}\}p\(w\)h\_\{a\}\(w\)=\\widetilde\{q\}\_\{a\},\\ a\\in S\.The Lagrangian is

ℒ​\(p,ξ,α\)=∑wp​\(w\)​log⁡p​\(w\)p0​\(w\)−ξ⊤​\(∑wp​\(w\)​hS​\(w\)−q~S\)\+α​\(∑wp​\(w\)−1\)\.\\mathcal\{L\}\(p,\\xi,\\alpha\)=\\sum\_\{w\}p\(w\)\\log\\frac\{p\(w\)\}\{p\_\{0\}\(w\)\}\-\\xi^\{\\top\}\\left\(\\sum\_\{w\}p\(w\)h\_\{S\}\(w\)\-\\widetilde\{q\}\_\{S\}\\right\)\+\\alpha\\left\(\\sum\_\{w\}p\(w\)\-1\\right\)\.Differentiating with respect top​\(w\)p\(w\)and setting the derivative to zero gives

log⁡p​\(w\)p0​\(w\)\+1−ξ⊤​hS​\(w\)\+α=0,\\log\\frac\{p\(w\)\}\{p\_\{0\}\(w\)\}\+1\-\\xi^\{\\top\}h\_\{S\}\(w\)\+\\alpha=0,and therefore

pξ​\(w\)=p0​\(w\)​exp⁡\{ξ⊤​hS​\(w\)\}Z​\(ξ\),Z​\(ξ\)=∑u∈𝒲p0​\(u\)​exp⁡\{ξ⊤​hS​\(u\)\}\.p\_\{\\xi\}\(w\)=\\frac\{p\_\{0\}\(w\)\\exp\\\{\\xi^\{\\top\}h\_\{S\}\(w\)\\\}\}\{Z\(\\xi\)\},\\qquad Z\(\\xi\)=\\sum\_\{u\\in\\mathcal\{W\}\}p\_\{0\}\(u\)\\exp\\\{\\xi^\{\\top\}h\_\{S\}\(u\)\\\}\.The dual objective can be written as

maxξ∈ℝ\|S\|⁡\{ξ⊤​q~S−log⁡Z​\(ξ\)\},\\max\_\{\\xi\\in\\mathbb\{R\}^\{\|S\|\}\}\\left\\\{\\xi^\{\\top\}\\widetilde\{q\}\_\{S\}\-\\log Z\(\\xi\)\\right\\\},whose gradient isq~S−𝔼pξ​\[hS​\(W\)\]\\widetilde\{q\}\_\{S\}\-\\mathbb\{E\}\_\{p\_\{\\xi\}\}\[h\_\{S\}\(W\)\]and whose Hessian is minus the covariance ofhS​\(W\)h\_\{S\}\(W\)underpξp\_\{\\xi\}\. Thus optimizing the dual searches for multipliersξ\\xiwhose exponential\-family distribution matches the released noisy moments\.

Becauseq~S\\widetilde\{q\}\_\{S\}contains DP noise, exact feasibility is not guaranteed\. For example, noise can make a released cell mass slightly negative, or make several marginal constraints mutually inconsistent\. Implementations therefore solve a relaxed weighted calibration problem such as

minP∈𝒫⁡12​‖ΣS−1/2​ΠS​\{q​\(P\)−q~\}‖22\+ρ​KL​\(P∥P0\),\\min\_\{P\\in\\mathcal\{P\}\}\\frac\{1\}\{2\}\\left\\lVert\\Sigma\_\{S\}^\{\-1/2\}\\Pi\_\{S\}\\\{q\(P\)\-\\widetilde\{q\}\\\}\\right\\rVert\_\{2\}^\{2\}\+\\rho\\,\\mathrm\{KL\}\(P\\\|P\_\{0\}\),whereΣS\\Sigma\_\{S\}is the known DP noise covariance on the measured coordinates, so high\-noise coordinates are automatically downweighted\. Equivalently, one can solve the exact dual only on retained high\-signal coordinates and stop when residuals are at the noise scale, e\.g\.\|qa​\(P\)−q~a\|≤ccal​σa\|q\_\{a\}\(P\)\-\\widetilde\{q\}\_\{a\}\|\\leq c\_\{\\mathrm\{cal\}\}\\sigma\_\{a\}\. This is the sense in which Algorithm[2](https://arxiv.org/html/2607.08122#alg2)uses Private\-PGM\-style reconstruction: it fits a structured exponential\-family / graphical\-model distribution to noisy measured moments rather than trying to estimate the full joint table without structure, then samples synthetic records from the fitted distribution\.

Algorithm 2Fixed Causal\-Workload Synthesis Procedure1:Confidential data

DD, query functions

\{ha\}a=1m\\\{h\_\{a\}\\\}\_\{a=1\}^\{m\}, privacy budget

\(ε,δ\)\(\\varepsilon,\\delta\), base distribution

P0P\_\{0\}, synthetic size

nsynn\_\{\\mathrm\{syn\}\}
2:Compute exact empirical answers

qa​\(D\)=n−1​∑iha​\(Wi\)q\_\{a\}\(D\)=n^\{\-1\}\\sum\_\{i\}h\_\{a\}\(W\_\{i\}\)for

a=1,…,ma=1,\\ldots,m
3:Compute the sensitivity bound

Δ2\\Delta\_\{2\}and Gaussian scale

σ=Δ2​2​log⁡\(1\.25/δ\)/ε\\sigma=\\Delta\_\{2\}\\sqrt\{2\\log\(1\.25/\\delta\)\}/\\varepsilon
4:Release

q~=q​\(D\)\+Z\\widetilde\{q\}=q\(D\)\+Z, where

Z∼𝒩​\(0,σ2​Im\)Z\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\_\{m\}\)
5:Optionally retain high\-signal coordinates

Sτ=\{a:\|q~a\|/σa≥τSNR\}S\_\{\\tau\}=\\\{a:\|\\widetilde\{q\}\_\{a\}\|/\\sigma\_\{a\}\\geq\\tau\_\{\\mathrm\{SNR\}\}\\\}; otherwise set

Sτ=\[m\]S\_\{\\tau\}=\[m\]
6:Fit

PsynP^\{\\mathrm\{syn\}\}by solving the relaxed maximum\-entropy calibration problem on

ΠSτ​q~\\Pi\_\{S\_\{\\tau\}\}\\widetilde\{q\}
7:Draw

nsynn\_\{\\mathrm\{syn\}\}records

DsynD^\{\\mathrm\{syn\}\}from

PsynP^\{\\mathrm\{syn\}\}
8:Synthetic dataset

DsynD^\{\\mathrm\{syn\}\}and recorded noise scale

σ\\sigma

## Appendix DCausal\-AIMSelection Details

At each roundkk,Causal\-AIMcomputes an internal utility for each unmeasured candidate grouprr, estimating the reduction in ATE error if its coordinate blockI​\(r\)⊂\[m\]I\(r\)\\subset\[m\]were added to the measured workload\. This is an adaptive greedy selection rule with private randomized selection: candidates are scored against the current reconstructed model, one group is selected through the exponential mechanism, its coordinates are measured once, and the model is refit before the next round\. Previously measured groups are removed from the candidate set and are not re\-measured; later rounds reuse their stored noisy answers\.

The score is motivated by the orthogonal\-score residual

Δr=‖𝔼Pk−1​\[ψ​\(W;θ^k−1,ζ^k−1\)⋅ϕr​\(X\)\]‖22,\\Delta\_\{r\}=\\left\\lVert\\mathbb\{E\}\_\{P\_\{k\-1\}\}\[\\psi\(W;\\hat\{\\theta\}\_\{k\-1\},\\hat\{\\zeta\}\_\{k\-1\}\)\\cdot\\phi\_\{r\}\(X\)\]\\right\\rVert\_\{2\}^\{2\},whereψ\\psiis the influence function for the ATE and\(θ^k−1,ζ^k−1\)\(\\hat\{\\theta\}\_\{k\-1\},\\hat\{\\zeta\}\_\{k\-1\}\)are the current nuisance estimates underPk−1P\_\{k\-1\}\. HereW=\(X,T,Y\)W=\(X,T,Y\); for the ATE, with nuisancesζ=\(m0,m1,e\)\\zeta=\(m\_\{0\},m\_\{1\},e\)and targetτ\\tau, a standard orthogonal score is

ψ​\(W;τ,ζ\)=m1​\(X\)−m0​\(X\)\+T​\{Y−m1​\(X\)\}e​\(X\)−\(1−T\)​\{Y−m0​\(X\)\}1−e​\(X\)−τ\.\\psi\(W;\\tau,\\zeta\)=m\_\{1\}\(X\)\-m\_\{0\}\(X\)\+\\frac\{T\\\{Y\-m\_\{1\}\(X\)\\\}\}\{e\(X\)\}\-\\frac\{\(1\-T\)\\\{Y\-m\_\{0\}\(X\)\\\}\}\{1\-e\(X\)\}\-\\tau\.In the implementation, this is operationalized by a moment\-space proxy\. Letq\(k\)q^\{\(k\)\}be the current noisy moment vector assembled from the coordinates measured so far and from the public base distribution on unmeasured coordinates\. A typical utility is

uk​\(r\)=L^k​‖ΠI​\(r\)​\{q​\(D\)−q\(k−1\)\}‖2,u\_\{k\}\(r\)=\\widehat\{L\}\_\{k\}\\,\\bigl\\lVert\\Pi\_\{I\(r\)\}\\\{q\(D\)\-q^\{\(k\-1\)\}\\\}\\bigr\\rVert\_\{2\},whereL^k\\widehat\{L\}\_\{k\}is an empirical Lipschitz/overlap factor andΠI​\(r\)\\Pi\_\{I\(r\)\}projects onto the coordinates for grouprr\. Large utility means that the current reconstruction leaves a causally weighted moment block poorly matched\. The utilities are used only inside the selection mechanism and are not released\. IfΔu\\Delta\_\{u\}bounds the sensitivity ofuku\_\{k\}, the exponential mechanism selects

Pr⁡\(rk=r\)∝exp⁡\{εscore,k​uk​\(r\)2​Δu\},r∈Ak−1\.\\Pr\(r\_\{k\}=r\)\\propto\\exp\\\!\\left\\\{\\frac\{\\varepsilon\_\{\\mathrm\{score\},k\}\\,u\_\{k\}\(r\)\}\{2\\Delta\_\{u\}\}\\right\\\},\\qquad r\\in A\_\{k\-1\}\.Thus larger utility means larger selection probability; if one instead defines a loss, the utility is the negative loss\.

