PTOPOFL: Topology-Guided Personalised Federated Learning with Persistent-Homology Client Descriptors

Kelly L Vomo-Donfack; Adryel Hoszu; Grégory Ginot; Ian Morilla

cs.LGMar 4, 2026

PTOPOFL: Topology-Guided Personalised Federated Learning with Persistent-Homology Client Descriptors

Authors: Kelly L Vomo-Donfack, Adryel Hoszu, Grégory Ginot, Ian Morilla

Organizations: Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, Laboratoire d’excellence Infibrex, F-93430 Villetaneuse, France · Instituto de Hortofruticultura Subtropical y Mediterránea La Mayora (IHSM), Universidad de Málaga–Consejo Superior de Investigaciones Científicas, Málaga, Spain

Abstract

Clustered and personalised federated learning (FL) needs a signal telling the server which clients should share a model. Most methods derive it from losses or model updates that must be recomputed during training. We study pTopoFL, which derives it once, before training, from a compact persistent-homology (PH) descriptor of each client's feature distribution: the server clusters clients on these descriptors, weights them within clusters, and runs blended clustered local gradient descent, returning one personalised model per cluster. For strongly convex, smooth losses we analyse the complete procedure: for any partition, one round is a contraction with factor (1-eta*mu)^tau; the fixed point is characterised by a local-step drift bound, an exact formula for the effective strength of blending (which yields a calibration rule) and a stochastic floor exposing a bias-variance trade-off of clustering; and personalisation lowers the heterogeneity floor in proportion to a measurable share of the dispersion of client optima. Experiments on synthetic federations, a label-skew benchmark and a real four-hospital dataset confirm the theory and delimit the method. When feature geometry and decision rules co-vary, the PH partition recovers the latent client groups, including differences a linear-subspace descriptor misses, and pTopoFL is competitive with the strongest clustered baselines; on the real federation it lowers the test log-loss of FedAvg from 0.463 to 0.438, on par with the best personalised methods. When heterogeneity is invisible in feature geometry, it offers no advantage over FedAvg. pTopoFL transmits model parameters and is no more private than FedAvg; we give a record-level differential-privacy accounting and show that privatising the descriptor destroys the partition. Code reproduces every number.

Figures & tables

Panel of Figure 1 — Figure 1: pTopoFL protocol. Every client sends two kinds of message. Green: the descriptor $\phi_{k}$ , sent once , before training. Solid blue: its updated local model $\theta_{k}^{t+1}\in\mathbb{R}^{p}$ , sent in every round. Dashed blue: the server returns to client $k$ the personalised model of its cluster, $\theta_{c(k)}^{t+1}$ . The server uses descriptors only to build the partition and the weights, and uses models only through the per-cluster weighted sums $u_{j}$ . Arrows are drawn for clients $1$ and $K$ ; all clients behave alike.

Figure 2: Numerical checks of Section 4 on federation (A). (a) The distance to the fixed point decays at least as fast as

q^{t}

(Theorem 3 ); dashed lines show the bound. (b) The drift bias at

\alpha=0

grows linearly in

\eta(\tau-1)

and lies below the bound of Theorem 4 (dotted). (c) Fraction of the between-cluster spread retained by blending; the lines are the predictions of Proposition 5 at the smallest (dashed) and median (dotted) Hessian eigenvalue.

	Synthetic (clustered)		Label skew		Fed-Heart-Disease
Method	Log-loss $\downarrow$	AUC $\uparrow$	Log-loss $\downarrow$	AUC $\uparrow$	Log-loss $\downarrow$	AUC $\uparrow$
Local	$0.396\pm 0.044$	$0.878\pm 0.022$	$0.625\pm 0.050$	$0.765\pm 0.040$	$0.440\pm 0.031$	$0.879\pm 0.016$
FedAvg	$0.451\pm 0.050^{+}$	$0.830\pm 0.026$	$0.590\pm 0.018$	$0.743\pm 0.021$	$0.463\pm 0.030^{+}$	$0.864\pm 0.018$
FedProx	$0.450\pm 0.050^{+}$	$0.831\pm 0.026$	$0.589\pm 0.017$	$0.744\pm 0.021$	$0.463\pm 0.030^{+}$	$0.864\pm 0.018$
SCAFFOLD	$0.448\pm 0.053^{+}$	$0.831\pm 0.027$	$0.589\pm 0.017$	$0.743\pm 0.021$	$0.459\pm 0.027^{+}$	$0.866\pm 0.016$
pFedMe	$0.446\pm 0.051^{+}$	$0.832\pm 0.026$	$0.587\pm 0.017$	$0.746\pm 0.021$	$0.457\pm 0.027^{+}$	$0.867\pm 0.017$

