Faithful Federated Tabular Data Synthesis

Zilong Cao; Hai Zhang; Xuan Bi

cs.LGOct 24, 2026

Faithful Federated Tabular Data Synthesis

Authors: Zilong Cao, Hai Zhang, Xuan Bi

Organizations: School of Mathematics, Northwest University · Carlson School of Management, University of Minnesota

Abstract

Synthetic data is an important data source in the AI era. This paper studies the generation of mixed-type tabular synthetic data in federated settings, where two key challenges arise: ensuring distributional fidelity and effectively handling mixed-type data. We propose Federated Integrative Two-Step Synthesis (FITS), a framework for mixed-type tabular data synthesis. At its core is a two-step federated algorithm for estimating conditional cumulative distribution functions (CDFs) using local linear regression. Building on this algorithm, we develop a federated synthesis procedure based on the probability chain rule to ensure distributional fidelity. To handle mixed-type data, FITS introduces a continualization method that lifts discrete variables into a continuous space and recovers their discrete distributions without loss at both the density-function and CDF levels. This transformation provides a unified continuous representation of mixed-type variables, facilitating the application of local linear regression and synthetic data generation. Our theoretical analysis characterizes the conditional density and cumulative distribution functions of the continualized mixed-type variables and establishes the uniform convergence of the resulting federated conditional CDF estimators. Experiments demonstrate that FITS-generated synthetic data improves downstream task performance for all clients, with the benefits depending on how the synthetic data is used.

Figures & tables

Figure 2: FITS: generation of a single synthetic sample by the chain rule.

\begin{cases}\widehat{\bm{F}}(\bm{g}_{1})\text{ by Eq.~\eqref{eq:first_dim_kde}},&j=1,\\[2.84526pt] \widehat{\bm{F}}(\bm{g}_{j}\mid\hat{{\bm{x}}}_{b,:j-1}=(\hat{x}_{b,1},...,\hat{x}_{b,j-1}))\text{ by Algorithm~\ref{alg:two_step_fllr}},&j>1.\end{cases}

Algorithm 2 FITS: Federated Synthetic Data Generation

Panel of Figure 3 — Figure 3: Illustration of binary continualization in a bivariate model with one continuous $X_{1}$ and one binary variable $X_{2}\in\{0,1\}$ . Panel (a) shows the lifted density $g(x_{1},y)=f(x_{1},0)\mathbb{I}\{{y\in(-1,0]}\}+f(x_{1},1)\mathbb{I}\{y\in(0,1]\}$ , where $f(x_{1},0)=pf(x_{1}\mid 0)$ and $f(x_{1},1)=(1-p)f(x_{1}\mid 1)$ . The line $y=0$ marks the internal block interface across which the lifted density may jump. Panel (b) shows the original joint CDF $F_{\mathrm{orig}}(x_{1},t)=\mathbb{P}(X_{1}\leq x_{1},X_{2}\leq t)$ , which is stepwise in the discrete coordinate. Panel (c) shows the lifted CDF after setting $Y=X_{2}-U$ , where $U\sim\mathrm{Unif}(0,1)$ is independent of $(X_{1},X_{2})$ ; on $(-1,0]$ and $(0,1]$ , it is affine in $y$ , with forms $p\bm{\phi}_{0}(x_{1})(1+y)$ and $p\bm{\phi}_{0}(x_{1})+(1-p)\bm{\phi}_{1}(x_{1})y$ , respectively.

	Simulation	Bike Sharing	Adult Income
Task	Regression and classification	Regression	Classification
Data source	Synthetic DGP	UCI Bike Sharing	UCI Adult Income
Full sample size	$10{,}000$	$17{,}379$	$45{,}222$
Training sample size	$6{,}000$	$13{,}903$	$36{,}177$
Test sample size	$4{,}000$	$3{,}476$	$9{,}045$
Generated sample size	$3{,}000$	$5{,}000$	$20{,}000$

Table 1: Experimental designs for the simulation and real-data studies.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)
	HeteroFedSyn	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)	4.6669 (0.3037)
	FedTabDiff	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)	1.3846 (0.5311)
M2	None	0.0751 (0.0329)	0.0930 (0.0385)	0.1302 (0.0568)	0.1446 (0.0698)	0.1310 (0.0616)	0.1541 (0.0704)	0.1751 (0.0725)	0.2063 (0.0856)	0.2198 (0.1036)	0.4730 (0.1856)
M3	FITS-G	0.1060 (0.0435)	0.1225 (0.0549)	0.1350 (0.0622)	0.1406 (0.0620)	0.1405 (0.0638)	0.1413 (0.0629)	0.1438 (0.0654)	0.1467 (0.0663)	0.1501 (0.0669)	0.1545 (0.0682)

Table 2: Simulation: linear-regression parameter estimation error.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)
	HeteroFedSyn	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)	3.7460 (0.5029)
	FedTabDiff	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)	2.1627 (0.7893)
M2	None	0.3923 (0.1852)	0.5028 (0.2397)	0.6821 (0.3112)	0.8635 (0.3364)	0.7619 (0.3377)	0.8787 (0.4005)	0.8757 (0.4659)	1.0606 (0.5028)	1.1961 (0.6224)	10.6726 (16.6535)
M3	FITS-G	0.5117 (0.1249)	0.5694 (0.1246)	0.6281 (0.1209)	0.6370 (0.1343)	0.6446 (0.1351)	0.6602 (0.1370)	0.6615 (0.1381)	0.6714 (0.1420)	0.6681 (0.1364)	0.6983 (0.1363)

Table 3: Simulation: logistic-regression parameter estimation error.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)
	HeteroFedSyn	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)	0.7650 (0.0221)
	FedTabDiff	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)	0.8582 (0.0075)
M2	None	0.8547 (0.0101)	0.8406 (0.0093)	0.8280 (0.0105)	0.8278 (0.0115)	0.8237 (0.0127)	0.8242 (0.0123)	0.8259 (0.0122)	0.8246 (0.0135)	0.8234 (0.0134)	0.8193 (0.0235)
M3	FITS-G	0.8692 (0.0080)	0.8672 (0.0070)	0.8644 (0.0074)	0.8645 (0.0075)	0.8629 (0.0082)	0.8638 (0.0072)	0.8635 (0.0084)	0.8634 (0.0081)	0.8617 (0.0079)	0.8603 (0.0081)

Table 4: Simulation: FT-Transformer classification accuracy on the test data.