Algorithm 3Causal\-AIMSelection and Measurement1:Data

DD, groups

ℛ\\mathcal\{R\}, coordinate blocks

\{I​\(r\)\}r∈ℛ\\\{I\(r\)\\\}\_\{r\\in\\mathcal\{R\}\}, budget

\(ε,δ\)\(\\varepsilon,\\delta\), rounds

KK, public base moment vector

q\(0\)=q​\(P0\)q^\{\(0\)\}=q\(P\_\{0\}\)
2:Split the budget into scoring and measurement portions, and split each portion across rounds by sequential composition

3:Initialize measured coordinates

S0←∅S\_\{0\}\\leftarrow\\emptyset, available groups

A0←ℛA\_\{0\}\\leftarrow\\mathcal\{R\}, and current moment proxy

q\(0\)q^\{\(0\)\}
4:for

k=1,…,min⁡\(K,\|ℛ\|\)k=1,\\ldots,\\min\(K,\|\\mathcal\{R\}\|\)do

5:For each

r∈Ak−1r\\in A\_\{k\-1\}, compute an internal utility

uk​\(r\)u\_\{k\}\(r\)for adding coordinate block

I​\(r\)I\(r\)
6:Select

rkr\_\{k\}from

Ak−1A\_\{k\-1\}with the exponential mechanism, with probability proportional to

exp⁡\{εscore,k​uk​\(r\)/\(2​Δu\)\}\\exp\\\{\\varepsilon\_\{\\mathrm\{score\},k\}u\_\{k\}\(r\)/\(2\\Delta\_\{u\}\)\\\}
7:Release noisy answers

q~I​\(rk\)=qI​\(rk\)​\(D\)\+Zk\\widetilde\{q\}\_\{I\(r\_\{k\}\)\}=q\_\{I\(r\_\{k\}\)\}\(D\)\+Z\_\{k\}with the Gaussian mechanism

8:Update

Sk←Sk−1∪I​\(rk\)S\_\{k\}\\leftarrow S\_\{k\-1\}\\cup I\(r\_\{k\}\)and

Ak←Ak−1∖\{rk\}A\_\{k\}\\leftarrow A\_\{k\-1\}\\setminus\\\{r\_\{k\}\\\}
9:Update

q\(k\)q^\{\(k\)\}by replacing coordinates

I​\(rk\)I\(r\_\{k\}\)with

q~I​\(rk\)\\widetilde\{q\}\_\{I\(r\_\{k\}\)\}and retaining all earlier noisy measurements

10:Refit

PkP\_\{k\}by maximum\-entropy calibration to the stored noisy answers on

SkS\_\{k\}
11:endfor

12:Output stored noisy moments

q~SK\\widetilde\{q\}\_\{S\_\{K\}\}; for synthetic release, output

PKP\_\{K\}and sample synthetic records from

PKP\_\{K\}

## Appendix EProofs

### E\.1Weighted ATEs: ATT and Subgroup Effects

The ATE decomposition extends to weighted effects whenever the weighting rule is represented by the retained workload coordinates\. For a nonnegative weight functionω\\omega, define

τω:=𝔼​\[ω​\(X\)​\{m1​\(X\)−m0​\(X\)\}\]𝔼​\[ω​\(X\)\]\.\\tau\_\{\\omega\}:=\\frac\{\\mathbb\{E\}\[\\omega\(X\)\\\{m\_\{1\}\(X\)\-m\_\{0\}\(X\)\\\}\]\}\{\\mathbb\{E\}\[\\omega\(X\)\]\}\.This includes subgroup effects withω​\(X\)=𝟙​\{X∈G\}\\omega\(X\)=\\mathbbm\{1\}\\\{X\\in G\\\}, and ATT\-type effects withω​\(X\)=e​\(X\)\\omega\(X\)=e\(X\), provided the subgroup or propensity weights are represented by the workload\.

###### Proposition 3\(Weighted\-effect moment stability\)\.

Letτ^ω​\(q\)=Nω​\(q\)/Dω​\(q\)\\widehat\{\\tau\}\_\{\\omega\}\(q\)=N\_\{\\omega\}\(q\)/D\_\{\\omega\}\(q\)be a direct moment\-map estimator using only retained workload coordinates\. Assume that, on the admissible workload domain,Dω​\(q\)≥ρω\>0D\_\{\\omega\}\(q\)\\geq\\rho\_\{\\omega\}\>0,\|Nω​\(q\)\|≤Mω​Dω​\(q\)\|N\_\{\\omega\}\(q\)\|\\leq M\_\{\\omega\}D\_\{\\omega\}\(q\), and

\|Nω​\(q\)−Nω​\(q′\)\|≤LN,ω​‖q−q′‖∞,\|Dω​\(q\)−Dω​\(q′\)\|≤LD,ω​‖q−q′‖∞\.\|N\_\{\\omega\}\(q\)\-N\_\{\\omega\}\(q^\{\\prime\}\)\|\\leq L\_\{N,\\omega\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\},\\qquad\|D\_\{\\omega\}\(q\)\-D\_\{\\omega\}\(q^\{\\prime\}\)\|\\leq L\_\{D,\\omega\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}\.Thenτ^ω\\widehat\{\\tau\}\_\{\\omega\}is Lipschitz in the workload moments:

\|τ^ω​\(q\)−τ^ω​\(q′\)\|≤Lω​‖q−q′‖∞,Lω=LN,ω\+Mω​LD,ωρω\.\|\\widehat\{\\tau\}\_\{\\omega\}\(q\)\-\\widehat\{\\tau\}\_\{\\omega\}\(q^\{\\prime\}\)\|\\leq L\_\{\\omega\}\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\},\\qquad L\_\{\\omega\}=\\frac\{L\_\{N,\\omega\}\+M\_\{\\omega\}L\_\{D,\\omega\}\}\{\\rho\_\{\\omega\}\}\.Consequently, the proof of Theorem[3](https://arxiv.org/html/2607.08122#Thmtheorem3)applies toτω\\tau\_\{\\omega\}after replacingLestL\_\{\\mathrm\{est\}\}byLωL\_\{\\omega\}and replacingApprox​\(ϕ;S\)\\mathrm\{Approx\}\(\\phi;S\)by the weighted approximation errorApproxω​\(ϕ;S\)\\mathrm\{Approx\}\_\{\\omega\}\(\\phi;S\)\.

###### Proof\.

Letδq=‖q−q′‖∞\\delta\_\{q\}=\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}\. By adding and subtractingNω​\(q′\)/Dω​\(q\)N\_\{\\omega\}\(q^\{\\prime\}\)/D\_\{\\omega\}\(q\),

\|Nω​\(q\)Dω​\(q\)−Nω​\(q′\)Dω​\(q′\)\|≤\|Nω​\(q\)−Nω​\(q′\)\|Dω​\(q\)\+\|Nω​\(q′\)\|​\|1Dω​\(q\)−1Dω​\(q′\)\|\.\\left\|\\frac\{N\_\{\\omega\}\(q\)\}\{D\_\{\\omega\}\(q\)\}\-\\frac\{N\_\{\\omega\}\(q^\{\\prime\}\)\}\{D\_\{\\omega\}\(q^\{\\prime\}\)\}\\right\|\\leq\\frac\{\|N\_\{\\omega\}\(q\)\-N\_\{\\omega\}\(q^\{\\prime\}\)\|\}\{D\_\{\\omega\}\(q\)\}\+\|N\_\{\\omega\}\(q^\{\\prime\}\)\|\\left\|\\frac\{1\}\{D\_\{\\omega\}\(q\)\}\-\\frac\{1\}\{D\_\{\\omega\}\(q^\{\\prime\}\)\}\\right\|\.The first term is at mostLN,ω​δq/ρωL\_\{N,\\omega\}\\delta\_\{q\}/\\rho\_\{\\omega\}\. For the second term, use\|Nω​\(q′\)\|≤Mω​Dω​\(q′\)\|N\_\{\\omega\}\(q^\{\\prime\}\)\|\\leq M\_\{\\omega\}D\_\{\\omega\}\(q^\{\\prime\}\)andDω​\(q\)≥ρωD\_\{\\omega\}\(q\)\\geq\\rho\_\{\\omega\}to obtain

\|Nω​\(q′\)\|​\|Dω​\(q\)−Dω​\(q′\)\|Dω​\(q\)​Dω​\(q′\)≤Mω​LD,ωρω​δq\.\|N\_\{\\omega\}\(q^\{\\prime\}\)\|\\frac\{\|D\_\{\\omega\}\(q\)\-D\_\{\\omega\}\(q^\{\\prime\}\)\|\}\{D\_\{\\omega\}\(q\)D\_\{\\omega\}\(q^\{\\prime\}\)\}\\leq\\frac\{M\_\{\\omega\}L\_\{D,\\omega\}\}\{\\rho\_\{\\omega\}\}\\delta\_\{q\}\.Combining the two displays gives the Lipschitz constant\. The decomposition proof of Theorem[3](https://arxiv.org/html/2607.08122#Thmtheorem3)then applies verbatim with targetτω\\tau\_\{\\omega\}and represented targetτω,ϕ,S\\tau\_\{\\omega,\\phi,S\}\. ∎

For a subgroup effect,ω​\(X\)=𝟙​\{X∈G\}\\omega\(X\)=\\mathbbm\{1\}\\\{X\\in G\\\}is represented wheneverGGis a union of workload cells; the denominator condition is a minimum subgroup\-mass condition\. For ATT in a partition basis,ω​\(X\)=e​\(X\)\\omega\(X\)=e\(X\)is represented by the cell propensityej=p1​j/\(p0​j\+p1​j\)e\_\{j\}=p\_\{1j\}/\(p\_\{0j\}\+p\_\{1j\}\), computed fromq\(0\)q^\{\(0\)\}and clipped as in Appendix[E\.3](https://arxiv.org/html/2607.08122#A5.SS3); the same cell\-mass and clipping assumptions give the required Lipschitz constants\. This proposition does not claim a full pointwise CATE guarantee; pointwise CATE requires additional approximation assumptions at the target covariate value\.

### E\.2Proof of Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1): ATE Lipschitzness

Throughout,Gt=n−1​∑i𝟙​\{Ti=t\}​ϕ​\(Xi\)​ϕ​\(Xi\)⊤G\_\{t\}=n^\{\-1\}\\sum\_\{i\}\\mathbbm\{1\}\\\{T\_\{i\}=t\\\}\\phi\(X\_\{i\}\)\\phi\(X\_\{i\}\)^\{\\top\}\(equivalentlyqt\(2\)=vec⁡\(Gt\)q\_\{t\}^\{\(2\)\}=\\operatorname\{vec\}\(G\_\{t\}\)when released explicitly\) andrt=qt\(1\)r\_\{t\}=q\_\{t\}^\{\(1\)\}are the arm\-specific Gram and outcome moments,ϕ¯:=q0\(0\)\+q1\(0\)\\bar\{\\phi\}:=q\_\{0\}^\{\(0\)\}\+q\_\{1\}^\{\(0\)\}is the empirical feature mean, andu=\(G0,G1,r0,r1,ϕ¯\)u=\(G\_\{0\},G\_\{1\},r\_\{0\},r\_\{1\},\\bar\{\\phi\}\)collects the moment inputs\. The ATE plug\-in isτ^λ​\(u\)=ϕ¯⊤​\(β^1,λ−β^0,λ\)\\widehat\{\\tau\}\_\{\\lambda\}\(u\)=\\bar\{\\phi\}^\{\\top\}\(\\widehat\{\\beta\}\_\{1,\\lambda\}\-\\widehat\{\\beta\}\_\{0,\\lambda\}\)withβ^t,λ=\(Gt\+λ​I\)−1​rt\\widehat\{\\beta\}\_\{t,\\lambda\}=\(G\_\{t\}\+\\lambda I\)^\{\-1\}r\_\{t\}\.

