Proto-EVFL: Prototype-based VFL

• Proto-EVFL is a vertical federated learning framework that leverages per-party class prototypes and bi-level optimization to process extremely unaligned data.
• It integrates a probabilistic dual prototype learning scheme based on optimal transport, mixed prior guidance, and adaptive gated aggregation to mitigate intra- and inter-party class imbalance.
• Empirical results show substantial performance gains—up to 6.97% improvement over baselines—with a convergence rate of O(1/√T) in challenging VFL scenarios.

Proto-EVFL is a vertical federated learning framework for settings in which aligned samples are scarce but each party owns a large pool of locally unaligned samples. It addresses the regime of extremely unaligned data, where heterogeneous feature spaces and heterogeneous sample spaces coexist with a common label space, and where unaligned samples are highly class-imbalanced both within each party and across parties. The method combines per-party class prototypes, a probabilistic dual prototype learning scheme based on conditional optimal transport cost with class prior probability, a mixed prior guided module, and an adaptive gated feature aggregation strategy. It is presented as the first bi-level optimization framework in VFL, and its convergence analysis yields a rate of $1/\sqrt{T}$ (Guo et al., 30 Jul 2025).

1. VFL setting and the imbalance structure

Proto-EVFL considers $M$ parties, indexed by $m=1,\dots,M$, where each party $m$ owns a feature matrix $X^m \in \mathbb{R}^{N^m \times D^m}$. One active party, denoted $m=1$, holds labels for the aligned subset $Y_a \in \{0,1\}^{N_a \times Z}$, while the remaining parties are passive. Each party has an aligned subset $X_a^m \in \mathbb{R}^{N_a \times D^m}$ and an unaligned subset $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$, with $N^m = N_a + N_u^m$. The formal heterogeneity assumption is

$M$0

so the parties differ in feature space and sample IDs but share the same underlying label set (Guo et al., 30 Jul 2025).

The paper distinguishes two imbalance mechanisms. Intra-party class imbalance refers to skewed class distributions within a single party’s local data, causing local extractor bias and active-party classifier bias. Inter-party class imbalance refers to the fact that different parties may be strong on different classes, producing feature contribution inconsistency during aggregation. Under extremely unaligned data, these two forms of imbalance limit feature representation, shrink the effective prediction space, and make conventional VFL methods that use unaligned data in a purely self-supervised or semi-supervised manner inadequate.

2. Framework composition and information flow

Proto-EVFL augments VFL with four coupled components: class prototypes, probabilistic dual prototype learning, mixed priors, and adaptive gated aggregation. Each party has a local extractor $M$1 that maps inputs to latent features, and the active party coordinates prototype updates and global classification (Guo et al., 30 Jul 2025).

Component	Core object	Function
Class prototypes	$M$2	Per-party latent semantic centers for class $M$3
Probabilistic dual prototype learning	$M$4	Prior-aware matching between unaligned features and prototypes
Mixed prior guided module	$M$5	Combines local and global class priors
Adaptive gated feature aggregation	$M$6	Dynamically weights party contributions on aligned data

A communication round proceeds as follows. The active party maintains and updates prototypes based on aligned data, then transmits prototypes and global priors to the passive parties. Each passive party updates its extractor on unaligned samples through the local prototype-based objective, recomputes aligned representations, and sends aligned features and local priors back to the active party. The active party then uses adaptors and a gating network to aggregate party-specific aligned representations, optimizes the classifier on cross-entropy loss, updates prototypes again, and recomputes global priors. No raw data or gradients are exchanged.

The expression “dual prototypes” refers to the two directional uses of prototypes in the optimal-transport-style matching scheme: from features to prototypes and from prototypes to features. This duality is central to the framework’s handling of unaligned unlabeled data.

3. Prototype geometry and probabilistic dual prototype learning

For each party $M$7 and class $M$8, Proto-EVFL maintains a class prototype $M$9 in the latent space of $m=1,\dots,M$0. These prototypes are learnable and are updated centrally at the active party using aligned data and the adapted aligned representations $m=1,\dots,M$1 as specified in Eq. (15). The paper describes them as class-specific semantic centers that allow the system to model relationships between classes in latent space and to support prediction for unseen classes (Guo et al., 30 Jul 2025).

Let $m=1,\dots,M$2 denote the feature of an unaligned sample. The conditional distribution over prototypes is

$m=1,\dots,M$3

This is a prior-weighted soft assignment of a feature to class prototypes. Using a cosine-dissimilarity cost $m=1,\dots,M$4, the expected feature-to-prototype loss is

$m=1,\dots,M$5

A reverse prototype-to-feature conditional distribution is defined in Eq. (4), yielding the complementary loss $m=1,\dots,M$6 in Eq. (5). The local optimization objective at party $m=1,\dots,M$7 is

$m=1,\dots,M$8

The paper interprets these two losses as a conditional optimal transport cost. In the $m=1,\dots,M$9 direction, feature mass is transported to prototypes according to posterior assignment; in the $m$0 direction, every prototype must also find support among local features. This suppresses prototype omission for minority classes and avoids a one-sided clustering effect. A plausible implication is that the method does not merely reweight unaligned samples; it reshapes local feature geometry so that minority-class prototypes remain active during representation learning.