Variables	HeteroFedSyn	FedTabDiff	FITS-G
$X$	0.0702 (0.0084)	0.1215 (0.0493)	0.0392 (0.0079)
$(X,Y_{\mathrm{lin}})$	0.1074 (0.0073)	0.1260 (0.0492)	0.0429 (0.0076)
$(X,Y_{\mathrm{log}})$	0.0752 (0.0074)	0.1231 (0.0490)	0.0409 (0.0079)

Table 5: Simulation: Energy Distance between synthetic and full data.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)
	HeteroFedSyn	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)	0.3068 (0.0503)
	FedTabDiff	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)	0.3077 (0.0708)
M2	None	0.1059 (0.0711)	0.1397 (0.0924)	0.2044 (0.1343)	0.2043 (0.1359)	0.2331 (0.1749)	0.2088 (0.1111)	0.2900 (0.1840)	0.3137 (0.1962)	0.3291 (0.2270)	0.6422 (0.4136)
M3	FITS-G	0.0933 (0.0370)	0.0976 (0.0402)	0.1138 (0.0442)	0.1117 (0.0460)	0.1126 (0.0413)	0.1114 (0.0425)	0.1186 (0.0406)	0.1169 (0.0424)	0.1173 (0.0399)	0.1224 (0.0445)

Table 6: Bike: linear-regression parameter estimation error.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)
	HeteroFedSyn	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)	0.7025 (0.0331)
	FedTabDiff	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)	0.6709 (0.0267)
M2	None	0.6099 (0.0189)	0.6106 (0.0189)	0.6135 (0.0193)	0.6140 (0.0190)	0.6157 (0.0189)	0.6152 (0.0196)	0.6172 (0.0198)	0.6192 (0.0200)	0.6216 (0.0209)	0.6582 (0.0274)
M3	FITS-G	0.6098 (0.0192)	0.6101 (0.0192)	0.6109 (0.0194)	0.6107 (0.0195)	0.6110 (0.0194)	0.6110 (0.0194)	0.6111 (0.0194)	0.6111 (0.0195)	0.6113 (0.0194)	0.6116 (0.0194)

Table 7: Bike: linear-regression MSE on the test data.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)
	HeteroFedSyn	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)	0.8725 (0.0907)
	FedTabDiff	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)	0.6221 (0.0552)
M2	None	0.0687 (0.0060)	0.0727 (0.0041)	0.0896 (0.0065)	0.0944 (0.0056)	0.0990 (0.0070)	0.1079 (0.0079)	0.1154 (0.0107)	0.1262 (0.0142)	0.1522 (0.0192)	0.4313 (0.0741)
M3	FITS-G	0.1150 (0.0122)	0.1324 (0.0143)	0.1636 (0.0177)	0.1697 (0.0191)	0.1732 (0.0152)	0.1790 (0.0210)	0.1840 (0.0212)	0.1908 (0.0158)	0.1984 (0.0210)	0.2213 (0.0199)

Table 8: Bike: FT-Transformer regression MSE on the test data.

Variables	HeteroFedSyn	FedTabDiff	FITS-G
$X$	0.0903 (0.0093)	0.1247 (0.0260)	0.0595 (0.0042)
$X,Y$	0.1008 (0.0072)	0.1244 (0.0253)	0.0599 (0.0044)

Table 9: Bike: Energy Distance between synthetic and full real data.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)
	HeteroFedSyn	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)	0.7452 (0.2609)
	FedTabDiff	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)	2.3245 (0.1224)
M2	None	0.2712 (0.1082)	0.2810 (0.1239)	0.4893 (0.2531)	0.5538 (0.2660)	0.5820 (0.2656)	0.5930 (0.2823)	0.6766 (0.2996)	0.7805 (0.3195)	0.7778 (0.4350)	1.6560 (0.8013)
M3	FITS-G	0.7996 (0.1013)	0.8844 (0.1078)	0.9800 (0.1166)	0.9865 (0.1158)	0.9973 (0.1187)	1.0069 (0.1231)	1.0121 (0.1186)	1.0249 (0.1155)	1.0422 (0.1198)	1.0822 (0.1207)

Table 10: Adult: logistic-regression parameter estimation error.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)
	HeteroFedSyn	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)	0.8179 (0.0036)
	FedTabDiff	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)	0.7821 (0.0083)
M2	None	0.8204 (0.0031)	0.8207 (0.0032)	0.8204 (0.0036)	0.8199 (0.0034)	0.8202 (0.0028)	0.8194 (0.0038)	0.8196 (0.0034)	0.8195 (0.0033)	0.8189 (0.0038)	0.8131 (0.0054)
M3	FITS-G	0.8201 (0.0029)	0.8198 (0.0029)	0.8195 (0.0032)	0.8193 (0.0030)	0.8193 (0.0030)	0.8192 (0.0029)	0.8192 (0.0029)	0.8190 (0.0030)	0.8190 (0.0030)	0.8188 (0.0030)

Table 11: Adult: logistic-regression classification accuracy on the test data.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)
	HeteroFedSyn	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)	0.8291 (0.0079)
	FedTabDiff	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)	0.8333 (0.0065)
M2	None	0.8513 (0.0045)	0.8506 (0.0040)	0.8394 (0.0076)	0.8353 (0.0086)	0.8304 (0.0081)	0.8290 (0.0079)	0.8231 (0.0091)	0.8176 (0.0068)	0.8098 (0.0075)	0.8015 (0.0102)
M3	FITS-G	0.8513 (0.0042)	0.8513 (0.0039)	0.8498 (0.0043)	0.8498 (0.0044)	0.8496 (0.0039)	0.8493 (0.0045)	0.8496 (0.0044)	0.8491 (0.0045)	0.8474 (0.0053)	0.8485 (0.0048)

Table 12: Adult: FT-Transformer classification accuracy on the test data.