###### Lemma 1\(Ridge perturbation bound, full form\)\.

Under the assumptions of Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1), consider two inputsu,u′u,u^\{\\prime\}withGt′\+λ​I⪰\(κ\+λ\)​I/2G^\{\\prime\}\_\{t\}\+\\lambda I\\succeq\(\\kappa\+\\lambda\)I/2and‖β^t,λ​\(u\)‖2∨‖β^t,λ​\(u′\)‖2≤Rβ\\left\\lVert\\widehat\{\\beta\}\_\{t,\\lambda\}\(u\)\\right\\rVert\_\{2\}\\vee\\left\\lVert\\widehat\{\\beta\}\_\{t,\\lambda\}\(u^\{\\prime\}\)\\right\\rVert\_\{2\}\\leq R\_\{\\beta\}fort=0,1t=0,1\. Then

\|τ^λ​\(u\)−τ^λ​\(u′\)\|\\displaystyle\|\\widehat\{\\tau\}\_\{\\lambda\}\(u\)\-\\widehat\{\\tau\}\_\{\\lambda\}\(u^\{\\prime\}\)\|≤2​Rβ​‖ϕ¯−ϕ¯′‖2\\displaystyle\\leq 2R\_\{\\beta\}\\left\\lVert\\bar\{\\phi\}\-\\bar\{\\phi\}^\{\\prime\}\\right\\rVert\_\{2\}\+2​ϕmaxκ\+λ​∑t=01\{‖rt−rt′‖2\+Rβ​‖Gt−Gt′‖op\}\.\\displaystyle\\hskip\-40\.00006pt\+\\frac\{2\\phi\_\{\\max\}\}\{\\kappa\+\\lambda\}\\sum\_\{t=0\}^\{1\}\\left\\\{\\left\\lVert r\_\{t\}\-r^\{\\prime\}\_\{t\}\\right\\rVert\_\{2\}\+R\_\{\\beta\}\\left\\lVert G\_\{t\}\-G^\{\\prime\}\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\\right\\\}\.\(6\)

###### Proof\.

LetAt=Gt\+λ​IA\_\{t\}=G\_\{t\}\+\\lambda IandAt′=Gt′\+λ​IA^\{\\prime\}\_\{t\}=G^\{\\prime\}\_\{t\}\+\\lambda I, so‖At−1‖op≤1/\(κ\+λ\)\\left\\lVert A\_\{t\}^\{\-1\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq 1/\(\\kappa\+\\lambda\)and‖\(At′\)−1‖op≤2/\(κ\+λ\)\\left\\lVert\(A^\{\\prime\}\_\{t\}\)^\{\-1\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq 2/\(\\kappa\+\\lambda\)\. The identityAt−1​rt−\(At′\)−1​rt′=At−1​\(rt−rt′\)\+At−1​\(Gt′−Gt\)​\(At′\)−1​rt′A\_\{t\}^\{\-1\}r\_\{t\}\-\(A^\{\\prime\}\_\{t\}\)^\{\-1\}r^\{\\prime\}\_\{t\}=A\_\{t\}^\{\-1\}\(r\_\{t\}\-r^\{\\prime\}\_\{t\}\)\+A\_\{t\}^\{\-1\}\(G^\{\\prime\}\_\{t\}\-G\_\{t\}\)\(A^\{\\prime\}\_\{t\}\)^\{\-1\}r^\{\\prime\}\_\{t\}gives

‖β^t,λ​\(u\)−β^t,λ​\(u′\)‖2≤‖rt−rt′‖2\+Rβ​‖Gt−Gt′‖opκ\+λ,\\left\\lVert\\widehat\{\\beta\}\_\{t,\\lambda\}\(u\)\-\\widehat\{\\beta\}\_\{t,\\lambda\}\(u^\{\\prime\}\)\\right\\rVert\_\{2\}\\leq\\frac\{\\left\\lVert r\_\{t\}\-r^\{\\prime\}\_\{t\}\\right\\rVert\_\{2\}\+R\_\{\\beta\}\\left\\lVert G\_\{t\}\-G^\{\\prime\}\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\}\{\\kappa\+\\lambda\},with constants at most doubled when only the lower bound onAt′A^\{\\prime\}\_\{t\}is used\. Splitting the plug\-in difference,

\|ϕ¯⊤​\(β^1,λ​\(u\)−β^0,λ​\(u\)\)−\(ϕ¯′\)⊤​\(β^1,λ​\(u′\)−β^0,λ​\(u′\)\)\|\\displaystyle\|\\bar\{\\phi\}^\{\\top\}\(\\widehat\{\\beta\}\_\{1,\\lambda\}\(u\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(u\)\)\-\(\\bar\{\\phi\}^\{\\prime\}\)^\{\\top\}\(\\widehat\{\\beta\}\_\{1,\\lambda\}\(u^\{\\prime\}\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(u^\{\\prime\}\)\)\|≤‖ϕ¯−ϕ¯′‖2​‖β^1,λ​\(u\)−β^0,λ​\(u\)‖2\\displaystyle\\leq\\left\\lVert\\bar\{\\phi\}\-\\bar\{\\phi\}^\{\\prime\}\\right\\rVert\_\{2\}\\,\\left\\lVert\\widehat\{\\beta\}\_\{1,\\lambda\}\(u\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(u\)\\right\\rVert\_\{2\}\+‖ϕ¯′‖2​∑t=01‖β^t,λ​\(u\)−β^t,λ​\(u′\)‖2,\\displaystyle\\quad\+\\left\\lVert\\bar\{\\phi\}^\{\\prime\}\\right\\rVert\_\{2\}\\sum\_\{t=0\}^\{1\}\\left\\lVert\\widehat\{\\beta\}\_\{t,\\lambda\}\(u\)\-\\widehat\{\\beta\}\_\{t,\\lambda\}\(u^\{\\prime\}\)\\right\\rVert\_\{2\},where‖β^1,λ​\(u\)−β^0,λ​\(u\)‖2≤2​Rβ\\left\\lVert\\widehat\{\\beta\}\_\{1,\\lambda\}\(u\)\-\\widehat\{\\beta\}\_\{0,\\lambda\}\(u\)\\right\\rVert\_\{2\}\\leq 2R\_\{\\beta\}by the radius assumption and‖ϕ¯′‖2≤ϕmax\\left\\lVert\\bar\{\\phi\}^\{\\prime\}\\right\\rVert\_\{2\}\\leq\\phi\_\{\\max\}sinceϕ¯′\\bar\{\\phi\}^\{\\prime\}averages feature vectors bounded byϕmax\\phi\_\{\\max\}\. Substituting the coefficient bound proves \([6](https://arxiv.org/html/2607.08122#A5.E6)\)\. ∎

*Proof of Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1)\.*Letδq=‖q−q′‖∞\\delta\_\{q\}=\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}denote the maximum coordinatewise perturbation across the moment blocks enteringu=\(G0,G1,r0,r1,ϕ¯\)u=\(G\_\{0\},G\_\{1\},r\_\{0\},r\_\{1\},\\bar\{\\phi\}\)\. Then‖ϕ¯−ϕ¯′‖2≤p​δq\\left\\lVert\\bar\{\\phi\}\-\\bar\{\\phi\}^\{\\prime\}\\right\\rVert\_\{2\}\\leq\\sqrt\{p\}\\,\\delta\_\{q\},‖rt−rt′‖2≤p​δq\\left\\lVert r\_\{t\}\-r\_\{t\}^\{\\prime\}\\right\\rVert\_\{2\}\\leq\\sqrt\{p\}\\,\\delta\_\{q\}, and‖Gt−Gt′‖op≤p​δq\\left\\lVert G\_\{t\}\-G\_\{t\}^\{\\prime\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq p\\,\\delta\_\{q\}\. Substituting these inequalities into Lemma[1](https://arxiv.org/html/2607.08122#Thmlemma1)gives

\|τ^λ​\(q\)−τ^λ​\(q′\)\|≤\{2​Rβ​p\+4​ϕmax​\(p\+Rβ​p\)κ\+λ\}​δq\.\|\\widehat\{\\tau\}\_\{\\lambda\}\(q\)\-\\widehat\{\\tau\}\_\{\\lambda\}\(q^\{\\prime\}\)\|\\leq\\left\\\{2R\_\{\\beta\}\\sqrt\{p\}\+\\frac\{4\\phi\_\{\\max\}\(\\sqrt\{p\}\+R\_\{\\beta\}p\)\}\{\\kappa\+\\lambda\}\\right\\\}\\delta\_\{q\}\.Thus one valid main\-text choice is

Cϕ=2​Rβ​p\+4​ϕmax​\(p\+Rβ​p\),C\_\{\\phi\}=2R\_\{\\beta\}\\sqrt\{p\}\+4\\phi\_\{\\max\}\(\\sqrt\{p\}\+R\_\{\\beta\}p\),which yieldsLϕ≤Cϕ​\{1\+\(κ\+λ\)−1\}L\_\{\\phi\}\\leq C\_\{\\phi\}\\\{1\+\(\\kappa\+\\lambda\)^\{\-1\}\\\}\. For propensity\-clipped IPW and orthogonal\-score estimators, clippinge^​\(x\)\\widehat\{e\}\(x\)to\[η,1−η\]\[\\eta,1\-\\eta\]makesa↦1/aa\\mapsto 1/aanda↦1/\(1−a\)a\\mapsto 1/\(1\-a\)Lipschitz with constants of orderη−2\\eta^\{\-2\}on the clipped interval, and the same decomposition applies with additional polynomial factors in1/η1/\\eta\. The unregularized case follows by settingλ=0\\lambda=0whenκ\>0\\kappa\>0\. ∎

*Proof of Corollary[1](https://arxiv.org/html/2607.08122#Thmcorollary1)\.*By Gaussian tails and a union bound, with probability at least1−γ1\-\\gamma, the released workload vector obeys‖q~−q​\(D\)‖∞≤δm=c0​σ​log⁡\(m/γ\)\\left\\lVert\\widetilde\{q\}\-q\(D\)\\right\\rVert\_\{\\infty\}\\leq\\delta\_\{m\}=c\_\{0\}\\sigma\\sqrt\{\\log\(m/\\gamma\)\}\. In a full joint\-cell basis the Gram perturbation is diagonal, somaxt⁡‖G~t−Gt‖op≤δm\\max\_\{t\}\\left\\lVert\\widetilde\{G\}\_\{t\}\-G\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq\\delta\_\{m\}\. For explicitly released dense Gram blocks, the same conclusion follows under the stated operator\-norm event; a standard Gaussian\-matrix bound gives this event whenλ\\lambdais at least a constant multiple ofσ​p\+log⁡\(1/γ\)\\sigma\\sqrt\{p\+\\log\(1/\\gamma\)\}\.

λmin​\(G~t\+λ​I\)≥κ\+λ−‖G~t−Gt‖op≥κ\+λ/2\\lambda\_\{\\min\}\(\\widetilde\{G\}\_\{t\}\+\\lambda I\)\\geq\\kappa\+\\lambda\-\\left\\lVert\\widetilde\{G\}\_\{t\}\-G\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\\geq\\kappa\+\\lambda/2whenevermaxt⁡‖G~t−Gt‖op≤λ/2\\max\_\{t\}\\left\\lVert\\widetilde\{G\}\_\{t\}\-G\_\{t\}\\right\\rVert\_\{\\mathrm\{op\}\}\\leq\\lambda/2\. Theorem[1](https://arxiv.org/html/2607.08122#Thmtheorem1)then gives the displayed main\-text rate\. ∎

#### Conditioning intuition\.

With full joint\-cell indicators, an arm\-specific Gram matrix loses rank exactly when some cell contains few or no records from that treatment arm\. With concatenated bin indicators or other overlapping bases, the same issue appears through missing co\-occurrences or near\-collinear Gram blocks\. DP noise of scaleσ\\sigmafurther perturbs each released coordinate, so at smallε\\varepsilonand largeppthe effectiveκ\\kappacan be zero and the unregularized moment\-to\-ATE map is not Lipschitz\. Corollary[1](https://arxiv.org/html/2607.08122#Thmcorollary1)restores stability by settingλ\\lambdaat the privacy\-noise scale\. This plays a different role from SNR thresholding: SNR thresholding stops the max\-entropy solver from chasing moments that are indistinguishable from privacy noise, while ridge keeps the nuisance inverse stable when the retained moments still leave an arm’s Gram matrix near\-singular\.

### E\.3Stability of Clipped IPW and AIPW Scores

This appendix makes Proposition[1](https://arxiv.org/html/2607.08122#Thmproposition1)explicit for partition\-feature workloads\. The argument is standard perturbation calculus for clipped inverse\-propensity scores; it is included only to show how the same moment\-error bound propagates through IPW/AIPW post\-processing\.