4. Mixed priors and adaptive aggregation

Because unaligned data are unlabeled, class priors over local unaligned samples are estimated with an EM-like update. The local prior for class $m$1 at party $m$2 is

$m$3

with the posterior assignments defined from the previous round’s prototypes and priors in Eq. (10). The active party averages local priors to form the global prior $m$4 (Guo et al., 30 Jul 2025).

The mixed prior guided module then combines local and global information through

$m$5

where $m$6 is a personalized mixing coefficient. The role of this update is to avoid purely local prior estimation, which may amplify majority classes, while also avoiding blind imposition of a global prior on parties whose local minority-class evidence is extremely sparse. In the paper’s interpretation, the mixed prior guides the conditional optimal transport matching so that underrepresented classes can acquire higher posterior mass during sample selection.

Inter-party imbalance is addressed at the active party by adaptive gated feature aggregation. Each aligned representation is first transformed by an adaptor,

$m$7

and then weighted through a gating function

$m$8

where $m$9 is trainable. The active party aggregates the adapted local features into a joint representation $X^m \in \mathbb{R}^{N^m \times D^m}$0 and trains a classifier $X^m \in \mathbb{R}^{N^m \times D^m}$1 via

$X^m \in \mathbb{R}^{N^m \times D^m}$2

with $X^m \in \mathbb{R}^{N^m \times D^m}$3 the cross-entropy loss (Guo et al., 30 Jul 2025).

This aggregation strategy is explicitly designed to mitigate feature contribution inconsistency. Rather than treating all parties symmetrically, the active party learns to emphasize whichever party is more informative for the current sample or class. The paper characterizes this as a dynamic weighting-and-aggregation mechanism across parties.

5. Bi-level optimization and convergence

Proto-EVFL is formulated as a bi-level optimization problem. The upper level optimizes the active-party parameters

$X^m \in \mathbb{R}^{N^m \times D^m}$4

while the lower level optimizes the extractors

$X^m \in \mathbb{R}^{N^m \times D^m}$5

The resulting problem is

$X^m \in \mathbb{R}^{N^m \times D^m}$6

with expectations over aligned samples $X^m \in \mathbb{R}^{N^m \times D^m}$7 and unaligned samples $X^m \in \mathbb{R}^{N^m \times D^m}$8 given explicitly in Eq. (17) (Guo et al., 30 Jul 2025).

The convergence theorem assumes Lipschitz continuity of $X^m \in \mathbb{R}^{N^m \times D^m}$9, smoothness of $m=1$0, $m=1$1, and the joint objective, Lipschitz second derivatives for mixed and lower-level Hessian terms, bounded stochastic-gradient variance, and bounded extractor domain. Under these assumptions, with

$m=1$2

and $m=1$3, Theorem 1 provides the stationarity bound in Eq. (18). When $m=1$4, the convergence rate becomes $m=1$5.

The theoretical significance assigned to this result is twofold. First, it treats the extractor updates on unaligned data and the classifier-side updates on aligned data within one coupled optimization picture. Second, it places prototype-guided VFL with heterogeneous sample spaces in a standard stochastic nonconvex convergence framework. This suggests that Proto-EVFL is intended not only as an empirical modification of VFL, but as a structurally distinct training paradigm.

6. Empirical performance and ablation evidence

The experimental study covers four datasets: ModelNet-10, Fashion-MNIST, Credit, and Adult. The evaluation includes normal, few-shot, and zero-shot scenarios, with aligned sample counts $m=1$6, $m=1$7, and $m=1$8. Baselines include Local Model, Vanilla VFL, SS-VFL, FedHSSL, and an Upper Boundary centralized model (Guo et al., 30 Jul 2025).

Representative results show that Proto-EVFL consistently improves over VFL baselines, with especially large gains when aligned supervision is extremely scarce.