Table 1: Personalised performance of honest clients (mean

\pm

std over

10

seeds; lower log-loss and higher AUC are better). Bold marks the best deployable method. + / - : significantly worse/better than pTopoFL (paired Wilcoxon, Holm,

0.05

). ⋄ : non-deployable references with the same clustered training as pTopoFL .

Figure 3: Training curves : personalised test log-loss of honest clients (mean

\pm

std over seeds). All methods are genuinely iterative and converge within

T=150

rounds.

	Aligned (dim.)	Aligned (tails)	Concept only	Geometry only
ARI, PH-descriptor partition	$0.95$	$1.00$	$-0.02$	$0.95$
ARI, subspace partition	$0.87$	$-0.01$	$-0.01$	$0.90$
$\rho(\mathcal{C})$ , PH descriptor	$0.46$	$0.37$	$0.86$	$0.78$
$\rho(\mathcal{C})$ , random	$0.81$	$0.84$	$0.85$	$0.88$
$\rho(\mathcal{C})$ , oracle ⋄	$0.45$	$0.37$	$0.36$	$0.74$
FedAvg	$0.451\pm 0.050$	$0.506\pm 0.042$	$0.477\pm 0.028$	$0.379\pm 0.026$

Table 2: Alignment between geometry and decision rules (synthetic,

K=8

). The first rows show recovery of the latent groups (ARI) and the within-cluster share

\rho(\mathcal{C})

of the dispersion of client optima (lower is better). The bottom rows show the personalised log-loss. ⋄ : non-deployable.

Figure 4: Gain over FedAvg against the between-cluster share

1-\rho

of the descriptor partition, for every regime and seed. Corollary 7 predicts a monotone relation; the correlation is

r=0.87

Figure 5: Corrupted clients : log-loss of the honest clients as the fraction of corrupted clients grows (

K=10

, federation A). Dashed lines are pTopoFL variants without trust weights and with the mean/std trust rule.

Figure 6: Privacy. (a) Agreement between the partitions obtained from noisy and clean descriptors, as a function of the noise level; the dashed line is the radius of Corollary 10 . (b) Personalised log-loss under record-level

(\varepsilon,10^{-5})

-DP (Proposition 2 ). (c) Membership-inference AUC on the models clients transmit.

Variant (Synthetic, clustered)	Log-loss $\downarrow$
Full pTopoFL ( $\bar{\alpha}=0.3$ , $\tau=10$ )	$0.379\pm 0.045$
No clustering ( $M=1$ )	$0.455\pm 0.053$
Uniform within-cluster weights ( $w_{k}\propto n_{k}$ )	$0.378\pm 0.044$
No trust weights	$0.378\pm 0.044$
v1 trust rule (mean/std $z$ )	$0.380\pm 0.044$
Uncalibrated $\alpha=0.3$ (v1 default)	$0.405\pm 0.049$

Table 3: Ablation on federation (A) (log-loss, mean

\pm

std over seeds).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Method	uplink / client / round	downlink / client / round
FedAvg, FedProx, Ditto	$p$	$p$
SCAFFOLD	$2p$	$2p$
pFedMe	$p$	$p$
IFCA ( $M$ clusters)	$p$ (+ cluster id)	$Mp$
pTopoFL	$p$ (+ $46$ floats once)	$p$

Appendix

Table 4: Communication per client and per round (in floats);

p

is the model dimension.

Hyper-parameter	Value
$\lambda$ (L2, all parameters)	0.01
$\tau$ (local GD steps)	10
$T$ (rounds)	150
$\eta$	$1/\max_{k}L_{k}$
$M$ (clusters)	2
$\bar{\alpha}$ (target effective blending)	0.3

Appendix

Table 5: Hyper-parameters, fixed before the experiments and shared by all federations.