Variables	HeteroFedSyn	FedTabDiff	FITS-G
$X$	0.0551 (0.0017)	0.2021 (0.0605)	0.0543 (0.0023)
$X,Y$	0.0552 (0.0017)	0.2020 (0.0594)	0.0539 (0.0023)

Table 13: Adult: Energy Distance between synthetic and full real data.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Specification	Description
Covariates	$(X_{B},X_{G_{1}},X_{G_{2}},X_{O},X_{L},X_{G_{3}})$	Chain order used to generate the synthetic covariate table.
Linear task	$(X_{B},X_{G_{1}},X_{G_{2}},X_{O},X_{L},X_{G_{3}},Y_{\rm lin})$	The response $Y_{\rm lin}$ is generated after the covariates.
Logistic task	$(X_{B},X_{G_{1}},X_{G_{2}},X_{O},X_{L},X_{G_{3}},Y_{\rm log})$	The response $Y_{\rm log}$ is generated after the same covariates.
Parameter order	$(\mathrm{intercept},X_{B},X_{G_{1}},X_{G_{2}},X_{O},X_{L},$ $X_{G_{3}})$	Coefficients are aligned with this ordering.
True linear-model parameter $\beta^{\star}$	$(3.75,1.5,1.2,-0.8,-1.1,0.5,0.7)^{\top}$	Ground-truth coefficient vector for the linear response model.
True logistic-model parameter $\beta^{\star}$	$(0,1.5,1.2,-0.8,-1.1,0.5,0.7)^{\top}$	Ground-truth coefficient vector for the logistic response model.

Appendix

Table S1 : Chain order, task-specific parameters, and evaluation metrics.

	Bike Sharing	Adult Income
Response	$Y=$ cnt , standardized	$Y=\mathbf{1}\{\texttt{income}>\texttt{50K}\}$
Full sample size	$17{,}379$	$45{,}222$
Train/test split	$13{,}903/3{,}476$	$36{,}177/9{,}045$
Variable mapping	$X_{1}$ : temp $X_{2}$ : atemp $X_{3}$ : hum $X_{4}$ : windspeed $X_{5}$ : season $X_{6}$ : yr $X_{7}$ : mnth $X_{8}$ : hr $X_{9}$ : holiday $X_{10}$ : weekday $X_{11}$ : workingday $X_{12}$ : weathersit	$X_{1}$ : age $X_{2}$ : fnlwgt $X_{3}$ : capital-gain $X_{4}$ : capital-loss $X_{5}$ : hours-per-week $X_{6}$ : education-num $X_{7}$ : workclass $X_{8}$ : education $X_{9}$ : marital-status $X_{10}$ : occupation $X_{11}$ : relationship $X_{12}$ : race $X_{13}$ : sex $X_{14}$ : native-country
Continuous predictors	$X_{1},X_{2},X_{3},X_{4}$	$X_{1},X_{2},X_{3},X_{4},X_{5}$
Discrete predictors	$X_{5},X_{6},X_{7},X_{8},X_{9},X_{10},X_{11},X_{12}$	$X_{6},X_{7},X_{8},X_{9},X_{10},X_{11},X_{12},X_{13},X_{14}$

Appendix

Table S2 : Preprocessing and variable-type assignments for the UCI datasets.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)
M1	FITS-K	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)	0.1721 (0.0580)
M2	None	0.0751 (0.0329)	0.0930 (0.0385)	0.1302 (0.0568)	0.1446 (0.0698)	0.1310 (0.0616)	0.1541 (0.0704)	0.1751 (0.0725)	0.2063 (0.0856)	0.2198 (0.1036)	0.4730 (0.1856)
M3	FITS-G	0.1060 (0.0435)	0.1225 (0.0549)	0.1350 (0.0622)	0.1406 (0.0620)	0.1405 (0.0638)	0.1413 (0.0629)	0.1438 (0.0654)	0.1467 (0.0663)	0.1501 (0.0669)	0.1545 (0.0682)
M3	FITS-K	0.1140 (0.0398)	0.1315 (0.0493)	0.1462 (0.0526)	0.1500 (0.0538)	0.1511 (0.0542)	0.1522 (0.0547)	0.1565 (0.0560)	0.1584 (0.0554)	0.1635 (0.0550)	0.1680 (0.0554)

Appendix

Table S3 : Simulation: linear-regression parameter estimation error (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)
M1	FITS-K	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)	0.7289 (0.1278)
M2	None	0.3923 (0.1852)	0.5028 (0.2397)	0.6821 (0.3112)	0.8635 (0.3364)	0.7619 (0.3377)	0.8787 (0.4005)	0.8757 (0.4659)	1.0606 (0.5028)	1.1961 (0.6224)	10.6726 (16.6535)
M3	FITS-G	0.5117 (0.1249)	0.5694 (0.1246)	0.6281 (0.1209)	0.6370 (0.1343)	0.6446 (0.1351)	0.6602 (0.1370)	0.6615 (0.1381)	0.6714 (0.1420)	0.6681 (0.1364)	0.6983 (0.1363)
M3	FITS-K	0.5252 (0.1168)	0.5823 (0.1136)	0.6448 (0.1178)	0.6529 (0.1194)	0.6603 (0.1237)	0.6752 (0.1200)	0.6779 (0.1286)	0.6896 (0.1315)	0.6856 (0.1328)	0.7172 (0.1244)

Appendix

Table S4 : Simulation: logistic-regression parameter estimation error (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)
M1	FITS-K	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)	0.8603 (0.0082)
M2	None	0.8547 (0.0101)	0.8406 (0.0093)	0.8280 (0.0105)	0.8278 (0.0115)	0.8237 (0.0127)	0.8242 (0.0123)	0.8259 (0.0122)	0.8246 (0.0135)	0.8234 (0.0134)	0.8193 (0.0235)
M3	FITS-G	0.8692 (0.0080)	0.8672 (0.0070)	0.8644 (0.0074)	0.8645 (0.0075)	0.8629 (0.0082)	0.8638 (0.0072)	0.8635 (0.0084)	0.8634 (0.0081)	0.8617 (0.0079)	0.8603 (0.0081)
M3	FITS-K	0.8695 (0.0084)	0.8671 (0.0076)	0.8638 (0.0080)	0.8656 (0.0078)	0.8637 (0.0076)	0.8627 (0.0085)	0.8640 (0.0077)	0.8633 (0.0078)	0.8608 (0.0083)	0.8615 (0.0080)

