Faithful Federated Tabular Data Synthesis

1 Introduction

• Federated tabular synthesis remains limited, while existing approaches face substantial communication costs, fidelity concerns, and inefficiencies with mixed-type data.Deep models can require extensive communication; marginal methods may miss high-dimensional structure, and common discrete-variable encodings can increase dimension or compromise interpretability.Source: §1 · p. 2
• FITS estimates conditional CDFs through a two-step federated procedure and uses the probability chain rule to synthesize multivariate mixed-type tabular data.Clients first send weighted Hessian-type matrices; they then send local response-smoothed CDF contributions used to construct global conditional CDFs for inverse sampling.Source: §1 · p. 2
• FITS lifts discrete variables into a continuous space while preserving inversion to the original discrete domain without increasing data dimension.The framework also uses client-designed kernels and a penalty-adaptive kernel to accommodate client-specific characteristics and category mismatches after continualization.Source: §1 · pp. 2–3
• The paper characterizes continualized variables at the density and distribution-function levels and establishes uniform convergence for federated conditional-CDF estimation.It also develops a modified procedure to restore consistency near blockwise boundaries and establishes uniform consistency for the per-step CDF estimates used in chain-rule generation.Source: §1 · p. 3
• FITS improves downstream performance and achieves the best distributional fidelity among the compared methods across simulated and real datasets, regression and classification tasks, and three downstream models.The comparisons include HeteroFedSyn and FedTabDiff.Source: §1 · p. 3

2.1 Federated setup and notation

• The estimand is the conditional CDF F(z | x) = P(Z ≤ z | X = x), from which conditional functionals can be derived and samples generated by inverse-CDF sampling.The setup assumes observations across clients are i.i.d. draws from a common joint distribution.Source: §2.1 · p. 4
• Clients do not transmit raw samples, response vectors, or local design matrices; the estimator instead communicates aggregated design summaries and CDF contribution vectors.The setup allows synchronous communication between clients and a central server.Source: §2.1 · p. 4

2.2 Local-linear formulation of the conditional CDF

• Local linear regression estimates the conditional CDF as an intercept, expressible as a linear combination of smoothed response indicators with weights that depend on the query and design, but not on z.This separability enables the two-step federated procedure; local linear estimation also removes the leading covariate-density bias term and is bias-corrected at support boundaries.Source: §2.2 · pp. 4–5

3 The Proposed Method

• FITS combines federated conditional-CDF estimation and chain-rule sampling as its framework for continuous-data synthesis.The two-step aggregation procedure supplies the conditional CDF estimates used by the generation procedure.Source: §3 · p. 5, §3.1 · p. 5, §3.3 · p. 10

3.1 Two-step federated aggregation

• The federated estimator exactly reformulates the pooled local-linear estimator when the weights match, ridge regularization is absent, and the pooled design matrix is invertible.The procedure aggregates kernel-weighted Hessians, then client CDF estimates; its per-query communication is O(d^2 + L) per client, independent of local sample size.Source: §3.1 · p. 8

3.2 Client-designed kernel

• Clients may use fixed-bandwidth fitting on all local samples or KNN fitting on selected neighbors, with local standardization performed without transmitting client scales.Standardized coordinates determine kernel weights, while the local-linear design remains in the original coordinates to target the population conditional distribution there.Source: §3.2 · pp. 8–9, §3.1 · p. 8

3.3 Federated Synthetic Data Generation

• Synthetic records are generated coordinate by coordinate by inverse-sampling conditional CDFs given previously sampled coordinates.The first variable uses an unconditional CDF; estimated CDF values between grid points are linearly interpolated.Source: §3.3 · p. 10

4 Generalization to Mixed-Type Data

• The proposed continualization lifts discrete variables into a continuous space and permits exact inversion to the original discrete values without information loss.It is presented as a way to unify mixed-type data within a continuous framework for density-based methods.Source: §4 · pp. 10–11

4.1 Continualization of discrete variables

• Continualization spreads each discrete category's probability mass uniformly over its associated interval, preserving category mass and enabling exact recovery.The resulting density is piecewise constant and the distribution function is piecewise affine.Source: §4.1 · p. 12

4.2 Multidimensional continualization and block structure

• In multiple dimensions, the lifted density inherits conditional smoothness within each category block but may jump across block boundaries.The paper identifies these interfaces as a covariate jumping set; the blockwise structure supports its uniform-convergence analysis.Source: §4.2 · p. 13