###### Lemma 2\(Clipped IPW/AIPW perturbation\)\.

Letϕj​\(x\)=𝟙​\{x∈Aj\}\\phi\_\{j\}\(x\)=\\mathbbm\{1\}\\\{x\\in A\_\{j\}\\\},j=1,…,pj=1,\\ldots,p, be partition indicators,\|Y\|≤B\|Y\|\\leq B, and definept​j​\(q\)=qt​j\(0\)p\_\{tj\}\(q\)=q\_\{tj\}^\{\(0\)\},sj​\(q\)=p0​j​\(q\)\+p1​j​\(q\)s\_\{j\}\(q\)=p\_\{0j\}\(q\)\+p\_\{1j\}\(q\), andrt​j​\(q\)=qt​j\(1\)r\_\{tj\}\(q\)=q\_\{tj\}^\{\(1\)\}\. For two admissible moment vectorsq,q′q,q^\{\\prime\}, setδq=‖q−q′‖∞\\delta\_\{q\}=\\left\\lVert q\-q^\{\\prime\}\\right\\rVert\_\{\\infty\}andΔe=maxj⁡\|e^j​\(q\)−e^j​\(q′\)\|\\Delta\_\{e\}=\\max\_\{j\}\|\\widehat\{e\}\_\{j\}\(q\)\-\\widehat\{e\}\_\{j\}\(q^\{\\prime\}\)\|, wheree^j​\(⋅\)∈\[η,1−η\]\\widehat\{e\}\_\{j\}\(\\cdot\)\\in\[\\eta,1\-\\eta\]\. Then the clipped IPW functional

τ^IPW​\(q\)=∑j=1p\{r1​j​\(q\)e^j​\(q\)−r0​j​\(q\)1−e^j​\(q\)\}\\widehat\{\\tau\}\_\{\\mathrm\{IPW\}\}\(q\)=\\sum\_\{j=1\}^\{p\}\\left\\\{\\frac\{r\_\{1j\}\(q\)\}\{\\widehat\{e\}\_\{j\}\(q\)\}\-\\frac\{r\_\{0j\}\(q\)\}\{1\-\\widehat\{e\}\_\{j\}\(q\)\}\\right\\\}satisfies

\|τ^IPW​\(q\)−τ^IPW​\(q′\)\|≤2​pη​δq\+Bη2​Δe\.\|\\widehat\{\\tau\}\_\{\\mathrm\{IPW\}\}\(q\)\-\\widehat\{\\tau\}\_\{\\mathrm\{IPW\}\}\(q^\{\\prime\}\)\|\\leq\\frac\{2p\}\{\\eta\}\\delta\_\{q\}\+\\frac\{B\}\{\\eta^\{2\}\}\\Delta\_\{e\}\.Ife^j​\(q\)=clip⁡\{p1​j​\(q\)/sj​\(q\),η,1−η\}\\widehat\{e\}\_\{j\}\(q\)=\\operatorname\{clip\}\\\{p\_\{1j\}\(q\)/s\_\{j\}\(q\),\\eta,1\-\\eta\\\}andminj,q∘∈\{q,q′\}⁡sj​\(q∘\)≥ρ\\min\_\{j,q\_\{\\circ\}\\in\\\{q,q^\{\\prime\}\\\}\}s\_\{j\}\(q\_\{\\circ\}\)\\geq\\rho, thenΔe≤3​δq/ρ\\Delta\_\{e\}\\leq 3\\delta\_\{q\}/\\rho\.

Now suppose\|m^t​j​\(q\)\|≤Bm\|\\widehat\{m\}\_\{tj\}\(q\)\|\\leq B\_\{m\}and setΔm=maxt,j⁡\|m^t​j​\(q\)−m^t​j​\(q′\)\|\\Delta\_\{m\}=\\max\_\{t,j\}\|\\widehat\{m\}\_\{tj\}\(q\)\-\\widehat\{m\}\_\{tj\}\(q^\{\\prime\}\)\|\. The partition AIPW functional

τ^AIPW​\(q\)\\displaystyle\\widehat\{\\tau\}\_\{\\mathrm\{AIPW\}\}\(q\)=∑j=1psj​\(q\)​\{m^1​j​\(q\)−m^0​j​\(q\)\}\\displaystyle=\\sum\_\{j=1\}^\{p\}s\_\{j\}\(q\)\\\{\\widehat\{m\}\_\{1j\}\(q\)\-\\widehat\{m\}\_\{0j\}\(q\)\\\}\+∑j=1pr1​j​\(q\)−p1​j​\(q\)​m^1​j​\(q\)e^j​\(q\)−∑j=1pr0​j​\(q\)−p0​j​\(q\)​m^0​j​\(q\)1−e^j​\(q\)\\displaystyle\\quad\+\\sum\_\{j=1\}^\{p\}\\frac\{r\_\{1j\}\(q\)\-p\_\{1j\}\(q\)\\widehat\{m\}\_\{1j\}\(q\)\}\{\\widehat\{e\}\_\{j\}\(q\)\}\-\\sum\_\{j=1\}^\{p\}\\frac\{r\_\{0j\}\(q\)\-p\_\{0j\}\(q\)\\widehat\{m\}\_\{0j\}\(q\)\}\{1\-\\widehat\{e\}\_\{j\}\(q\)\}satisfies

\|τ^AIPW​\(q\)−τ^AIPW​\(q′\)\|≤\(4​Bm​p\+2​p​\(1\+Bm\)η\)​δq\+2​\(1\+1η\)​Δm\+B\+Bmη2​Δe\.\|\\widehat\{\\tau\}\_\{\\mathrm\{AIPW\}\}\(q\)\-\\widehat\{\\tau\}\_\{\\mathrm\{AIPW\}\}\(q^\{\\prime\}\)\|\\leq\\left\(4B\_\{m\}p\+\\frac\{2p\(1\+B\_\{m\}\)\}\{\\eta\}\\right\)\\delta\_\{q\}\+2\\left\(1\+\\frac\{1\}\{\\eta\}\\right\)\\Delta\_\{m\}\+\\frac\{B\+B\_\{m\}\}\{\\eta^\{2\}\}\\Delta\_\{e\}\.Consequently, if the fitted nuisance maps obeyΔe≤Le​δq\\Delta\_\{e\}\\leq L\_\{e\}\\delta\_\{q\}andΔm≤Lm​δq\\Delta\_\{m\}\\leq L\_\{m\}\\delta\_\{q\}, clipped IPW and AIPW are Lipschitz in the released moment vector with the displayed estimator\-specific constants\.

*Proof\.*Foru,v∈\[η,1−η\]u,v\\in\[\\eta,1\-\\eta\],\|1/u−1/v\|≤\|u−v\|/η2\|1/u\-1/v\|\\leq\|u\-v\|/\\eta^\{2\}and\|1/\(1−u\)−1/\(1−v\)\|≤\|u−v\|/η2\|1/\(1\-u\)\-1/\(1\-v\)\|\\leq\|u\-v\|/\\eta^\{2\}\. Apply these inequalities after adding and subtracting terms with either the moments held fixed or the nuisance functions held fixed\. The partition basis gives∑j\|r1​j\|\+∑j\|r0​j\|≤B\\sum\_\{j\}\|r\_\{1j\}\|\+\\sum\_\{j\}\|r\_\{0j\}\|\\leq B,∑jp1​j\+∑jp0​j≤1\\sum\_\{j\}p\_\{1j\}\+\\sum\_\{j\}p\_\{0j\}\\leq 1, and∑j\|sj​\(q\)−sj​\(q′\)\|≤2​p​δq\\sum\_\{j\}\|s\_\{j\}\(q\)\-s\_\{j\}\(q^\{\\prime\}\)\|\\leq 2p\\delta\_\{q\}\. These bounds yield the IPW and AIPW displays\. For the cell propensity ratio, clipping is nonexpansive and the mapp1​j/sjp\_\{1j\}/s\_\{j\}has perturbation at most3​δq/ρ3\\delta\_\{q\}/\\rhowhensj≥ρs\_\{j\}\\geq\\rho\. ∎

### E\.4Proof of Theorem[2](https://arxiv.org/html/2607.08122#Thmtheorem2): DP Moment Accuracy

Writeq​\(D\)=n−1​∑ih​\(Wi\)q\(D\)=n^\{\-1\}\\sum\_\{i\}h\(W\_\{i\}\)with‖h​\(Wi\)‖2≤Hq\\left\\lVert h\(W\_\{i\}\)\\right\\rVert\_\{2\}\\leq H\_\{q\}\. Under replacement adjacency, changing one record changesq​\(D\)q\(D\)by at most

Δ2≤2​Hqn\.\\Delta\_\{2\}\\leq\\frac\{2H\_\{q\}\}\{n\}\.
Applying the Gaussian mechanism toq​\(D\)q\(D\)addsZ∼𝒩​\(0,σ2​Im\)Z\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\_\{m\}\)withσ=Δ2​2​log⁡\(1\.25/δ\)/ε\\sigma=\\Delta\_\{2\}\\sqrt\{2\\log\(1\.25/\\delta\)\}/\\varepsilon\. By a standard Gaussian tail bound and a union bound over themmcoordinates, with probability at least1−γ1\-\\gamma,

‖Z‖∞≤σ​2​log⁡\(m/γ\)\.\\left\\lVert Z\\right\\rVert\_\{\\infty\}\\leq\\sigma\\sqrt\{2\\log\(m/\\gamma\)\}\.Substituting the expression forσ\\sigmayields the stated rate\. For the base workload,‖h​\(W\)‖22=\(1\+Y2\)​‖ϕ​\(X\)‖22≤ϕmax2​\(1\+B2\)\\left\\lVert h\(W\)\\right\\rVert\_\{2\}^\{2\}=\(1\+Y^\{2\}\)\\left\\lVert\\phi\(X\)\\right\\rVert\_\{2\}^\{2\}\\leq\\phi\_\{\\max\}^\{2\}\(1\+B^\{2\}\)\. For an optional Gram\-augmented workload, one additional active Gram block contributes‖ϕ​\(X\)​ϕ​\(X\)⊤‖F2=‖ϕ​\(X\)‖24≤ϕmax4\\left\\lVert\\phi\(X\)\\phi\(X\)^\{\\top\}\\right\\rVert\_\{F\}^\{2\}=\\left\\lVert\\phi\(X\)\\right\\rVert\_\{2\}^\{4\}\\leq\\phi\_\{\\max\}^\{4\}, givingHq2≤ϕmax2​\(1\+B2\)\+ϕmax4H\_\{q\}^\{2\}\\leq\\phi\_\{\\max\}^\{2\}\(1\+B^\{2\}\)\+\\phi\_\{\\max\}^\{4\}\. In the experiments, we useB=5B=5\(outcomes clipped to\[−5,5\]\[\-5,5\]\)\. ∎

### E\.5Proof of Theorem[3](https://arxiv.org/html/2607.08122#Thmtheorem3): ATE Error Decomposition

LetPsynP^\{\\mathrm\{syn\}\}be the actual solver output\. Writeτ^​\(q\)\\widehat\{\\tau\}\(q\)for the population moment\-map estimator andτ^syn\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}for its empirical version computed fromnsynn\_\{\\mathrm\{syn\}\}synthetic draws\. Decompose the error via the triangle inequality:

\|τ^syn−τ\|\\displaystyle\|\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}\-\\tau\|≤\|τ−τϕ,S\|⏟\(iii\) approximation\+\|τϕ,S−τ^​\(q⋆\)\|⏟= 0 by definition\+\|τ^​\(q⋆\)−τ^​\(q​\(D\)\)\|⏟\(i\) sampling\\displaystyle\\leq\\underbrace\{\|\\tau\-\\tau\_\{\\phi,S\}\|\}\_\{\\text\{\(iii\) approximation\}\}\+\\underbrace\{\|\\tau\_\{\\phi,S\}\-\\widehat\{\\tau\}\(q^\{\\star\}\)\|\}\_\{\\text\{= 0 by definition\}\}\+\\underbrace\{\|\\widehat\{\\tau\}\(q^\{\\star\}\)\-\\widehat\{\\tau\}\(q\(D\)\)\|\}\_\{\\text\{\(i\) sampling\}\}\+\|τ^​\(q​\(D\)\)−τ^​\(q~\)\|⏟\(ii\) privacy\+\|τ^​\(q~\)−τ^​\(q​\(Psyn\)\)\|⏟\(v\) calibration\+\|τ^​\(q​\(Psyn\)\)−τ^syn\|⏟\(iv\) Monte Carlo,\\displaystyle\\quad\+\\underbrace\{\|\\widehat\{\\tau\}\(q\(D\)\)\-\\widehat\{\\tau\}\(\\widetilde\{q\}\)\|\}\_\{\\text\{\(ii\) privacy\}\}\+\\underbrace\{\|\\widehat\{\\tau\}\(\\widetilde\{q\}\)\-\\widehat\{\\tau\}\(q\(P^\{\\mathrm\{syn\}\}\)\)\|\}\_\{\\text\{\(v\) calibration\}\}\+\\underbrace\{\|\\widehat\{\\tau\}\(q\(P^\{\\mathrm\{syn\}\}\)\)\-\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}\|\}\_\{\\text\{\(iv\) Monte Carlo\}\},\(7\)whereτϕ,S\\tau\_\{\\phi,S\}is the target represented by the retained workload coordinates\.