Appendix

Table S5 : Simulation: FT-Transformer classification accuracy on the test data (FITS-G vs FITS-K).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)	0.1584 (0.0698)
M2	–	0.0751 (0.0329)	0.0930 (0.0385)	0.1302 (0.0568)	0.1446 (0.0698)	0.1310 (0.0616)	0.1541 (0.0704)	0.1751 (0.0725)	0.2063 (0.0856)	0.2198 (0.1036)	0.4730 (0.1856)
M3	$r=0$	0.0751 (0.0329)	0.0930 (0.0385)	0.1302 (0.0568)	0.1446 (0.0698)	0.1310 (0.0616)	0.1541 (0.0704)	0.1751 (0.0725)	0.2063 (0.0856)	0.2198 (0.1036)	0.4730 (0.1856)
	$r=0.5$	0.0938 (0.0376)	0.1082 (0.0464)	0.1364 (0.0651)	0.1582 (0.0761)	0.1490 (0.0804)	0.1616 (0.0748)	0.1836 (0.0678)	0.2218 (0.1041)	0.2308 (0.0934)	0.4392 (0.2013)
	$r=1$	0.0999 (0.0401)	0.1184 (0.0481)	0.1491 (0.0679)	0.1544 (0.0687)	0.1642 (0.0853)	0.1580 (0.0759)	0.1823 (0.0736)	0.2009 (0.1022)	0.2129 (0.0876)	0.3802 (0.1548)

Appendix

Table S6 : Simulation: linear-regression parameter estimation error (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)	0.7111 (0.1401)
M2	–	0.3923 (0.1852)	0.5028 (0.2397)	0.6821 (0.3112)	0.8635 (0.3364)	0.7619 (0.3377)	0.8787 (0.4005)	0.8757 (0.4659)	1.0606 (0.5028)	1.1961 (0.6224)	10.6726 (16.6535)
M3	$r=0$	0.3923 (0.1852)	0.5028 (0.2397)	0.6821 (0.3112)	0.8635 (0.3364)	0.7619 (0.3377)	0.8787 (0.4005)	0.8757 (0.4659)	1.0606 (0.5028)	1.1961 (0.6224)	10.6726 (16.6535)
	$r=0.5$	0.4041 (0.1665)	0.4679 (0.1653)	0.5697 (0.2247)	0.6907 (0.2742)	0.6489 (0.2302)	0.6532 (0.2362)	0.7458 (0.3929)	0.8008 (0.3280)	0.8787 (0.3414)	3.0192 (3.4075)
	$r=1$	0.4546 (0.1308)	0.4850 (0.1498)	0.5876 (0.2121)	0.6201 (0.2102)	0.6232 (0.2326)	0.6382 (0.2552)	0.6440 (0.3432)	0.7573 (0.2600)	0.7669 (0.2690)	1.7311 (0.8331)

Appendix

Table S7 : Simulation: logistic-regression parameter estimation error (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)	0.8608 (0.0079)
M2	–	0.8547 (0.0101)	0.8406 (0.0093)	0.8280 (0.0105)	0.8278 (0.0115)	0.8237 (0.0127)	0.8242 (0.0123)	0.8259 (0.0122)	0.8246 (0.0135)	0.8234 (0.0134)	0.8193 (0.0235)
M3	$r=0$	0.8547 (0.0101)	0.8406 (0.0093)	0.8280 (0.0105)	0.8278 (0.0115)	0.8237 (0.0127)	0.8242 (0.0123)	0.8259 (0.0122)	0.8246 (0.0135)	0.8234 (0.0134)	0.8193 (0.0235)
	$r=0.5$	0.8639 (0.0067)	0.8486 (0.0097)	0.8306 (0.0111)	0.8251 (0.0121)	0.8214 (0.0137)	0.8187 (0.0139)	0.8210 (0.0114)	0.8208 (0.0128)	0.8179 (0.0137)	0.8115 (0.0229)
	$r=1$	0.8661 (0.0070)	0.8566 (0.0088)	0.8323 (0.0111)	0.8295 (0.0097)	0.8232 (0.0111)	0.8191 (0.0111)	0.8206 (0.0116)	0.8167 (0.0138)	0.8120 (0.0160)	0.8078 (0.0217)

Appendix

Table S8 : Simulation: FT-Transformer classification accuracy on the test data (FITS-G: all modes).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)
M1	FITS-K	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)	0.1577 (0.0603)
M2	None	0.1059 (0.0711)	0.1397 (0.0924)	0.2044 (0.1343)	0.2043 (0.1359)	0.2331 (0.1749)	0.2088 (0.1111)	0.2900 (0.1840)	0.3137 (0.1962)	0.3291 (0.2270)	0.6422 (0.4136)
M3	FITS-G	0.0933 (0.0370)	0.0976 (0.0402)	0.1138 (0.0442)	0.1117 (0.0460)	0.1126 (0.0413)	0.1114 (0.0425)	0.1186 (0.0406)	0.1169 (0.0424)	0.1173 (0.0399)	0.1224 (0.0445)
M3	FITS-K	0.1095 (0.0516)	0.1169 (0.0508)	0.1347 (0.0557)	0.1350 (0.0556)	0.1377 (0.0552)	0.1373 (0.0568)	0.1428 (0.0572)	0.1440 (0.0595)	0.1460 (0.0563)	0.1551 (0.0602)

Appendix

Table S9 : Bike: linear-regression parameter estimation error (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)
M1	FITS-K	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)	0.6187 (0.0184)
M2	None	0.6099 (0.0189)	0.6106 (0.0189)	0.6135 (0.0193)	0.6140 (0.0190)	0.6157 (0.0189)	0.6152 (0.0196)	0.6172 (0.0198)	0.6192 (0.0200)	0.6216 (0.0209)	0.6582 (0.0274)
M3	FITS-G	0.6098 (0.0192)	0.6101 (0.0192)	0.6109 (0.0194)	0.6107 (0.0195)	0.6110 (0.0194)	0.6110 (0.0194)	0.6111 (0.0194)	0.6111 (0.0195)	0.6113 (0.0194)	0.6116 (0.0194)
M3	FITS-K	0.6119 (0.0185)	0.6130 (0.0186)	0.6151 (0.0184)	0.6153 (0.0186)	0.6158 (0.0184)	0.6158 (0.0185)	0.6162 (0.0185)	0.6166 (0.0185)	0.6169 (0.0184)	0.6182 (0.0185)