4.3 An illustrative bivariate example

• In the binary example, lifting changes the CDF from stepwise in the discrete coordinate to piecewise affine, while inverse categorization recovers the original labels.Within each block, the lifted conditional functions are continuous in the continuous coordinate and constant in the lifted coordinate.Source: Figure 3 · p. 14, §4.3 · p. 15

4.4 Penalty-adaptive kernel

• Near interval boundaries, geometrically close observations can belong to different categories, so FITS centers discrete query coordinates and applies category-aware kernel penalties.The penalty discounts ordinal mismatches according to category distance and nominal mismatches by a coding-independent factor.Source: §4 · p. 11, §4.4 · pp. 15–16

5.1 Assumptions

• Under compact-domain geometry, second-order smoothness, and Gaussian-kernel regularity assumptions, the analysis extends standard local linear conditions to unbounded-support kernels.The first two assumptions support uniform kernel approximation and local linear bias control; clients share a common covariate bandwidth.Source: §5.1 · pp. 17–18

5.2 Uniform convergence under continuous covariates

• For continuous covariates, the pooled-sample local linear estimator has uniform convergence over both covariates and all response values, with error decomposed into bias and stochastic fluctuation.The local linear construction removes the first-order covariate bias; the pointwise corollary gives a bound useful for downstream applications.Source: §5.2 · p. 18

5.3 Regularity of the lifted joint and conditional laws

• Continualization gives the lifted joint and conditional distributions piecewise regularity rather than global smoothness, providing the conditions needed to extend uniform convergence to mixed-type variables.The conditional CDF and its continuous-covariate derivatives may differ across adjacent blocks.Source: §5.3 · p. 19, §5.3.2 · p. 20

5.3.1 Lifted joint density: blockwise continuity and one-sided limits

• The continualized joint density is constant within each discrete-variable block, has one-sided boundary limits, and is piecewise smooth, with jumps when adjacent blockwise density values differ.Within-block derivatives in continuous covariates inherit the regularity of the original density sections, while derivatives in the continualized coordinate vanish.Source: §5.3.1 · pp. 19–20

5.3.2 Lifted conditional distribution: blockwise C2 structure

• The lifted conditional CDF is invariant to the continualized covariate coordinate within each block and is exactly affine in the continualized response coordinate for discrete responses.These properties follow from spreading probability uniformly within each continualized cell and enable blockwise convergence analysis.Source: §5.3.2 · pp. 20–21

5.4 Uniform convergence under continualized mixed-type variates

• The mixed-type convergence analysis applies the continuous-covariate result blockwise, while representative points recover conditional distributions at covariate interfaces where direct uniform control is unavailable.The representative-point extension includes excluded interface neighborhoods and jumping points at the shrinking-interior rate.Source: §5.4 · pp. 21–22
• The mixed-type analysis establishes uniform convergence on shrinking interior blocks and extends it to covariate interfaces, including jumping points, by evaluating at interior representatives.For discrete responses, conditional probability masses can be estimated by finite differences of the conditional CDF estimator, with a strictly sharper rate.Source: §5.4 · pp. 22–23
• Theorem 5.4 and Corollaries 5.3–5.4 establish a uniform rate of O_p(w^2 + |log w_x|/(M w d_x)) over the full continualized covariate domain for both continuous- and discrete-response cases.The rate applies to the federated estimator developed in Section 3.Source: Define the conditional mass estimator · p. 23

Define the conditional mass estimator

• The federated estimator developed in Section 3 attains the uniform rate Op(w2 | log wx|/(Mwdx)), established jointly by Theorem 5.4 and Corollaries 5.3 and 5.4.The rate holds on the full continualized covariate domain in both the continuous- and discrete-response cases.Source: Define the conditional mass estimator · p. 23

6 Experiments

• The experiments use ten clients with a fixed imbalanced split, comparing four data-use modes across simulated and real mixed-type tasks.The split ranges from about 30% of training observations at the largest client to about 1% at the smallest; modes cover synthetic-only, local-only, pooled, and loss-reweighted data.Source: §6 · p. 23, §6.1.1 · p. 24, §6.1.3 · p. 25

6.2.1 Simulation design

• The simulation uses six mixed-type covariates and task-specific regression and classification responses; the real studies use Bike for regression and Adult for binary classification.On real data, parameter error is measured against a full-data reference estimator because the true downstream parameters are unavailable.Source: §6.2.1 · p. 25, §6.3.1 · p. 28