*Term \(i\):*By the CLT,‖q​\(D\)−q⋆‖∞=Op​\(n−1/2\)\\left\\lVert q\(D\)\-q^\{\\star\}\\right\\rVert\_\{\\infty\}=O\_\{p\}\(n^\{\-1/2\}\)\(concentration of bounded averages\)\. By the estimator Lipschitz condition,\|τ^​\(q⋆\)−τ^​\(q​\(D\)\)\|≤Lest​Op​\(n−1/2\)\|\\widehat\{\\tau\}\(q^\{\\star\}\)\-\\widehat\{\\tau\}\(q\(D\)\)\|\\leq L\_\{\\mathrm\{est\}\}O\_\{p\}\(n^\{\-1/2\}\)\.

*Term \(ii\):*By Theorem[2](https://arxiv.org/html/2607.08122#Thmtheorem2),‖q~−q​\(D\)‖∞=Op​\(Hq​log⁡m⋅log⁡\(1/δ\)/\(n​ε\)\)\\left\\lVert\\widetilde\{q\}\-q\(D\)\\right\\rVert\_\{\\infty\}=O\_\{p\}\(H\_\{q\}\\sqrt\{\\log m\\cdot\\log\(1/\\delta\)\}/\(n\\varepsilon\)\)\. Applying the same Lipschitz condition gives the privacy term\.

*Term \(iii\):*This isApprox​\(ϕ;S\):=\|τ−τϕ,S\|\\mathrm\{Approx\}\(\\phi;S\):=\|\\tau\-\\tau\_\{\\phi,S\}\|, the bias from representing outcome and propensity models in a finite retained basis\. This term is zero whenmtm\_\{t\}andeeare linear inϕ\\phiand nonzero otherwise; its magnitude depends on the richness ofϕ\\phi\.

*Term \(iv\):*The synthetic data estimatorτ^syn\\widehat\{\\tau\}^\{\\mathrm\{syn\}\}is computed fromnsynn\_\{\\mathrm\{syn\}\}i\.i\.d\. draws fromPsynP^\{\\mathrm\{syn\}\}\. By the CLT for the synthetic population, this contributesOp​\(nsyn−1/2\)O\_\{p\}\(n\_\{\\mathrm\{syn\}\}^\{\-1/2\}\)\.

*Term \(v\), calibration:*If exact matching is infeasible because the measurements are noisy, define the relaxation residual directly asΔcal​\(S\):=ΠS​\{q~−q​\(Psyn\)\}\\Delta\_\{\\mathrm\{cal\}\}\(S\):=\\Pi\_\{S\}\\\{\\widetilde\{q\}\-q\(P^\{\\mathrm\{syn\}\}\)\\\}\. The estimator Lipschitz condition bounds the calibration term by

\|τ^​\(q~\)−τ^​\(q​\(Psyn\)\)\|≤Lest​‖Δcal​\(S\)‖2=CalGap​\(S,σ\)\.\|\\widehat\{\\tau\}\(\\widetilde\{q\}\)\-\\widehat\{\\tau\}\(q\(P^\{\\mathrm\{syn\}\}\)\)\|\\leq L\_\{\\mathrm\{est\}\}\\left\\lVert\\Delta\_\{\\mathrm\{cal\}\}\(S\)\\right\\rVert\_\{2\}=\\mathrm\{CalGap\}\(S,\\sigma\)\.
Combining all five terms yields the stated bound\. ∎

###### Corollary 3\(SNR thresholding controls the calibration gap\)\.

LetSτ=\{a:\|q~a\|/σa≥τSNR\}S\_\{\\tau\}=\\\{a:\\ \|\\widetilde\{q\}\_\{a\}\|/\\sigma\_\{a\}\\geq\\tau\_\{\\mathrm\{SNR\}\}\\\}be the retained set, whereσa\\sigma\_\{a\}is the Gaussian\-mechanism standard deviation of coordinateaa\. Suppose the max\-entropy solver is run only onSτS\_\{\\tau\}and stopped at noise\-level tolerance

\|q~a−qa​\(Psyn\)\|≤ccal​σafor all​a∈Sτ,\|\\widetilde\{q\}\_\{a\}\-q\_\{a\}\(P^\{\\mathrm\{syn\}\}\)\|\\leq c\_\{\\mathrm\{cal\}\}\\,\\sigma\_\{a\}\\qquad\\text\{for all \}a\\in S\_\{\\tau\},\(8\)for a fixed numerical constantccalc\_\{\\mathrm\{cal\}\}\. Ifσa≤σ¯\\sigma\_\{a\}\\leq\\bar\{\\sigma\}onSτS\_\{\\tau\}, thenCalGap​\(Sτ,σ\)≤Lest​ccal​σ¯​mkept\\mathrm\{CalGap\}\(S\_\{\\tau\},\\sigma\)\\leq L\_\{\\mathrm\{est\}\}c\_\{\\mathrm\{cal\}\}\\bar\{\\sigma\}\\sqrt\{m\_\{\\mathrm\{kept\}\}\}withmkept=\|Sτ\|m\_\{\\mathrm\{kept\}\}=\|S\_\{\\tau\}\|; in the homoskedastic caseσ¯=σ\\bar\{\\sigma\}=\\sigma, soCalGap​\(Sτ,σ\)=O​\(Lest​σ​mkept\)\\mathrm\{CalGap\}\(S\_\{\\tau\},\\sigma\)=O\(L\_\{\\mathrm\{est\}\}\\sigma\\sqrt\{m\_\{\\mathrm\{kept\}\}\}\)\.

###### Proof\.

By definition ofCalGap\\mathrm\{CalGap\}and the tolerance \([8](https://arxiv.org/html/2607.08122#A5.E8)\),

CalGap​\(Sτ,σ\)=Lest​\(∑a∈Sτ\|q~a−qa​\(Psyn\)\|2\)1/2≤Lest​ccal​\(∑a∈Sτσa2\)1/2,\\mathrm\{CalGap\}\(S\_\{\\tau\},\\sigma\)=L\_\{\\mathrm\{est\}\}\\Bigl\(\\sum\_\{a\\in S\_\{\\tau\}\}\|\\widetilde\{q\}\_\{a\}\-q\_\{a\}\(P^\{\\mathrm\{syn\}\}\)\|^\{2\}\\Bigr\)^\{1/2\}\\leq L\_\{\\mathrm\{est\}\}c\_\{\\mathrm\{cal\}\}\\Bigl\(\\sum\_\{a\\in S\_\{\\tau\}\}\\sigma\_\{a\}^\{2\}\\Bigr\)^\{1/2\},and the claim follows fromσa≤σ¯\\sigma\_\{a\}\\leq\\bar\{\\sigma\}and\|Sτ\|=mkept\|S\_\{\\tau\}\|=m\_\{\\mathrm\{kept\}\}\. The tolerance condition is essential: thresholding alone cannot bound arbitrary optimization error; the guarantee is for the SNR\-thresholded, noise\-tolerant calibration rule\. Moments excluded bySτS\_\{\\tau\}are not counted inCalGap​\(Sτ,σ\)\\mathrm\{CalGap\}\(S\_\{\\tau\},\\sigma\)because the solver no longer matches them; their effect moves intoApprox​\(ϕ;Sτ\)\\mathrm\{Approx\}\(\\phi;S\_\{\\tau\}\), which is the bias–variance tradeoff measured in the workload\-dimension ablation\. ∎

## Appendix FBias\-Aware NA\+MI Interval and the SNR Deployment Diagnostic

This section gives the coverage\-corrected interval motivated inAppendix[L](https://arxiv.org/html/2607.08122#A12); it turns the coverage–privacy paradox into a measurable diagnostic and a conservative correction\. Throughout,τ^NA\+MI\\widehat\{\\tau\}\_\{\\mathrm\{NA\+MI\}\}is the NA\+MI point estimate,σNA\+MI2\\sigma\_\{\\mathrm\{NA\+MI\}\}^\{2\}its Rubin\-rules variance \(which accounts for DP moment noise\), andApprox^​\(ϕ\)\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)an estimate of the workload approximation bias defined below\.

###### Proposition 4\(Bias\-aware NA\+MI interval\)\.

Suppose the estimator admits the expansionτ^NA\+MI−τ=bϕ\+σNA\+MI​Z\+op​\(σNA\+MI\+\|bϕ\|\)\\widehat\{\\tau\}\_\{\\mathrm\{NA\+MI\}\}\-\\tau=b\_\{\\phi\}\+\\sigma\_\{\\mathrm\{NA\+MI\}\}Z\+o\_\{p\}\(\\sigma\_\{\\mathrm\{NA\+MI\}\}\+\|b\_\{\\phi\}\|\)withZ⇒𝒩​\(0,1\)Z\\Rightarrow\\mathcal\{N\}\(0,1\), wherebϕb\_\{\\phi\}is the leading workload approximation bias, and that the diagnostic is conservative:Pr⁡\{\|bϕ\|≤Approx^​\(ϕ\)\+op​\(σNA\+MI\)\}→1\\Pr\\\{\|b\_\{\\phi\}\|\\leq\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)\+o\_\{p\}\(\\sigma\_\{\\mathrm\{NA\+MI\}\}\)\\\}\\to 1\. Defineσ~2=σNA\+MI2\+Approx^​\(ϕ\)2\\widetilde\{\\sigma\}^\{2\}=\\sigma\_\{\\mathrm\{NA\+MI\}\}^\{2\}\+\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)^\{2\}andCIcorr​\(α\)=τ^NA\+MI±z1−α/2​σ~\\mathrm\{CI\}\_\{\\mathrm\{corr\}\}\(\\alpha\)=\\widehat\{\\tau\}\_\{\\mathrm\{NA\+MI\}\}\\pm z\_\{1\-\\alpha/2\}\\,\\widetilde\{\\sigma\}\. Then for confidence levels withz1−α/2≥3z\_\{1\-\\alpha/2\}\\geq\\sqrt\{3\}\(including standard 95% intervals\),CIcorr​\(α\)\\mathrm\{CI\}\_\{\\mathrm\{corr\}\}\(\\alpha\)has asymptotic coverage at least1−α1\-\\alpha\. Whenbϕ=0b\_\{\\phi\}=0it reduces to the usual NA\+MI interval; when\|bϕ\|≫σNA\+MI\|b\_\{\\phi\}\|\\gg\\sigma\_\{\\mathrm\{NA\+MI\}\}it becomes conservative rather than collapsing around a biased center\.