Appendix

Table S10 : Bike: linear-regression test MSE (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)
M1	FITS-K	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)	0.2826 (0.0270)
M2	None	0.0687 (0.0060)	0.0727 (0.0041)	0.0896 (0.0065)	0.0944 (0.0056)	0.0990 (0.0070)	0.1079 (0.0079)	0.1154 (0.0107)	0.1262 (0.0142)	0.1522 (0.0192)	0.4313 (0.0741)
M3	FITS-G	0.1150 (0.0122)	0.1324 (0.0143)	0.1636 (0.0177)	0.1697 (0.0191)	0.1732 (0.0152)	0.1790 (0.0210)	0.1840 (0.0212)	0.1908 (0.0158)	0.1984 (0.0210)	0.2213 (0.0199)
M3	FITS-K	0.1253 (0.0128)	0.1524 (0.0140)	0.1911 (0.0196)	0.1988 (0.0215)	0.2061 (0.0163)	0.2148 (0.0225)	0.2220 (0.0261)	0.2336 (0.0237)	0.2414 (0.0241)	0.2753 (0.0256)

Appendix

Table S11 : Bike: FT-Transformer regression MSE on the test data (FITS-G vs FITS-K).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)	0.1240 (0.0443)
M2	–	0.1059 (0.0711)	0.1397 (0.0924)	0.2044 (0.1343)	0.2043 (0.1359)	0.2331 (0.1749)	0.2088 (0.1111)	0.2900 (0.1840)	0.3137 (0.1962)	0.3291 (0.2270)	0.6422 (0.4136)
M3	$r=0$	0.1059 (0.0711)	0.1397 (0.0924)	0.2044 (0.1343)	0.2043 (0.1359)	0.2331 (0.1749)	0.2088 (0.1111)	0.2900 (0.1840)	0.3137 (0.1962)	0.3291 (0.2270)	0.6422 (0.4136)
	$r=0.5$	0.0872 (0.0571)	0.1019 (0.0543)	0.1626 (0.0885)	0.1498 (0.0676)	0.1713 (0.1141)	0.1391 (0.0711)	0.1881 (0.1157)	0.2089 (0.1124)	0.2287 (0.1309)	0.4503 (0.2013)
	$r=1$	0.0871 (0.0389)	0.0983 (0.0587)	0.1493 (0.0802)	0.1302 (0.0707)	0.1179 (0.0663)	0.1328 (0.0521)	0.1719 (0.0813)	0.1727 (0.0921)	0.1821 (0.0818)	0.3670 (0.1706)

Appendix

Table S12 : Bike: linear-regression parameter estimation error (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)	0.6117 (0.0195)
M2	–	0.6099 (0.0189)	0.6106 (0.0189)	0.6135 (0.0193)	0.6140 (0.0190)	0.6157 (0.0189)	0.6152 (0.0196)	0.6172 (0.0198)	0.6192 (0.0200)	0.6216 (0.0209)	0.6582 (0.0274)
M3	$r=0$	0.6099 (0.0189)	0.6106 (0.0189)	0.6135 (0.0193)	0.6140 (0.0190)	0.6157 (0.0189)	0.6152 (0.0196)	0.6172 (0.0198)	0.6192 (0.0200)	0.6216 (0.0209)	0.6582 (0.0274)
	$r=0.5$	0.6095 (0.0191)	0.6102 (0.0193)	0.6125 (0.0195)	0.6123 (0.0196)	0.6133 (0.0190)	0.6134 (0.0192)	0.6144 (0.0196)	0.6156 (0.0191)	0.6176 (0.0194)	0.6388 (0.0234)
	$r=1$	0.6097 (0.0192)	0.6102 (0.0189)	0.6116 (0.0195)	0.6113 (0.0196)	0.6123 (0.0193)	0.6127 (0.0197)	0.6130 (0.0188)	0.6135 (0.0194)	0.6154 (0.0201)	0.6299 (0.0203)

Appendix

Table S13 : Bike: linear-regression test MSE (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)	0.2348 (0.0209)
M2	–	0.0687 (0.0060)	0.0727 (0.0041)	0.0896 (0.0065)	0.0944 (0.0056)	0.0990 (0.0070)	0.1079 (0.0079)	0.1154 (0.0107)	0.1262 (0.0142)	0.1522 (0.0192)	0.4313 (0.0741)
M3	$r=0$	0.0687 (0.0060)	0.0727 (0.0041)	0.0896 (0.0065)	0.0944 (0.0056)	0.0990 (0.0070)	0.1079 (0.0079)	0.1154 (0.0107)	0.1262 (0.0142)	0.1522 (0.0192)	0.4313 (0.0741)
	$r=0.5$	0.0935 (0.0091)	0.1047 (0.0092)	0.1467 (0.0123)	0.1597 (0.0171)	0.1661 (0.0204)	0.1760 (0.0186)	0.1970 (0.0202)	0.2112 (0.0231)	0.2334 (0.0240)	0.4408 (0.0804)
	$r=1$	0.1077 (0.0108)	0.1215 (0.0133)	0.1578 (0.0167)	0.1751 (0.0173)	0.1848 (0.0244)	0.1994 (0.0196)	0.2175 (0.0233)	0.2352 (0.0261)	0.2596 (0.0254)	0.4393 (0.0654)

Appendix

Table S14 : Bike: FT-Transformer regression test MSE (FITS-G: all modes).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)
M1	FITS-K	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)	1.2821 (0.0896)
M2	None	0.2712 (0.1082)	0.2810 (0.1239)	0.4893 (0.2531)	0.5538 (0.2660)	0.5820 (0.2656)	0.5930 (0.2823)	0.6766 (0.2996)	0.7805 (0.3195)	0.7778 (0.4350)	1.6560 (0.8013)
M3	FITS-G	0.7996 (0.1013)	0.8844 (0.1078)	0.9800 (0.1166)	0.9865 (0.1158)	0.9973 (0.1187)	1.0069 (0.1231)	1.0121 (0.1186)	1.0249 (0.1155)	1.0422 (0.1198)	1.0822 (0.1207)
M3	FITS-K	0.9506 (0.0771)	1.0472 (0.0796)	1.1552 (0.0915)	1.1634 (0.0879)	1.1752 (0.0891)	1.1857 (0.0927)	1.1962 (0.0903)	1.2082 (0.0856)	1.2248 (0.0852)	1.2697 (0.0898)