6.2.2 Baseline comparisons

• FITS-generated data generally outperforms the baselines; smaller clients benefit more, and M3/M4 generally outperform synthetic-only M1.For FT-Transformer classification, FITS improves accuracy across all clients, and M3/M4 also outperform local-only M2; on Adult logistic-regression parameter error, HeteroFedSyn performs better than FITS.Source: Table 2 · p. 25, §6.2.2 · p. 26, Table 3 · p. 26
• FITS-G and FITS-K have negligible performance differences across the downstream tasks, supporting the KNN-kernel scheme’s effectiveness.FITS-G uses a Gaussian-kernel scheme and FITS-K a KNN-kernel scheme.Source: §6.2.2 · p. 26
• M4 is an effective way to use FITS-G synthetic data when its synthetic-observation loss weight λ is chosen appropriately.M4 combines local and synthetic observations through a weighted loss; the experiments vary its parameters to assess data use.Source: §6.1.3 · p. 25, §6.2.2 · pp. 26–27

6.2.3 Distributional fidelity

• FITS-G has the lowest Energy Distance among the generators on the simulation data, indicating the best full-dimensional tabular distributional fidelity.Energy Distance is used to compare synthetic data with the full-data reference.Source: §6.2.3 · p. 27, Table 5 · p. 27

6.3.2 Bike dataset

• On Bike, FITS achieves the best performance across the regression tasks and models considered, while M4 with an appropriate λ is an effective way to combine synthetic and local data.The Energy Distance results also indicate that FITS more faithfully preserves the Bike joint distribution; M4 with λ = 1 is equivalent to M3 with r = ∞.Source: §6.3.2 · p. 28, Table 6 · p. 29, Table 7 · p. 29

6.3.3 Adult dataset

• On Adult, HeteroFedSyn has lower logistic-regression parameter error, the methods have comparable test classification accuracy, and FITS performs best with FT-Transformer.The authors suggest two-way marginals may explain HeteroFedSyn’s logistic-regression result, while higher-order dependencies may matter more for FT-Transformer; Energy Distance is consistent with FITS having the highest overall fidelity.Source: §6.3.3 · p. 30, Table 10 · p. 30, Table 11 · p. 30

7 Conclusion

• FITS combines a two-step federated local linear conditional-CDF procedure, discrete-variable continualization with an inverse mapping, and client-designed kernels for mixed-type data.The kernel designs include fixed-bandwidth and KNN-based schemes, plus a penalty-adaptive kernel for mismatched continualized covariate categories.Source: §7 · p. 31
• Continualization preserves probability mass while inducing piecewise continuous joint densities and conditional CDFs, and the federated conditional-CDF estimator has uniform convergence guarantees.The convergence result covers response and covariate directions under the continualized support structure and Gaussian-kernel smoothing.Source: §7 · p. 31
• FITS constructs its synthetic-data generator using the probability chain rule and inverse conditional-CDF sampling, then evaluates utility across several ways of combining local and synthetic data.The experiments assess downstream learning tasks.Source: §7 · p. 31
• For clients with limited data, synthetic-only training can outperform training solely on local observations; weighted M4 augmentation is also effective, with λ playing a crucial role.Utility depends on client sample size, downstream model capacity, and training mode.Source: §7 · pp. 31–32
• Future work includes principled weighting, characterizing when synthetic samples help or harm a client, stronger privacy guarantees, efficient grid construction, and chain-rule ordering strategies.These directions are proposed to clarify the statistical and practical scope of federated synthetic data.Source: §7 · p. 32

A Useful tools

• The appendix presents empirical-process tools used in Appendix B’s proofs without proof, referring readers to Xie [2023] for detailed arguments.It begins with lemmas on Euclidean function classes.Source: A Useful tools · p. 35
• A.1–A.3 state that bounded-variation function classes, uniformly bounded finite-dimensional function spaces, and products of Euclidean classes are Euclidean under the stated conditions.For products, the lemma also gives a uniform bound and coefficients for the resulting class.Source: A Useful tools · p. 35
• Under Assumptions 1 and 3, the eigenvalues of Ω(x, w_x) and Ξ(x, w_x) lie in [1/C, C] for all x and sufficiently small positive w_x.The bound is stated for a constant C > 0.Source: A Useful tools · p. 35

B Proof of Theorem 5.1

• Under matched kernel weights, the pooled centralized local linear estimator is the federated estimator, so the appendix’s rate results transfer to it.The proof works under Assumptions 1–3 with pooled sample size M = Σ_k m_k.Source: B Proof of Theorem 5.1 · p. 35