	Synthetic (clustered)			Label skew			Fed-Heart-Disease
Method	worse than FedAvg	worse than Local	worst client	worse than FedAvg	worse than Local	worst client	worse than FedAvg	worse than Local	worst client
Local	0.26	0.00	0.601	0.65	0.00	0.820	0.42	0.00	0.567
FedAvg	0.00	0.74	0.598	0.00	0.35	0.723	0.00	0.57	0.586
IFCA	0.16	0.42	0.519	0.30	0.17	0.695	0.38	0.62	0.576
Ditto	0.17	0.51	0.518	0.33	0.06	0.690	0.33	0.53	0.549
PACFL-style	0.11	0.45	0.520	0.51	0.34	0.741	0.42	0.65	0.573

Appendix

Table 6: Per-client view (all federations). Shown are the fraction of honest clients whose personalised log-loss is worse than under FedAvg and under local training, and the loss of the worst client, all averaged over seeds.

$\tau$ \ $\bar{\alpha}$	0	0.1	0.3	0.6	1
1	$0.377$	$0.378$	$0.379$	$0.383$	$0.448$
10	$0.377$	$0.378$	$0.379$	$0.383$	$0.455$
50	$0.379$	$0.379$	$0.380$	$0.388$	$0.479$

Appendix

Table 7: Blending and local steps (federation A, log-loss). Rows: number of local steps

\tau

, with the total number of local steps held equal. Columns: target effective blending

\bar{\alpha}

, with

\alpha

set by Eq. ( 6 ). With calibration, the results depend on

\bar{\alpha}

but hardly on

\tau

, as Proposition 5 (c) predicts.

Variant	Log-loss $\downarrow$	ARI
$M=1$	$0.455\pm 0.053$	$0.00$
$M=2$ (default)	$0.379\pm 0.045$	$0.95$
$M=3$	$0.383\pm 0.048$	$0.73$
$M=4$	$0.378\pm 0.048$	$0.52$
$n_{\mathrm{sub}}=40$	$0.379\pm 0.047$	$0.95$
$n_{\mathrm{sub}}=160$	$0.380\pm 0.045$	$0.95$

Appendix

Table 8: Sensitivity of pTopoFL on federation (A): log-loss and recovery of the latent groups (ARI).

Attack / score	10%	20%	30%	40%
shift + flips, robust	0.00 / 0.10	0.05 / 0.21	0.07 / 0.10	0.03 / 0.23
shift + flips, mean/std	0.10 / 0.14	0.05 / 0.21	0.10 / 0.19	0.05 / 0.25
heavy-tailed + flips, robust	0.30 / 0.02	0.35 / 0.04	0.23 / 0.06	0.15 / 0.12
heavy-tailed + flips, mean/std	0.50 / 0.13	0.45 / 0.07	0.37 / 0.17	0.15 / 0.25
label inversion, robust	0.00 / 0.12	0.10 / 0.09	0.17 / 0.19	0.10 / 0.12
label inversion, mean/std	0.00 / 0.17	0.20 / 0.16	0.20 / 0.16	0.23 / 0.23

Appendix

Table 9: Detection of corrupted clients by trust weights (federation A,

K=10

). Each entry gives the fraction of corrupted clients flagged (

t_{k}<1

) and the fraction of honest clients flagged, for each fraction of corrupted clients.

$\varepsilon$	$\infty$	64	16	8	4
DP-FedAvg log-loss	$0.423$	$0.424$	$0.428$	$0.438$	$0.483$
DP- pTopoFL log-loss	$0.359$	$0.410$	$0.434$	$0.519$	$1.127$
DP- pTopoFL (descriptor not private) log-loss	$0.359$	$0.359$	$0.364$	$0.379$	$0.444$
DP-FedAvg MIA AUC	$0.508$	$0.509$	$0.508$	$0.508$	$0.507$
DP- pTopoFL MIA AUC	$0.507$	$0.508$	$0.508$	$0.508$	$0.506$
DP- pTopoFL (descriptor not private) MIA AUC	$0.507$	$0.507$	$0.507$	$0.506$	$0.505$

Appendix

Table 10: Privacy experiment (Section 5.6 ): log-loss, membership-inference AUC, partition recovery and property-inference accuracy as functions of the record-level

\varepsilon

(

\delta=10^{-5}

Figure 7: Entropy coordinates of the normalised descriptor over

20

rounds. Dashed lines are the two drifting clients.

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