Appendix

Table S15 : Adult: logistic-regression parameter estimation error (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)
M1	FITS-K	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)	0.8174 (0.0030)
M2	None	0.8204 (0.0031)	0.8207 (0.0032)	0.8204 (0.0036)	0.8199 (0.0034)	0.8202 (0.0028)	0.8194 (0.0038)	0.8196 (0.0034)	0.8195 (0.0033)	0.8189 (0.0038)	0.8131 (0.0054)
M3	FITS-G	0.8201 (0.0029)	0.8198 (0.0029)	0.8195 (0.0032)	0.8193 (0.0030)	0.8193 (0.0030)	0.8192 (0.0029)	0.8192 (0.0029)	0.8190 (0.0030)	0.8190 (0.0030)	0.8188 (0.0030)
M3	FITS-K	0.8195 (0.0031)	0.8191 (0.0032)	0.8184 (0.0032)	0.8185 (0.0032)	0.8181 (0.0030)	0.8182 (0.0030)	0.8182 (0.0030)	0.8180 (0.0030)	0.8178 (0.0030)	0.8176 (0.0030)

Appendix

Table S16 : Adult: logistic-regression test accuracy (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)
M1	FITS-K	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)	0.8465 (0.0038)
M2	None	0.8513 (0.0045)	0.8506 (0.0040)	0.8394 (0.0076)	0.8353 (0.0086)	0.8304 (0.0081)	0.8290 (0.0079)	0.8231 (0.0091)	0.8176 (0.0068)	0.8098 (0.0075)	0.8015 (0.0102)
M3	FITS-G	0.8513 (0.0042)	0.8513 (0.0039)	0.8498 (0.0043)	0.8498 (0.0044)	0.8496 (0.0039)	0.8493 (0.0045)	0.8496 (0.0044)	0.8491 (0.0045)	0.8474 (0.0053)	0.8485 (0.0048)
M3	FITS-K	0.8507 (0.0038)	0.8499 (0.0048)	0.8487 (0.0039)	0.8484 (0.0043)	0.8475 (0.0044)	0.8477 (0.0049)	0.8482 (0.0040)	0.8476 (0.0052)	0.8461 (0.0042)	0.8467 (0.0042)

Appendix

Table S17 : Adult: FT-Transformer classification test accuracy (FITS-G vs FITS-K).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)	1.0930 (0.1225)
M2	–	0.2712 (0.1082)	0.2810 (0.1239)	0.4893 (0.2531)	0.5538 (0.2660)	0.5820 (0.2656)	0.5930 (0.2823)	0.6766 (0.2996)	0.7805 (0.3195)	0.7778 (0.4350)	1.6560 (0.8013)
M3	$r=0$	0.2712 (0.1082)	0.2810 (0.1239)	0.4893 (0.2531)	0.5538 (0.2660)	0.5820 (0.2656)	0.5930 (0.2823)	0.6766 (0.2996)	0.7805 (0.3195)	0.7778 (0.4350)	1.6560 (0.8013)
	$r=0.5$	0.4937 (0.1383)	0.5016 (0.1498)	0.6129 (0.2177)	0.5926 (0.2377)	0.6516 (0.2328)	0.6575 (0.2170)	0.6018 (0.2645)	0.6413 (0.2764)	0.7271 (0.3544)	1.3037 (0.5863)
	$r=1$	0.6709 (0.1233)	0.6605 (0.1319)	0.7212 (0.2112)	0.7154 (0.2037)	0.7412 (0.2095)	0.7044 (0.2303)	0.6906 (0.2636)	0.7160 (0.3141)	0.8068 (0.2605)	1.2527 (0.6042)

Appendix

Table S18 : Adult: logistic-regression parameter estimation error (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)	0.8187 (0.0030)
M2	–	0.8204 (0.0031)	0.8207 (0.0032)	0.8204 (0.0036)	0.8199 (0.0034)	0.8202 (0.0028)	0.8194 (0.0038)	0.8196 (0.0034)	0.8195 (0.0033)	0.8189 (0.0038)	0.8131 (0.0054)
M3	$r=0$	0.8204 (0.0031)	0.8207 (0.0032)	0.8204 (0.0036)	0.8199 (0.0034)	0.8202 (0.0028)	0.8194 (0.0038)	0.8196 (0.0034)	0.8195 (0.0033)	0.8189 (0.0038)	0.8131 (0.0054)
	$r=0.5$	0.8204 (0.0032)	0.8205 (0.0031)	0.8204 (0.0034)	0.8202 (0.0033)	0.8201 (0.0028)	0.8199 (0.0036)	0.8196 (0.0033)	0.8196 (0.0033)	0.8191 (0.0039)	0.8152 (0.0046)
	$r=1$	0.8202 (0.0030)	0.8204 (0.0031)	0.8202 (0.0034)	0.8199 (0.0033)	0.8199 (0.0030)	0.8195 (0.0035)	0.8198 (0.0032)	0.8193 (0.0036)	0.8188 (0.0039)	0.8163 (0.0047)

Appendix

Table S19 : Adult: logistic-regression test accuracy (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)	0.8478 (0.0039)
M2	–	0.8513 (0.0045)	0.8506 (0.0040)	0.8394 (0.0076)	0.8353 (0.0086)	0.8304 (0.0081)	0.8290 (0.0079)	0.8231 (0.0091)	0.8176 (0.0068)	0.8098 (0.0075)	0.8015 (0.0102)
M3	$r=0$	0.8513 (0.0045)	0.8506 (0.0040)	0.8394 (0.0076)	0.8353 (0.0086)	0.8304 (0.0081)	0.8290 (0.0079)	0.8231 (0.0091)	0.8176 (0.0068)	0.8098 (0.0075)	0.8015 (0.0102)
	$r=0.5$	0.8526 (0.0040)	0.8498 (0.0050)	0.8444 (0.0062)	0.8445 (0.0054)	0.8397 (0.0065)	0.8369 (0.0074)	0.8318 (0.0068)	0.8268 (0.0076)	0.8181 (0.0076)	0.7954 (0.0112)
	$r=1$	0.8514 (0.0049)	0.8501 (0.0044)	0.8475 (0.0041)	0.8451 (0.0047)	0.8438 (0.0051)	0.8425 (0.0051)	0.8382 (0.0056)	0.8329 (0.0083)	0.8242 (0.0087)	0.7942 (0.0106)