B.1 Setup and notation used throughout the proofs

• The proof decomposes estimation error into stochastic fluctuation around a deterministic pseudo-true parameter and smoothing-induced oracle bias, bounding the two components separately.The pseudo-true parameter converges to the target parameter as the response and covariate bandwidths shrink.Source: §B.1 · p. 37

B.2 Uniform bias bound

• The intercept’s oracle bias is O(w_x^2 + w_z^2), uniformly over z ∈ R and x ∈ X.Source: §B.2 · p. 42

B.3 Stochastic bound

• Under Assumptions 1–3, the proof establishes a uniform stochastic bound by controlling the empirical moment and design-matrix fluctuations over bounded Euclidean classes.It uses uniform invertibility of the population design matrix to translate those convergence rates into a bound for the estimator.Source: §B.3 · pp. 42–44
• With probability approaching one, the estimated local matrix eigenvalues lie uniformly in [1/(2C), 2C] over x ∈ X.The proof then uses this control to conclude its first claim; the intercept statement follows from the first coordinate of the vector bound.Source: §B.3 · p. 45

B.5 Proof of Lemma A.5

• The eigenvalues of Ξ(x, wx) are uniformly bounded above and below for x ∈ X and sufficiently small wx, using Gaussian-kernel integrability and the geometric condition in Assumption 1.Source: §B.5 · pp. 46–47

C.1 Proofs for the lifted joint and conditional laws

• The lifted-law proofs proceed from blockwise C0 regularity to Cr regularity, then derive the conditional-distribution structure using a probability-law identity.These steps correspond to Theorem 5.2, Corollary 5.2, and Theorem 5.3, respectively.Source: §C.1 · p. 47

C.2 Proof of Theorem 5.2

• The density g is continuous in the interior of the lifted support, and its discontinuity set is contained in the partition-boundary set.Within each block, g admits a continuous extension to the block’s closure.Source: §C.2 · pp. 48–49
• Within each block, the density’s continuous extension is determined by the corresponding original density value divided by the block measure.This yields piecewise continuity across the lifted support.Source: §C.2 · pp. 48–49
• The lifted joint density g is obtained from the continualized distribution and integrates to one, confirming it is a probability density on X_c × Y.The proof derives the density by disintegrating over the partition and using the conditional uniform distribution within each block.Source: §C.2 · pp. 49–50
• At a partition boundary, one-sided limits from adjacent blocks can differ, producing a jump when the corresponding blockwise density values differ.The one-sided limits are specified by the block’s density value divided by its interval measure.Source: §C.2 · pp. 49, 51

• Within each block interior, the lifted density is C^r in the relative product sense.The result also gives interior derivative formulas.Source: §C.3 · p. 52
• Each block restriction admits a C^r extension to its closed block, but adjacent extensions need not agree on shared boundaries.Source: §C.3 · pp. 52, 54
• Across a block boundary, one-sided limits of covariate derivatives coincide exactly when the corresponding blockwise derivative values are equal.Source: §C.3 · pp. 52, 54

C.4 Proof of Theorem 5.3

• Within each covariate block, the lifted CDF is independent of continualized discrete coordinates; at covariate boundaries, blockwise one-sided C2 limits exist but values and derivatives may differ.The continuous-response CDF is C2 within each block, with derivatives in the continualized discrete coordinates equal to zero.Source: §C.4 · pp. 54–55
• For continualized discrete responses, the conditional density is valid and the CDF is affine within response intervals, continuous across their interfaces, but its derivative may jump.The CDF is also independent of the continualized discrete covariates within each block.Source: §C.4 · pp. 55, 57

C.5 Proof of Theorem 5.4

• Theorem 5.4’s proof controls cross-block Gaussian-kernel contributions as asymptotically negligible and combines within-block expansions with stochastic bounds.The proof treats observations inside and outside the query block separately.Source: §C.5 · pp. 59, 61
• For discrete responses, symmetric smoothing cancels bias exactly within response intervals, while Gaussian-tail bounds control contributions from outside those intervals.The proof combines this response-smoothing bound with covariate expansions and stochastic bounds.Source: §C.5 · pp. 60–61

C.6 Proof of Corollary 5.4

• Combining this bound with (60) and (36) proves (37). For ϵ, the same result follows by evaluating bpi at the corresponding interior representative ˜x◦ ϵ.The conditional probability masses are invariant in y within each continualization block.Source: §C.6 · p. 62