###### Proof\.

Condition on the event\|bϕ\|≤A\+op​\(σNA\+MI\)\|b\_\{\\phi\}\|\\leq A\+o\_\{p\}\(\\sigma\_\{\\mathrm\{NA\+MI\}\}\)withA=Approx^​\(ϕ\)A=\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)\. Ignoring the lower\-order term, coverage is bounded below byPr⁡\{\|σNA\+MI​Z\+bϕ\|≤z1−α/2​σNA\+MI2\+A2\}\\Pr\\\{\|\\sigma\_\{\\mathrm\{NA\+MI\}\}Z\+b\_\{\\phi\}\|\\leq z\_\{1\-\\alpha/2\}\\sqrt\{\\sigma\_\{\\mathrm\{NA\+MI\}\}^\{2\}\+A^\{2\}\}\\\}, which for fixedAAis minimized over\|bϕ\|≤A\|b\_\{\\phi\}\|\\leq Aat\|bϕ\|=A\|b\_\{\\phi\}\|=A, where it equals

f​\(a\)=Φ​\(z​s​\(a\)−a\)\+Φ​\(z​s​\(a\)\+a\)−1,f\(a\)=\\Phi\\bigl\(zs\(a\)\-a\\bigr\)\+\\Phi\\bigl\(zs\(a\)\+a\\bigr\)\-1,withz=z1−α/2z=z\_\{1\-\\alpha/2\},a=A/σNA\+MIa=A/\\sigma\_\{\\mathrm\{NA\+MI\}\}, ands​\(a\)=1\+a2s\(a\)=\\sqrt\{1\+a^\{2\}\}\. Differentiating,

f′​\(a\)=ϕ​\(z​s​\(a\)−a\)​\(z​as​\(a\)−1\)\+ϕ​\(z​s​\(a\)\+a\)​\(z​as​\(a\)\+1\),f^\{\\prime\}\(a\)=\\phi\\bigl\(zs\(a\)\-a\\bigr\)\\Bigl\(\\tfrac\{za\}\{s\(a\)\}\-1\\Bigr\)\+\\phi\\bigl\(zs\(a\)\+a\\bigr\)\\Bigl\(\\tfrac\{za\}\{s\(a\)\}\+1\\Bigr\),whereϕ\\phidenotes the standard normal density\. Ifz​a/s​\(a\)≥1za/s\(a\)\\geq 1thenf′​\(a\)≥0f^\{\\prime\}\(a\)\\geq 0directly\. Otherwisef′​\(a\)≥0f^\{\\prime\}\(a\)\\geq 0is equivalent toexp⁡\{−2​z​a​s​\(a\)\}≥1−z​a/s​\(a\)1\+z​a/s​\(a\)\\exp\\\{\-2zas\(a\)\\\}\\geq\\frac\{1\-za/s\(a\)\}\{1\+za/s\(a\)\}, i\.e\., writingy=z​a/s​\(a\)y=za/s\(a\), toarctanh⁡\(y\)≥y/\(1−y2/z2\)\\operatorname\{arctanh\}\(y\)\\geq y/\(1\-y^\{2\}/z^\{2\}\)\. Forz2≥3z^\{2\}\\geq 3this holds termwise:arctanh⁡\(y\)=∑k≥0y2​k\+1/\(2​k\+1\)\\operatorname\{arctanh\}\(y\)=\\sum\_\{k\\geq 0\}y^\{2k\+1\}/\(2k\+1\),y/\(1−y2/z2\)=∑k≥0y2​k\+1/z2​ky/\(1\-y^\{2\}/z^\{2\}\)=\\sum\_\{k\\geq 0\}y^\{2k\+1\}/z^\{2k\}, andz2​k≥3k≥2​k\+1z^\{2k\}\\geq 3^\{k\}\\geq 2k\+1fork≥1k\\geq 1\. Henceffis nondecreasing ona≥0a\\geq 0wheneverz≥3z\\geq\\sqrt\{3\}, sof​\(a\)≥f​\(0\)=2​Φ​\(z\)−1=1−αf\(a\)\\geq f\(0\)=2\\Phi\(z\)\-1=1\-\\alpha\. ∎

#### EstimatingApprox^​\(ϕ\)\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)\.

Letψ^ϕ​\(W\)\\widehat\{\\psi\}\_\{\\phi\}\(W\)be the estimated orthogonal ATE score with nuisances fitted in theϕ\\phibasis, centered atτ^NA\+MI\\widehat\{\\tau\}\_\{\\mathrm\{NA\+MI\}\}, and letb​\(X\)∈ℝrb\(X\)\\in\\mathbb\{R\}^\{r\}be a fixed diagnostic dictionary richer thanϕ\\phi, scaled so that𝔼​\[b​\(X\)​b​\(X\)⊤\]≈I\\mathbb\{E\}\[b\(X\)b\(X\)^\{\\top\}\]\\approx Iunder the synthetic distribution\. With the residual score momentR^ϕ=‖nsyn−1​∑iψ^ϕ​\(Wisyn\)​b​\(Xisyn\)‖2\\widehat\{R\}\_\{\\phi\}=\\bigl\\lVert n\_\{\\mathrm\{syn\}\}^\{\-1\}\\sum\_\{i\}\\widehat\{\\psi\}\_\{\\phi\}\(W\_\{i\}^\{\\mathrm\{syn\}\}\)\\,b\(X\_\{i\}^\{\\mathrm\{syn\}\}\)\\bigr\\rVert\_\{2\}, a practical conservative choice isApprox^​\(ϕ\)=L^diag​R^ϕ\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)=\\widehat\{L\}\_\{\\mathrm\{diag\}\}\\widehat\{R\}\_\{\\phi\}, whereL^diag\\widehat\{L\}\_\{\\mathrm\{diag\}\}is the same empirical Lipschitz/overlap constant used to score candidate features inCausal\-AIM\. Large residual moments indicate that the current workload leaves systematic orthogonal\-score structure unexplainedin the regime at highε\\varepsilonwhereDP noise shrinks but approximation bias remains\. Both quantities are computable from the synthetic release and a public pilot or held\-out covariate sample, so the ratioApprox^​\(ϕ\)/σNA\+MI\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)/\\sigma\_\{\\mathrm\{NA\+MI\}\}can be tracked at deployment\.

#### Practical rule\.

Keep momentjjiffSNRj:=\|q~j\|/σj≥τSNR\\mathrm\{SNR\}\_\{j\}:=\|\\widetilde\{q\}\_\{j\}\|/\\sigma\_\{j\}\\geq\\tau\_\{\\mathrm\{SNR\}\}\(defaultτSNR=3\\tau\_\{\\mathrm\{SNR\}\}=3\) as the calibration filter, and report the corrected intervalτ^NA\+MI±z1−α/2​\(σNA\+MI2\+Approx^​\(ϕ\)2\)1/2\\widehat\{\\tau\}\_\{\\mathrm\{NA\+MI\}\}\\pm z\_\{1\-\\alpha/2\}\(\\sigma\_\{\\mathrm\{NA\+MI\}\}^\{2\}\+\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)^\{2\}\)^\{1/2\}\. IfApprox^​\(ϕ\)≪σNA\+MI\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)\\ll\\sigma\_\{\\mathrm\{NA\+MI\}\}, the correction is negligible and DP noise dominates; ifApprox^​\(ϕ\)≳σNA\+MI\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)\\gtrsim\\sigma\_\{\\mathrm\{NA\+MI\}\}, the analysis is approximation\-dominated, which is the diagnostic signatureof the coverage–privacy paradox at highε\\varepsilon\.

#### Connection to robust Bayes\.

The correction treats workload approximation as local model misspecification: NA\+MI accounts for posterior uncertainty induced by DP noise, while theApprox^​\(ϕ\)2\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)^\{2\}term enlarges the interval to cover discrepancy between the working workload model and the data\-generating law\. This correction is the additive, workload\-aware analogue of robust\-Bayes and misspecification\-aware posterior calibration\[bissiri2016genbayes,watson2016approxmodels,grunwald2017safebayes,miller2018coarsened\], which inflate posterior spread to remain valid under model misspecification\.

## Appendix GNA\+MI: Detailed Pseudocode

Algorithm[4](https://arxiv.org/html/2607.08122#alg4)gives an implementation\-level version ofthe noise\-aware multiple\-imputation procedure of Section[6](https://arxiv.org/html/2607.08122#S6)\.

Algorithm 4Noise\-Aware Multiple Imputation \(NA\+MI\)1:Measured DP moments

q~S\\widetilde\{q\}\_\{S\}on coordinates

SS, mechanism noise covariance

ΣS\\Sigma\_\{S\}, number of draws

MM, synthetic size

nsynn\_\{\\mathrm\{syn\}\}, level

α\\alpha, optional SNR threshold

τSNR\\tau\_\{\\mathrm\{SNR\}\}
2:Compute the retained set

Sτ=\{j∈S:\|q~j\|/σj≥τSNR\}S\_\{\\tau\}=\\\{j\\in S:\|\\widetilde\{q\}\_\{j\}\|/\\sigma\_\{j\}\\geq\\tau\_\{\\mathrm\{SNR\}\}\\\}once from

q~S\\widetilde\{q\}\_\{S\}\(take

Sτ=SS\_\{\\tau\}=Sif thresholding is off\)

3:for

ℓ=1,…,M\\ell=1,\\dots,Mdo

4:Draw

qS\(ℓ\)∼𝒩​\(q~S,ΣS\)q\_\{S\}^\{\(\\ell\)\}\\sim\\mathcal\{N\}\(\\widetilde\{q\}\_\{S\},\\Sigma\_\{S\}\)⊳\\trianglerightflat\-prior Gaussian posterior on measured coordinates

5:

Psyn,\(ℓ\)←P^\{\\mathrm\{syn\},\(\\ell\)\}\\leftarrowmax\-entropy calibration of \([3](https://arxiv.org/html/2607.08122#S4.E3)\) on

ΠSτ​qS\(ℓ\)\\Pi\_\{S\_\{\\tau\}\}q\_\{S\}^\{\(\\ell\)\}\(Algorithm[2](https://arxiv.org/html/2607.08122#alg2)\)

6:

Dsyn,\(ℓ\)←nsynD^\{\\mathrm\{syn\},\(\\ell\)\}\\leftarrow n\_\{\\mathrm\{syn\}\}ancestral\-sampling draws from

Psyn,\(ℓ\)P^\{\\mathrm\{syn\},\(\\ell\)\}
7:

\(τ^\(ℓ\),v^\(ℓ\)\)←\(\\widehat\{\\tau\}^\{\(\\ell\)\},\\widehat\{v\}^\{\(\\ell\)\}\)\\leftarrowdoubly robust ATE estimate and its variance estimate on

Dsyn,\(ℓ\)D^\{\\mathrm\{syn\},\(\\ell\)\}
8:endfor

9:

τ¯←M−1​∑ℓτ^\(ℓ\)\\bar\{\\tau\}\\leftarrow M^\{\-1\}\\sum\_\{\\ell\}\\widehat\{\\tau\}^\{\(\\ell\)\};

WM←M−1​∑ℓv^\(ℓ\)W\_\{M\}\\leftarrow M^\{\-1\}\\sum\_\{\\ell\}\\widehat\{v\}^\{\(\\ell\)\};

BM←\(M−1\)−1​∑ℓ\(τ^\(ℓ\)−τ¯\)2B\_\{M\}\\leftarrow\(M\-1\)^\{\-1\}\\sum\_\{\\ell\}\(\\widehat\{\\tau\}^\{\(\\ell\)\}\-\\bar\{\\tau\}\)^\{2\}
10:

TM←WM\+\(1\+1/M\)​BMT\_\{M\}\\leftarrow W\_\{M\}\+\(1\+1/M\)B\_\{M\};

νM←\(M−1\)​\(1\+WM\(1\+1/M\)​BM\)2\\nu\_\{M\}\\leftarrow\(M\-1\)\\bigl\(1\+\\tfrac\{W\_\{M\}\}\{\(1\+1/M\)B\_\{M\}\}\\bigr\)^\{2\}⊳\\trianglerightRubin’s rules\[rubin1987mi\]

11:Point estimate

τ¯\\bar\{\\tau\}and interval

τ¯±tνM,1−α/2​TM\\bar\{\\tau\}\\pm t\_\{\\nu\_\{M\},1\-\\alpha/2\}\\sqrt\{T\_\{M\}\}

The between\-imputation componentBMB\_\{M\}is what widens the interval asε\\varepsilondecreases: smallerε\\varepsilonmeans larger DP noise, more dispersed posterior drawsqS\(ℓ\)q\_\{S\}^\{\(\\ell\)\}, and hence more variableτ^\(ℓ\)\\widehat\{\\tau\}^\{\(\\ell\)\}\. The bias\-aware variant of Appendix[F](https://arxiv.org/html/2607.08122#A6)replacesTMT\_\{M\}byTM\+Approx^​\(ϕ\)2T\_\{M\}\+\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)^\{2\}\.