Appendix

Table S20 : Adult: FT-Transformer classification test accuracy (FITS-G: all modes).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)
	HeteroFedSyn	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)	3.5888 (0.3499)
	FedTabDiff	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)	0.7209 (0.1486)
M2	None	0.4926 (0.0113)	0.4942 (0.0111)	0.4968 (0.0115)	0.4970 (0.0120)	0.4975 (0.0116)	0.4997 (0.0123)	0.5015 (0.0135)	0.5032 (0.0133)	0.5057 (0.0126)	0.5602 (0.0375)
M3	FITS-G	0.4938 (0.0120)	0.4947 (0.0120)	0.4956 (0.0123)	0.4958 (0.0123)	0.4960 (0.0125)	0.4960 (0.0124)	0.4962 (0.0125)	0.4963 (0.0124)	0.4966 (0.0126)	0.4969 (0.0127)

Appendix

Table S21 : Simulation: linear-regression test MSE.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)
	HeteroFedSyn	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)	0.8147 (0.0184)
	FedTabDiff	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)	0.8687 (0.0058)
M2	None	0.8763 (0.0054)	0.8760 (0.0055)	0.8748 (0.0055)	0.8736 (0.0058)	0.8745 (0.0058)	0.8738 (0.0059)	0.8727 (0.0058)	0.8722 (0.0065)	0.8705 (0.0077)	0.8488 (0.0177)
M3	FITS-G	0.8763 (0.0056)	0.8761 (0.0056)	0.8762 (0.0058)	0.8758 (0.0059)	0.8760 (0.0059)	0.8760 (0.0057)	0.8762 (0.0058)	0.8759 (0.0059)	0.8760 (0.0060)	0.8760 (0.0059)

Appendix

Table S22 : Simulation: logistic-regression test accuracy.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)
	HeteroFedSyn	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)	5.5571 (0.9458)
	FedTabDiff	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)	0.7994 (0.1271)
M2	None	0.5567 (0.0273)	0.5889 (0.0319)	0.7232 (0.0644)	0.7731 (0.0580)	0.8112 (0.0694)	0.8747 (0.0853)	0.9386 (0.1001)	1.0237 (0.0881)	1.1248 (0.1095)	1.4740 (0.2824)
M3	FITS-G	0.5625 (0.0341)	0.5688 (0.0345)	0.5777 (0.0468)	0.5756 (0.0371)	0.5806 (0.0384)	0.5763 (0.0269)	0.5970 (0.0547)	0.5820 (0.0376)	0.5748 (0.0277)	0.5910 (0.0437)

Appendix

Table S23 : Simulation: FT-Transformer regression test MSE.

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)
M1	FITS-K	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)	0.4986 (0.0124)
M2	None	0.4926 (0.0113)	0.4942 (0.0111)	0.4968 (0.0115)	0.4970 (0.0120)	0.4975 (0.0116)	0.4997 (0.0123)	0.5015 (0.0135)	0.5032 (0.0133)	0.5057 (0.0126)	0.5602 (0.0375)
M3	FITS-G	0.4938 (0.0120)	0.4947 (0.0120)	0.4956 (0.0123)	0.4958 (0.0123)	0.4960 (0.0125)	0.4960 (0.0124)	0.4962 (0.0125)	0.4963 (0.0124)	0.4966 (0.0126)	0.4969 (0.0127)
M3	FITS-K	0.4944 (0.0118)	0.4955 (0.0118)	0.4966 (0.0120)	0.4967 (0.0120)	0.4969 (0.0122)	0.4971 (0.0121)	0.4974 (0.0121)	0.4975 (0.0121)	0.4978 (0.0123)	0.4982 (0.0124)

Appendix

Table S24 : Simulation: linear-regression test MSE (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)
M1	FITS-K	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)	0.8759 (0.0055)
M2	None	0.8763 (0.0054)	0.8760 (0.0055)	0.8748 (0.0055)	0.8736 (0.0058)	0.8745 (0.0058)	0.8738 (0.0059)	0.8727 (0.0058)	0.8722 (0.0065)	0.8705 (0.0077)	0.8488 (0.0177)
M3	FITS-G	0.8763 (0.0056)	0.8761 (0.0056)	0.8762 (0.0058)	0.8758 (0.0059)	0.8760 (0.0059)	0.8760 (0.0057)	0.8762 (0.0058)	0.8759 (0.0059)	0.8760 (0.0060)	0.8760 (0.0059)
M3	FITS-K	0.8765 (0.0054)	0.8763 (0.0053)	0.8763 (0.0054)	0.8761 (0.0057)	0.8762 (0.0053)	0.8761 (0.0054)	0.8761 (0.0052)	0.8761 (0.0056)	0.8761 (0.0055)	0.8759 (0.0054)

Appendix

Table S25 : Simulation: logistic-regression test accuracy (FITS-G vs FITS-K).

Mode	Baselines	Clients
Mode	Baselines	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	FITS-G	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)
M1	FITS-K	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)	0.6358 (0.0519)
M2	None	0.5567 (0.0273)	0.5889 (0.0319)	0.7232 (0.0644)	0.7731 (0.0580)	0.8112 (0.0694)	0.8747 (0.0853)	0.9386 (0.1001)	1.0237 (0.0881)	1.1248 (0.1095)	1.4740 (0.2824)
M3	FITS-G	0.5625 (0.0341)	0.5688 (0.0345)	0.5777 (0.0468)	0.5756 (0.0371)	0.5806 (0.0384)	0.5763 (0.0269)	0.5970 (0.0547)	0.5820 (0.0376)	0.5748 (0.0277)	0.5910 (0.0437)
M3	FITS-K	0.5714 (0.0423)	0.5757 (0.0258)	0.5883 (0.0368)	0.5934 (0.0364)	0.6013 (0.0440)	0.5951 (0.0382)	0.6050 (0.0395)	0.6055 (0.0391)	0.5987 (0.0392)	0.6189 (0.0497)

Appendix

Table S26 : Simulation: FT-Transformer regression test MSE (FITS-G vs FITS-K).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)	0.4971 (0.0127)
M2	–	0.4926 (0.0113)	0.4942 (0.0111)	0.4968 (0.0115)	0.4970 (0.0120)	0.4975 (0.0116)	0.4997 (0.0123)	0.5015 (0.0135)	0.5032 (0.0133)	0.5057 (0.0126)	0.5602 (0.0375)
M3	$r=0$	0.4926 (0.0113)	0.4942 (0.0111)	0.4968 (0.0115)	0.4970 (0.0120)	0.4975 (0.0116)	0.4997 (0.0123)	0.5015 (0.0135)	0.5032 (0.0133)	0.5057 (0.0126)	0.5602 (0.0375)
	$r=0.5$	0.4934 (0.0117)	0.4947 (0.0112)	0.4978 (0.0119)	0.4979 (0.0118)	0.4979 (0.0118)	0.5005 (0.0120)	0.5023 (0.0146)	0.5037 (0.0133)	0.5074 (0.0138)	0.5500 (0.0358)
	$r=1$	0.4937 (0.0121)	0.4953 (0.0118)	0.4964 (0.0114)	0.4979 (0.0128)	0.4982 (0.0132)	0.4994 (0.0126)	0.5017 (0.0144)	0.5024 (0.0122)	0.5057 (0.0142)	0.5400 (0.0294)