Dataset and setting	Baseline	Proto-EVFL
ModelNet-10, $m=1$9, zero-shot	FedHSSL 64.93%	76.72%
ModelNet-10, $Y_a \in \{0,1\}^{N_a \times Z}$0, few-shot	FedHSSL 32.63%	77.52%
Fashion-MNIST, $Y_a \in \{0,1\}^{N_a \times Z}$1, zero-shot	Vanilla VFL 55.60%	65.28%
Credit, $Y_a \in \{0,1\}^{N_a \times Z}$2	FedHSSL 60.41%	77.96%
Adult, $Y_a \in \{0,1\}^{N_a \times Z}$3	FedHSSL 76.92%	79.22%

The paper states that, even in a zero-shot scenario with one unseen class, Proto-EVFL outperforms baselines by at least $Y_a \in \{0,1\}^{N_a \times Z}$4. On ModelNet-10 with $Y_a \in \{0,1\}^{N_a \times Z}$5 aligned samples, Proto-EVFL reaches $Y_a \in \{0,1\}^{N_a \times Z}$6, close to the upper boundary of $Y_a \in \{0,1\}^{N_a \times Z}$7. On Fashion-MNIST with $Y_a \in \{0,1\}^{N_a \times Z}$8 aligned samples, it reaches $Y_a \in \{0,1\}^{N_a \times Z}$9 versus $X_a^m \in \mathbb{R}^{N_a \times D^m}$0 for FedHSSL.

Ablation studies identify the main performance sources. On ModelNet-10, removing prototype updating lowers accuracy from $X_a^m \in \mathbb{R}^{N_a \times D^m}$1 to $X_a^m \in \mathbb{R}^{N_a \times D^m}$2 at $X_a^m \in \mathbb{R}^{N_a \times D^m}$3, removing the probabilistic dual prototype learning scheme lowers it to $X_a^m \in \mathbb{R}^{N_a \times D^m}$4, removing the mixed prior module lowers it to $X_a^m \in \mathbb{R}^{N_a \times D^m}$5, and removing adaptive gated feature aggregation lowers it to $X_a^m \in \mathbb{R}^{N_a \times D^m}$6. A separate comparison of unaligned-data selection strategies shows $X_a^m \in \mathbb{R}^{N_a \times D^m}$7 for full PDTC at $X_a^m \in \mathbb{R}^{N_a \times D^m}$8, compared with $X_a^m \in \mathbb{R}^{N_a \times D^m}$9 for cosine similarity, $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$0 for traditional OT, and $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$1 for the single-direction PTDC variant.

Scalability experiments with 6 and 8 parties retain a large advantage. On ModelNet-10 with $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$2 aligned samples, FedHSSL drops to $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$3 for 6 parties and $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$4 for 8 parties, whereas Proto-EVFL yields $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$5 and $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$6, respectively. This suggests that the prototype-and-gating design remains effective even as the contribution inconsistency problem intensifies with additional parties.

7. Privacy profile, communication properties, and limitations

Proto-EVFL does not transmit raw data or raw gradients. The active party sends class prototypes and global priors; passive parties send aligned intermediate representations and local priors. The paper presents this as reducing some privacy risks relative to gradient-sharing VFL, though it does not claim a cryptographic privacy guarantee (Guo et al., 30 Jul 2025).

The privacy discussion is explicitly algorithmic rather than cryptographic. Intermediate representations and prototypes may still be privacy-sensitive. In a label-inference attack based on cosine similarity between class prototypes and aligned data, the reported attack accuracy is $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$7 for Proto-EVFL, compared with $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$8 for Vanilla VFL and $X_u^m \in \mathbb{R}^{N_u^m \times D^m}$9 for FedHSSL. With Gaussian noise added to representations or prototypes, moderate noise levels $N^m = N_a + N_u^m$0 still preserve an advantage over Vanilla VFL, whereas large noise $N^m = N_a + N_u^m$1 can reduce accuracy substantially, including a reported drop to $N^m = N_a + N_u^m$2.

Communication and runtime measurements also favor the method. The paper reports fewer communication rounds than SS-VFL and FedHSSL, with an average of about $N^m = N_a + N_u^m$3 rounds versus $N^m = N_a + N_u^m$4 for Vanilla VFL, $N^m = N_a + N_u^m$5 for SS-VFL, and $N^m = N_a + N_u^m$6 for FedHSSL on certain setups. On ModelNet-10, training time is reported as $N^m = N_a + N_u^m$7 hours for Proto-EVFL with $N^m = N_a + N_u^m$8 aligned samples and $N^m = N_a + N_u^m$9 hours with $M$00, compared with $M$01 and $M$02 hours for Vanilla VFL.

The stated limitations are equally specific. Proto-EVFL assumes a trusted active party under GDPR-style regulation; it does not implement secure aggregation, homomorphic encryption, or formal differential privacy; communication overhead still grows with the number of classes $M$03 and latent dimension $M$04; EM-based prior estimation depends on feature quality and prototype separation; and the convergence theory relies on smoothness, bounded variance, and bounded-domain assumptions. A common misconception would be to interpret Proto-EVFL as a generic secure VFL protocol. The paper does not support that reading. It instead proposes a class-imbalance-aware learning framework for VFL with extremely unaligned data, centered on prototype-guided representation learning and adaptive aggregation.