## Appendix HExperimental Protocol and Supporting Results

### H\.1Feature Construction and Defaults

For each dataset, continuous covariates are discretized intoL=5L=5quantile bins and the resulting indicator variables defineϕ\\phi\. When quantile edges coincide, bins collapse, so the effective dimension is data\-dependent; IHDP yieldsp=48p=48atL=5L=5\. The small\-LLdefault is consistent with practice in DP quantile estimation, where a handful of privately estimated quantiles per variable is reliable at moderate budgets\[gillenwater2021dpquantiles,kaplan2022dpapproxquantiles,lei2011dpmest\]\.

Unless stated otherwise, experiments usensyn=10​nn\_\{\\mathrm\{syn\}\}=10nsynthetic records per release,M=20M=20MI draws, no max\-entropy calibration ridge \(αcal=0\\alpha\_\{\\mathrm\{cal\}\}=0\), and no SNR thresholding in the main pipeline\. Outcomes are standardized by fixed public constants and clipped atB=5B=5; constants and clip fractions are reported in Appendix[I](https://arxiv.org/html/2607.08122#A9)\. The non\-private oracle uses unstandardized, unclipped outcomes\. Estimated propensities in the DR analysis are clipped to\[0\.02,0\.98\]\[0\.02,0\.98\]\(a conservative implementation bound, distinct from the positivity constantη\\etaof Assumption[1](https://arxiv.org/html/2607.08122#Thmassumption1)and from the DGP overlap parameter swept in the overlap ablation\); continuous\-outcome DR nuisance fits use a fixed ridge linear model, and binary outcomes use logistic regression\. Main benchmark experiments usenrep=500n\_\{\\mathrm\{rep\}\}=500replications, adaptive and ACS studies use100100, and ablations use200200\. Every run logs its full configuration, code version, and data source\.

### H\.2Metrics

All metrics are computed overnrepn\_\{\\mathrm\{rep\}\}independent replications\. Letτ^\(r\)\\widehat\{\\tau\}^\{\(r\)\}be the ATE estimate,CI\(r\)\\mathrm\{CI\}^\{\(r\)\}its 95% confidence interval, andτ\\tauthe true ATE\.

- •ATE bias:Bias=\|nrep−1​∑rτ^\(r\)−τ\|\\mathrm\{Bias\}=\\bigl\|n\_\{\\mathrm\{rep\}\}^\{\-1\}\\sum\_\{r\}\\widehat\{\\tau\}^\{\(r\)\}\-\\tau\\bigr\|\.
- •ATE RMSE:RMSE=\{nrep−1​∑r\(τ^\(r\)−τ\)2\}1/2\\mathrm\{RMSE\}=\\\{n\_\{\\mathrm\{rep\}\}^\{\-1\}\\sum\_\{r\}\(\\widehat\{\\tau\}^\{\(r\)\}\-\\tau\)^\{2\}\\\}^\{1/2\}\.
- •CI coverage:Cov=nrep−1​∑r𝟙​\{τ∈CI\(r\)\}\\mathrm\{Cov\}=n\_\{\\mathrm\{rep\}\}^\{\-1\}\\sum\_\{r\}\\mathbbm\{1\}\\\{\\tau\\in\\mathrm\{CI\}^\{\(r\)\}\\\}, targeting the nominal 95% level\.
- •CI length:Len=nrep−1​∑r\|CI\(r\)\|\\mathrm\{Len\}=n\_\{\\mathrm\{rep\}\}^\{\-1\}\\sum\_\{r\}\|\\mathrm\{CI\}^\{\(r\)\}\|\.
- •Marginal fidelity:TVD=d−1​∑j=1dTV​\(P^j,P^jsyn\)\\mathrm\{TVD\}=d^\{\-1\}\\sum\_\{j=1\}^\{d\}\\mathrm\{TV\}\(\\hat\{P\}\_\{j\},\\hat\{P\}\_\{j\}^\{\\mathrm\{syn\}\}\), the average total\-variation distance across one\-dimensional marginals\.

### H\.3Experiment Grid and Ablations

- •Causal versus generic workloads: compare ATE RMSE and bias for causal workload synthesis, MST, and AIM overε∈\{0\.5,1,2,5\}\\varepsilon\\in\\\{0\.5,1,2,5\\\}withδ=1/n2\\delta=1/n^\{2\}\.
- •Coverage calibration: compare naive intervals on generic and causal synthetic data against NA\+MI intervals under the same privacy budgets\.
- •Adaptive selection: compareCausal\-AIMagainst the fixed causal workload at the same total privacy budget, sweepingK∈\{1,3,5,10\}K\\in\\\{1,3,5,10\\\}adaptive rounds\.
- •Workload dimension: vary quantile\-bin granularityL∈\{3,5,10,20\}L\\in\\\{3,5,10,20\\\}, with and without SNR thresholding atτSNR=3\\tau\_\{\\mathrm\{SNR\}\}=3\.
- •MI draws: varyM∈\{5,10,20,50,100\}M\\in\\\{5,10,20,50,100\\\}\.
- •Synthetic sample size: varynsyn/n∈\{1,2,5,10,50\}n\_\{\\mathrm\{syn\}\}/n\\in\\\{1,2,5,10,50\\\}\.
- •Overlap: vary the positivity constantη\\etain the ACIC DGP and measure the effect on RMSE and coverage\.

## Appendix IACS Semi\-Synthetic DGP Details

#### Outcome standardization constants\.

For DP measurement, each dataset’s outcome is standardized as\(Y−mpub\)/spub\(Y\-m\_\{\\mathrm\{pub\}\}\)/s\_\{\\mathrm\{pub\}\}using the fixed public constants below \(treated as public benchmark metadata, consistent with the public clip boundBB\), then clipped atB=5B=5; all reported estimates and intervals are converted back to original units\. The non\-private oracle uses unstandardized, unclipped outcomes\.

We use 20 demographic covariates from the 2018 California ACS viafolktables\[ding2021retiring\]: age, education, marital status, relationship, disability, employment status, citizenship, migration, Hispanic origin, ancestry, nativity, deafness, blindness, cognitive difficulty, sex, race, PUMA, state, class of worker, and place of birth; structurally missing categorical codes are encoded as their own category\.

*Treatment assignment\.*LetXnX\_\{n\}denote column\-standardized covariates\. We drawβ∼𝒩​\(0,Id\)\\beta\\sim\\mathcal\{N\}\(0,I\_\{d\}\), normalize to unitℓ2\\ell\_\{2\}norm, and set the propensity scoree​\(x\)=expit⁡\(β⊤​x\+0\.8​sin⁡\(x1\)−0\.4​x2​x3\)e\(x\)=\\operatorname\{expit\}\\\!\\bigl\(\\beta^\{\\top\}x\+0\.8\\sin\(x\_\{1\}\)\-0\.4\\,x\_\{2\}x\_\{3\}\\bigr\), rescaled to\[0\.1,0\.9\]\[0\.1,0\.9\]to enforce overlap\. Treatment is drawn asTi∼Bernoulli​\(e​\(Xi\)\)T\_\{i\}\\sim\\mathrm\{Bernoulli\}\(e\(X\_\{i\}\)\)\.

*Outcome model\.*We drawg∼𝒩​\(0,Id\)g\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\(normalized\) and define the baseline outcomeμ0​\(x\)=g⊤​x\+0\.6​cos⁡\(x3\)\+0\.3​\(x12−x2\)\\mu\_\{0\}\(x\)=g^\{\\top\}x\+0\.6\\cos\(x\_\{3\}\)\+0\.3\(x\_\{1\}^\{2\}\-x\_\{2\}\)\. Individual treatment effects areτ​\(x\)=0\.8\+0\.5​tanh⁡\(x1\)\+0\.25​sin⁡\(x4−x5\)\\tau\(x\)=0\.8\+0\.5\\tanh\(x\_\{1\}\)\+0\.25\\sin\(x\_\{4\}\-x\_\{5\}\)\. Observed outcomes areYi=μ0​\(Xi\)\+τ​\(Xi\)​Ti\+εiY\_\{i\}=\\mu\_\{0\}\(X\_\{i\}\)\+\\tau\(X\_\{i\}\)\\,T\_\{i\}\+\\varepsilon\_\{i\}withεi∼𝒩​\(0,1\.12\)\\varepsilon\_\{i\}\\sim\\mathcal\{N\}\(0,1\.1^\{2\}\)\. The ground\-truth ATE is the mean of the sampled potential\-outcome differences,τ=n−1​∑i\{Yi​\(1\)−Yi​\(0\)\}\\tau=n^\{\-1\}\\sum\_\{i\}\\\{Y\_\{i\}\(1\)\-Y\_\{i\}\(0\)\\\}\.

## Appendix JSupplementary Experimental Figures

All figures below supplement the main\-text results in Section[7](https://arxiv.org/html/2607.08122#S7); they are ordered as benchmark diagnostics, ACS validation, design ablations, and the fidelity\-versus\-causal\-utility diagnostic\. Bias patterns for the main benchmark comparison appear in Figure[6](https://arxiv.org/html/2607.08122#A10.F6)\.