Appendix

Table S27 : Simulation: linear-regression test MSE (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)	0.8759 (0.0059)
M2	–	0.8763 (0.0054)	0.8760 (0.0055)	0.8748 (0.0055)	0.8736 (0.0058)	0.8745 (0.0058)	0.8738 (0.0059)	0.8727 (0.0058)	0.8722 (0.0065)	0.8705 (0.0077)	0.8488 (0.0177)
M3	$r=0$	0.8763 (0.0054)	0.8760 (0.0055)	0.8748 (0.0055)	0.8736 (0.0058)	0.8745 (0.0058)	0.8738 (0.0059)	0.8727 (0.0058)	0.8722 (0.0065)	0.8705 (0.0077)	0.8488 (0.0177)
	$r=0.5$	0.8763 (0.0054)	0.8761 (0.0055)	0.8753 (0.0057)	0.8743 (0.0055)	0.8749 (0.0059)	0.8746 (0.0059)	0.8731 (0.0060)	0.8732 (0.0061)	0.8719 (0.0067)	0.8570 (0.0114)
	$r=1$	0.8764 (0.0056)	0.8762 (0.0054)	0.8758 (0.0058)	0.8750 (0.0059)	0.8753 (0.0055)	0.8751 (0.0060)	0.8745 (0.0055)	0.8742 (0.0054)	0.8733 (0.0067)	0.8632 (0.0115)

Appendix

Table S28 : Simulation: logistic-regression test accuracy (FITS-G: all modes).

Mode	Parameters	Clients
Mode	Parameters	Client 0 (30%)	Client 1 (20%)	Client 2 (10%)	Client 3 (9%)	Client 4 (8%)	Client 5 (7%)	Client 6 (6%)	Client 7 (5%)	Client 8 (4%)	Client 9 (1%)
M1	–	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)	0.6028 (0.0433)
M2	–	0.5567 (0.0273)	0.5889 (0.0319)	0.7232 (0.0644)	0.7731 (0.0580)	0.8112 (0.0694)	0.8747 (0.0853)	0.9386 (0.1001)	1.0237 (0.0881)	1.1248 (0.1095)	1.4740 (0.2824)
M3	$r=0$	0.5567 (0.0273)	0.5889 (0.0319)	0.7232 (0.0644)	0.7731 (0.0580)	0.8112 (0.0694)	0.8747 (0.0853)	0.9386 (0.1001)	1.0237 (0.0881)	1.1248 (0.1095)	1.4740 (0.2824)
	$r=0.5$	0.5646 (0.0365)	0.5899 (0.0343)	0.6999 (0.0600)	0.7414 (0.0689)	0.7943 (0.0727)	0.8534 (0.1054)	0.9549 (0.1172)	1.0441 (0.1100)	1.2009 (0.1289)	1.5537 (0.2274)
	$r=1$	0.5654 (0.0329)	0.5804 (0.0322)	0.6707 (0.0498)	0.6957 (0.0524)	0.7449 (0.0674)	0.7833 (0.0765)	0.8586 (0.1190)	0.9823 (0.1009)	1.1235 (0.1313)	1.6559 (0.2200)

Appendix

Table S29 : Simulation: FT-Transformer regression test MSE (FITS-G: all modes).

Synthetic Data Generation

Synthetic data generation creates artificial examples, simulations, and datasets to train, test, or protect machine-learning systems.

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Concordia: Self-Improving Synthetic Tables for Federated LLMs
May 11, 2026Jimin Huang, Duanyu Feng, Nuo Chen +8LLM Fine-Tuning Synthetic Data Generation
University of Manchester · The Fin AI · National University of Singapore +5
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Jun 30, 2026Vasileios C. Pezoulas, Nikolaos S. Tachos, Eleni Georga +3Bayesian Mixture Models Bayesian Inference
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Aug 14, 2026Hao Yan, Lisa Pilgram, Dan Liu +3Synthetic Tabular Data Generation Tabular Diffusion Models
Children’s Hospital of Eastern Ontario Research Institute, Ontario, Canada · School of Epidemiology and Public Health, University of Ottawa, Ontario, Canada · Department of Nephrology and Medical Intensive Care, Charité - Universitaetsmedizin Berlin, Berlin, Germany +2

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Concordia: Self-Improving Synthetic Tables for Federated LLMs

Jimin Huang, Duanyu Feng, Nuo Chen, Xiaoyu Wang+7

University of Manchester · The Fin AI · National University of Singapore+5

LLM Fine-Tuning Synthetic Data Generation Synthetic Tabular Data Generation Non-IID Federated Learning Federated Fine-Tuning Federated Learning

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Federated learning (FL) enables training large language models (LLMs) without sharing raw data, but adapting LLMs under strict data isolation and non-IID client distributions remains challenging in practice. Synthetic data offers a natural privacy-preserving surrogate for local training, yet existing federated pipelines typically treat synthetic generation as static or loosely coupled with downstream optimization, leading to rapidly diminishing utility under heterogeneous clients. We study federated adaptation of LLMs on tabular tasks where raw records and validation data cannot be shared, and local training must rely entirely on synthetic tables. We propose Concordia, a tri-level optimization framework that aligns synthetic data generation with federated validation utility despite these constraints. At the client level, models are adapted via parameter-efficient LoRA training on synthetic tables. Clients additionally learn lightweight utility scorers from private validation feedback to reweight synthetic samples during local training. At the outer level, each client refines its own synthetic table generator using group-relative policy optimization (GRPO), guided by an ensemble of heterogeneous scorers shared across clients, without aggregating generator parameters or exposing validation data. Experiments on privacy-sensitive tabular benchmarks from finance and healthcare demonstrate that Concordia consistently improves federated performance, cross-client stability, and robustness to distribution shift compared to static and decoupled synthetic-data baselines.

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Synthetic Data Generation

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