D.1 Data usage modes for downstream learning

• The downstream comparison framework distinguishes synthetic-only, local-only, and mixed-data learning, with mixed-data learning pooling synthetic observations and local data.The local-only mode is the non-federated baseline; the mixed-data synthetic subset is sized relative to the client’s local sample.Source: §D.1 · pp. 62–63
• In the sample-count approach, synthetic and local observations contribute equally to empirical loss, while r controls the number of synthetic observations used.Setting r = ∞ uses all synthetic data.Source: §D.1 · p. 63
• Mode 4 adjusts the relative influence of synthetic and local data at the loss-function level through a tunable synthetic-loss weight λ ∈ [0, +∞).The weight controls the trade-off between the two data sources.Source: §D.1 · p. 63
• The four modes replace local data with synthetic data, ignore synthetic data, control its sample count relative to m_k, or tune its loss contribution with λ.Mode 4 combines synthetic and local data; λ = 0 recovers Mode 2, while λ = 1 recovers Mode 3 using the full synthetic dataset.Source: §D.1 · p. 63

D.2 Simulation data

• The simulation generates covariates with dependencies from the binary variable through the Gaussian pair, ordinal variable, and response, while a third continuous covariate is independent.Table S1 covers chain order, task-specific parameters, and evaluation metrics.Source: §D.2 · p. 64, Table S1 · p. 64

D.3 Real data

• Table S2 lists preprocessing and variable-type assignments for the UCI datasets.Source: Table S2 · p. 65

E Baseline methods

• FITS is compared with HeteroFedSyn and FedTabDiff, with FT-Transformer included as an additional downstream evaluation model.Source: E Baseline methods · p. 65

E.1 HeteroFedSyn

• The HeteroFedSyn implementation uses an adaptive marginal-selection variant and a deliberately weak privacy setting to emphasize synthesis utility.Pair selection uses dependency scores relative to expected release-noise error.Source: §E.1 · pp. 65–66

E.2 FedTabDiff

• FedTabDiff trains a denoising diffusion model on encoded mixed-type data, with all clients participating and sample-size-weighted FedAvg.The implementation uses 500 diffusion steps and 1,000 FedAvg rounds.Source: §E.2 · p. 66

• Evaluation models use covariates only, standardizing continuous features and mapping nominal and ordinal predictors to categorical indices.Source: §E.3 · p. 67
• The FT-Transformer uses 32-dimensional feature tokens and a learned [CLS] token, followed by two Transformer blocks and a scalar-output head.Each block uses four attention heads and a width-64 feed-forward network.Source: §E.3 · p. 67
• Models are trained with identical settings across methods, and performance is reported as original-scale regression test MSE or classification test accuracy.Binary predictions use a 0.5 probability threshold.Source: §E.3 · p. 67

• Simulation tables compare FITS-G and FITS-K on linear-regression parameter error, logistic-regression parameter error, and FT-Transformer classification accuracy.Source: Table S3 · p. 67, Table S4 · p. 68, Table S5 · p. 68
• Simulation tables report FITS-G results across modes for linear-regression parameter error, logistic-regression parameter error, and FT-Transformer classification accuracy.Source: Table S6 · p. 68, Table S7 · p. 68, Table S8 · p. 69
• Bike tables compare FITS-G and FITS-K on linear-regression parameter error, linear-regression test MSE, and FT-Transformer regression MSE.Source: Table S9 · p. 69, Table S10 · p. 69, Table S11 · p. 69
• Bike tables report FITS-G results across modes for linear-regression parameter error, linear-regression test MSE, and FT-Transformer regression test MSE.Source: Table S12 · p. 70, Table S13 · p. 70, Table S14 · p. 70
• Adult tables compare FITS-G and FITS-K on logistic-regression parameter error and test accuracy, plus FT-Transformer classification test accuracy.Source: Table S15 · p. 70, Table S16 · p. 71, Table S17 · p. 71
• Adult tables report FITS-G results across modes for logistic-regression parameter error and test accuracy, plus FT-Transformer classification test accuracy.Source: Table S18 · p. 71, Table S19 · p. 71, Table S20 · p. 72
• Simulation tables report linear-regression test MSE, logistic-regression test accuracy, and FT-Transformer regression test MSE.Source: Table S21 · p. 72, Table S22 · p. 72, Table S23 · p. 72
• Additional simulation tables compare FITS-G with FITS-K, and report FITS-G results across modes, for regression MSE and logistic accuracy.Source: Table S24 · p. 73, Table S25 · p. 73, Table S26 · p. 73