![Refer to caption](https://arxiv.org/html/2607.08122v1/x6.png)Figure 6:Benchmark bias comparison: Mean absolute ATE error \(nrep−1​∑r\|τ^\(r\)−τ\|n\_\{\\mathrm\{rep\}\}^\{\-1\}\\sum\_\{r\}\|\\hat\{\\tau\}^\{\(r\)\}\-\\tau\|\) across privacy budgets and four benchmark datasets \(nrep=500n\_\{\\mathrm\{rep\}\}=500\)\. Bias patterns by dataset and privacy level mirror the RMSE rankings in Figure[2](https://arxiv.org/html/2607.08122#S7.F2)\. At lowε\\varepsilon\(≤1\\leq 1\), all methods show substantial bias relative to the non\-private oracle; MST \+ naive DR has the lowest bias on most datasets, with Causal \+ NA\+MI lower on IHDP \(and on LaLonde atε=0\.5\\varepsilon=0\.5\)\. Asε\\varepsilongrows, all methods converge toward the oracle\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x7.png)Figure 7:ACS study: ATE RMSE heatmaps on the ACS semi\-synthetic study \(nrep=100n\_\{\\mathrm\{rep\}\}=100\)\. Rows index sample sizen∈\{1000,5000,20000\}n\\in\\\{1000,5000,20000\\\}and columns indexε∈\{0\.5,1,2,5\}\\varepsilon\\in\\\{0\.5,1,2,5\\\}\. Each panel shows one method\. At smallnnand lowε\\varepsilon, all private methods have elevated RMSE; at largennandε\\varepsilon, all converge toward the non\-private oracle level\. NA\+MI is competitive on RMSE at strict budgets \(winning atn=1000n\{=\}1000,ε=0\.5\\varepsilon\{=\}0\.5\) and is the only method with valid coverage \(Figure[8](https://arxiv.org/html/2607.08122#A10.F8)\)\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x8.png)Figure 8:ACS study: Empirical 95% CI coverage on the same ACS grid as Figure[7](https://arxiv.org/html/2607.08122#A10.F7)\(nrep=100n\_\{\\mathrm\{rep\}\}=100\)\. Causal workload \+ NA\+MI achieves coverage1\.001\.00at\(n=1000,ε=0\.5\)\(n\{=\}1000,\\,\\varepsilon\{=\}0\.5\), while MST \+ naive DR achieves only0\.070\.07\. The coverage advantage of NA\+MI is most pronounced at lowε\\varepsilonand moderatenn, where DP noise dominates the error\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x9.png)Figure 9:Workload dimension on IHDP atε=1\\varepsilon=1\(nrep=200n\_\{\\mathrm\{rep\}\}=200\), with and without SNR thresholding \(τSNR=3\\tau\_\{\\mathrm\{SNR\}\}=3\)\. Without thresholding, RMSE decreases withppin the tested range \(p≤101p\\leq 101\); withτSNR=3\\tau\_\{\\mathrm\{SNR\}\}=3, RMSE is roughly flat, with lower bias and more variance; the bias\-dominated regime of Remark[2](https://arxiv.org/html/2607.08122#Thmremark2)is not reached on this dataset\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x10.png)Figure 10:MI draws ablation: empirical 95% CI coverage versus number of imputation drawsMMon IHDP atε=1\\varepsilon=1\(nrep=200n\_\{\\mathrm\{rep\}\}=200\)\. Coverage is at or above0\.9950\.995for everyMMtested \(0\.9950\.995atM=5M\{=\}5,1\.0001\.000forM≥10M\\geq 10\): the interval is near\-nominal already at smallMM\. We useM=20M\{=\}20as a conservative default for noise\-aware MI\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x11.png)Figure 11:Synthetic sample size ablation: ATE RMSE versus the rationsyn/nn\_\{\\mathrm\{syn\}\}/non IHDP atε=1\\varepsilon=1\(nrep=200n\_\{\\mathrm\{rep\}\}=200\)\. RMSE varies modestly \(range12\.812\.8–15\.515\.5\) across ratios from11to5050, with no benefit from oversampling: the Monte Carlo termOp​\(nsyn−1/2\)O\_\{p\}\(n\_\{\\mathrm\{syn\}\}^\{\-1/2\}\)in Theorem[3](https://arxiv.org/html/2607.08122#Thmtheorem3)is dominated by DP noise already atnsyn=nn\_\{\\mathrm\{syn\}\}=n\. In practice,nsyn=nn\_\{\\mathrm\{syn\}\}=nsuffices\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x12.png)Figure 12:Overlap ablation on ACIC DGP 7 atε=1\\varepsilon=1\(nrep=200n\_\{\\mathrm\{rep\}\}=200\)\. The positivity constantη\\etavaries from0\.30\.3\(strong overlap\) to0\.010\.01\(near violation\)\. As overlap weakens, MST \+ naive DR maintains low RMSE \(0\.600\.60–0\.620\.62\) but poor coverage \(0\.210\.21–0\.280\.28\), while Causal workload \+ NA\+MI has higher RMSE \(7\.57\.5–8\.38\.3\) with near\-nominal coverage \(0\.9950\.995–1\.001\.00\) at everyη\\eta\. This confirms that the coverage–RMSE tradeoff persists across overlap regimes and that NA\+MI remains the only method with usable confidence intervals\.![Refer to caption](https://arxiv.org/html/2607.08122v1/x13.png)Figure 13:Marginal fidelity versus causal utility across all method–dataset–ε\\varepsilonconfigurations \(nrep=500n\_\{\\mathrm\{rep\}\}=500\)\. Each point represents one experimental cell; thexx\-axis is average per\-marginal TVD and theyy\-axis is ATE RMSE in standardized outcome units \(log scale\), so datasets with different outcome scales are comparable\. Generic workloads \(MST\) dominate on marginal fidelity \(TVD\) in nearly every configuration and on RMSE atε≥2\\varepsilon\\geq 2; atε=0\.5\\varepsilon=0\.5, causal workloads with NA\+MI match or beat MST on RMSE on half the benchmarks \(and on IHDP atε=1\\varepsilon=1\) despite worse TVD\. Distributional fidelity therefore does not order methods by causal utility; the case for causal workloads rests on noise\-aware uncertainty quantification \(Figure[3](https://arxiv.org/html/2607.08122#S7.F3)\)\.
## Appendix KSupplementary Empirical Analyses

This appendix reports supplementary experiments; all use the defaults of Section[7](https://arxiv.org/html/2607.08122#S7)atnrep=200n\_\{\\mathrm\{rep\}\}=200\(scalability:nrep=100n\_\{\\mathrm\{rep\}\}=100\) unless noted\.

### K\.1Multi\-Estimand Reuse from One Synthetic Release

From a*single*DP synthetic release, we estimate the ATE, ATT, and a subgroup effect with NA\+MI intervals and no additional privacy spending\. All eighteen dataset–budget–estimand cells attain nominal coverage \(Table[2](https://arxiv.org/html/2607.08122#A11.T2)\)\. Direct DP ATE estimators cannot offer this reuse: each answered query spends additional privacy budget\.

Table 2:Multi\-estimand reuse from a single DP release: 95% CI coverage \(RMSE, original outcome units\) of NA\+MI intervals for ATE, ATT, and a subgroup effect, all computed from one synthetic dataset per replication \(nrep=200n\_\{\\mathrm\{rep\}\}=200\)\.
### K\.2Direct DP ATE Proxy Comparison

Table 3:RMSE and coverage against output\-perturbation*proxies*of direct DP ATE estimators\. The proxies apply DP only at the output rather than to the propensity fit, a weaker DP regime than the published methods\[schroder2025private,ohnishi2024covbal\]; these results should therefore be read as a favorable contextual approximation of the direct\-estimator family\. Cells are RMSE / coverage,nrep=200n\_\{\\mathrm\{rep\}\}=200\.The comparison \(Table[3](https://arxiv.org/html/2607.08122#A11.T3)\) ismixed: the proxies win on single\-estimand RMSE on ACIC, while Causal \+ NA\+MI wins on IHDP and attains full coverage everywhere; and the synthetic\-data release supports multiple estimands at shared privacy cost \(Table[2](https://arxiv.org/html/2607.08122#A11.T2)\)\.

### K\.3Hybrid Causal and Generic Workloads

A 50/50 split of the privacy budget between causal moments and generic one\-way marginals \(nrep=200n\_\{\\mathrm\{rep\}\}=200; RMSE / coverage\):

The hybrid Pareto\-improves on the fixed causal workload on ACIC atε=1\\varepsilon=1\(lower RMSE at equal coverage\) but trails on IHDP; it is an alternative when both point accuracy and coverage matter, not a new default\.

### K\.4Adaptive Selection Operating Point

On IHDP \(K∈\{1,2,3,5,10\}K\\in\\\{1,2,3,5,10\\\},nrep=200n\_\{\\mathrm\{rep\}\}=200; RMSE / coverage\), noKKrecovers the fixed workload’s calibration:

Contrast with ACIC \(Figure[5](https://arxiv.org/html/2607.08122#S7.F5)\): whetherCausal\-AIMhelps is dataset\-dependent, which is the operating\-point message of Section[7](https://arxiv.org/html/2607.08122#S7)\.

### K\.5Scalability in Covariate Dimension

On ACS atn=5000n=5000,ε=1\\varepsilon=1, andnrep=100n\_\{\\mathrm\{rep\}\}=100, increasing the number of covariates raises both RMSE and synthesis time, with mirror descent on theϕ\\phi\-basis as the dominant cost\. Thed=40d=40andd=60d=60configurations extend the 20 base ACS covariates with deterministic derived features \(squares, pairwise interactions, and threshold indicators\)\.

### K\.6Calibration\-Ridge Sensitivity

On IHDP atε=1\\varepsilon=1\(nrep=200n\_\{\\mathrm\{rep\}\}=200\), RMSE / coverage across max\-entropy calibration ridge valuesαcal∈\{0\.001,0\.01,0\.1,1\.0\}\\alpha\_\{\\mathrm\{cal\}\}\\in\\\{0\.001,0\.01,0\.1,1\.0\\\}:14\.9/1\.0014\.9/1\.00,14\.2/1\.0014\.2/1\.00,15\.3/1\.0015\.3/1\.00,18\.6/1\.0018\.6/1\.00\. Performance is insensitive to this solver regularization over three orders of magnitude \(mild degradation only atαcal=1\\alpha\_\{\\mathrm\{cal\}\}=1\); this ablation is distinct from the theoretical ridge parameterλ\\lambdain Corollary[1](https://arxiv.org/html/2607.08122#Thmcorollary1), and SNR thresholding remains the main calibration filter \(Remark[2](https://arxiv.org/html/2607.08122#Thmremark2)\)\.

## Appendix LAdditional Discussion, Limitations, and Future Work

#### Why generic fidelity is insufficient\.

Causal inference requires preserving conditional relationships such asY∣T,XY\\mid T,X, not only low\-dimensional distributional marginals\. A synthetic dataset that matchesP​\(X,T\)P\(X,T\)andP​\(X,Y\)P\(X,Y\)but distortsP​\(Y∣T,X\)P\(Y\\mid T,X\)can have good marginal fidelity and poor ATE accuracy\. The causal workload makes the target\-specific requirement explicit by measuring the moments that identify the orthogonal score in the chosen feature basis\.

#### Practical guidance\.

A practitioner should specify the target estimand, derive the corresponding orthogonal score and moment requirements, choose a feature mapϕ\\phithat approximates those moments, generate synthetic data with a fixed causal workload orCausal\-AIM, and report NA\+MI uncertainty\. On our benchmarks, richer feature maps helped throughout the tested range \(LLup to 20 bins on IHDP\), so the practical default is the richestϕ\\phithe budget supports, using SNR thresholding and the measuredCalGap\\mathrm\{CalGap\}as safeguards when moments approach the noise floor\.

#### Coverage–RMSE tradeoff\.

The experiments reveal a tradeoff: MST often leads on point RMSE at moderate\-to\-loose budgets, but its naive confidence intervals are far below nominal coverage at strict budgets\. Causal workload \+ NA\+MI is the conservative recommendation when calibrated uncertainty quantification is essential\.Causal\-AIM\+ NA\+MI is useful when lower RMSE is prioritized and the operating point is favorable, while hybrid causal \+ generic workloads can help when both point accuracy and coverage matter \(Appendix[K](https://arxiv.org/html/2607.08122#A11)\)\.

#### Coverage–privacy paradox\.

NA\+MI coverage can decrease asε\\varepsilonincreases because DP noise shrinks while workload approximation bias remains\. In ACIC, Causal \+ NA\+MI coverage drops from1\.001\.00atε≤1\\varepsilon\\leq 1to0\.900\.90atε=2\\varepsilon=2and0\.170\.17atε=5\\varepsilon=5, while the other three main benchmarks remain at0\.99−1\.000\.99\{\-\}1\.00\. The diagnostic ratioApprox^​\(ϕ\)/σNA\+MI\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)/\\sigma\_\{\\mathrm\{NA\+MI\}\}identifies this approximation\-dominated regime, and Appendix[F](https://arxiv.org/html/2607.08122#A6)proves a conservative bias\-aware interval that inflates variance byApprox^​\(ϕ\)2\\widehat\{\\mathrm\{Approx\}\}\(\\phi\)^\{2\}\.

#### Limitations\.

The implementation instantiates the framework on the AIM/Private\-PGM family, with MST as the generic comparator; the workload\-design principle should transfer to other select–measure–reconstruct generators, but we have not evaluated modern generator families beyond this class\. The feature mapϕ\\phiand discretization of continuous covariates introduce approximation error, especially when the true outcome surface is highly nonlinear\. Theorem[3](https://arxiv.org/html/2607.08122#Thmtheorem3)gives a useful moment\-level decomposition, but constants can be loose and the theorem does not fully analyze every downstream fitted\-nuisance routine run on sampled synthetic rows\. Finally, valid NA\+MI intervals can be wide at strict privacy budgets, especially on small benchmarks such as LaLonde/NSW\.

#### Future work\.

Natural extensions include structured graphical\-model solvers and convex relaxations for better calibration; continuous covariates through private kernel mean embeddings or random Fourier features; subgroup\-specific and CATE workloads; automated DP\-aware selection ofϕ\\phi; and information\-theoretic lower bounds linking overlap, workload dimension, and unavoidable privacy distortion\.