Deep Latent Variable Model based Vertical Federated Learning with Flexible Alignment and Labeling Scenarios
Authors: Kihun Hong
Sejun Park
Ganguk Hwang
Venue: ICLR 2026 Poster (2026)
Keywords: Vertical Federated Learning
Deep Latent Variable Model
missing mechanism
MCAR
MAR
MNAR
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We propose a vertical federated learning method that accommodates both training and inference under arbitrary alignment and labeling scenarios.

Abstract: Federated learning (FL) has attracted significant attention for enabling collaborative learning without exposing private data. 
  Among the primary variants of FL, vertical federated learning (VFL) addresses feature-partitioned data held by multiple institutions, each holding complementary information for the same set of users. 
  However, existing VFL methods often impose restrictive assumptions such as a small number of participating parties, fully aligned data, or only using labeled data. 
  In this work, we reinterpret alignment gaps in VFL as missing data problems and propose a unified framework that accommodates both training and inference under arbitrary alignment and labeling scenarios, while supporting diverse missingness mechanisms. 
  In the experiments on 168 configurations spanning four benchmark datasets, six training-time missingness patterns, and seven testing-time missingness patterns, our method outperforms all baselines in 160 cases with an average gap of 9.6 percentage points over the next-best competitors.
  To the best of our knowledge, this is the first VFL framework to jointly handle arbitrary data alignment, unlabeled data, and multi-party collaboration all at once.

Published as a conference paper at ICLR 2026
DEEP LATENT VARIABLE MODEL BASED VERTICAL FEDERATED LEARNING WITH FLEXIBLE ALIGNMENT AND LABELING SCENARIOS
Kihun Hong Sejun Park Ganguk Hwang'
Korea Advanced Institute of Science and Technology (KAIST) [nuri9911, sejunpark, guhwang) @kaist.ac. wal.st.cc.k
ABSTRACT
Federated learning (FL) has attracted significant attention for enabling collaborative learning without exposing private data. Among the primary variants of FL. verti- cal federated learning (VFL) addresses catuure-prritiooeed data held by multiple institutions, each holding complementary information for the same set of users. However, existing VFL methods often impose restrictive assumptions such as a small number of participating parties, fully aligned data, only using labeled data. In this work, we reinterpret alignment gaps in VFL as missing data problems and propose unified framework that accommodates both training and inference under arbitrary alignment and labeling scenarios, while supporting diverse missingness mechanisms. In the experiments on 168 configurations spanning four benchmark datasets, six training-time missingness patterns, and seven testing-time missingness patterns, our method outperforms all baselines in 160 cases with an average gap of 9.6 percentage points over the next-best competitors. To the best of our knowledge, this is the first VFL framework to jointly handle arbitrary data alignment, unlabeled data, and multi-party collaboration all at once.
INTRODUCTION
With the rapid development of technology, concerns about data privacy and security have been rising, highlighting the importance of research these areas. Although individuals are reluctant to disclose their personal information for model training, it is indispensable for building more accurate machine learning models. To reconcile these competing interests, federated learning (FL) (McMahan et al.. 2017 Zhang al. 2021 has emerged promising paradigm for privacy-preservinn model training, leading to the development numerous methods that address a variety of related challenges.
From broader perspective, FL be categorized into three main types: horizontal federated learning (HFL), vertical federated learning (VFL), and federated transfer learning (FTL). Whereas HFL. with scenarios in which data distributed across different samples, VFI. (Liu et al. 2024 Yang et al. 2023) focuses on cases where data is partitioned by features rather than by samples. For instance, financial institutions and e-commerce platforms, hospitals and pharmaceutical companies, or insurance firms and automobile manufacturers often hold distinct yet complementary information about the same set of users. In such situations where collaborative learning desired achieve their respective goals, VFL provides good framework for training joint model without directly revealing raw data, thereby preserving privacy.
Building the VFL paradigm, researchers have explored various strategies to enhance its utility
1022|iiu Wu et al., (022), strengthening privacy through advanced security mechanisms (Fu et al. 2022 Kang
such as improving communication efficiency (Castiglia et al.] [023a by Feng 2022 Liu Jiu et al. 2022
ct al. 2022a Sun et al. 2024; Zou et al. (022), and, crucially, boosting the effectiveness (Ganguli
al. 2023 He et al. 2024 Huang al. [222 Kang et al. 2022b Li et al. (022). A key part of
this effort involves fully leveraging available information under restricted data. Unlike conventional machine learning algorithms, VFL typically requires data be aligned, which is condition that
becomes harder to satisfy as more institutions participate. Although there may be overlap in users
"Corresponding author

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Published as a conference paper at ICLR 2026
across different institutions, their overall user sets are rarely identical, leaving some unaligned and even limiting the number of aligned samples. A recent survey on VFL (Wu et al. 2025) reports that only 0.2% of potential VFL pairs fully alignable. Moreover, in healthcare, banking, or insurance sectors that demand VFL. acquiring labeled data is particularly challenging due to strict privacy regulations and the sensitive nature of the information. Hence, effectively utilizing unaligned or unlabeled data becomes crucial.
Several studies have attempted address these issues, but frequently rely on restrictive assump- tions. Some works are tailored to a small and fixed number of parties (most commonly two) (Kang
al. 2022a b;
Li al. 2022; Yang et al., 2022). while others do not fully exploit partially aligned
data (Feng 2022 He et al. 2024 Zuang et al.
2023), thereby wasting potentially valuable infor-
mation. addition, many approaches restrict inference to fully aligned data or specific parties 2023; Kang et al., (022b), limiting real-world
only (Feng, 2022; He et al., 2024 Huang et al.]
applicability. On the other hand, some researches (Ganguli al. 2023] Sun et al. (024) focus on inference with any type of unaligned data but do not consider training unaligned data.
Before introducing our method, we distinguish two kinds unaligned data: (a) Potentially alignable yet currently unlinked records caused by imperfect identifiers, which can often be addressed via
record- -onkaee techniques (Hardy et al. [017] Nock et al.) 2021]] | Wu et al.)
2024); and (b) Inherently unalignable records that have genuine counterparts across parties, making alignment fundamentally impossible. Our primary focus the latter that parallels blockwise missingness in the classical missing data literature. This distinction motivates the following discussion on missingness mechanisms.
Beyond VFL, many studies in machine learning have investigated tackle missing incomplete data. A fundamental first step in this line of work is to identify the underlying missingness mechanism, generally categorized into three types (Ghahramani Jordan
1995 Little & Rubin 2019) :
MCAR (Missing Completely Random): Missingness occurs entirely independent of both observed and unobserved values.
MAR (Missing at Random): Missingness may depend on observed values but not on unobserved ones.
MNAR (Missing Not at Random): Missingness may depend on both observed and unob- served values.
Viewing alignment in VFL through the lens of missingness, we can treat unaligned data as in standard missing data scenario. For example, recent work (Valdeira et al., 2024) applies the MCAR assumption to address arbitrary types of unaligned data, but it does not incorporate unlabeled data. Inspired by prior research (Ipsen et al. 2022]. we propose a novel algorithm, FALSE-VEL (Flexible Alignment and Labeling Scenarios Enabled Vertical Federated Learning), unified framework for VFL under diverse alignment and labeling scenarios.
Contributions. WALSE-VFLL (i) supports both training and inference under arbitrary alignment and labeling conditions in multi-party VFL, (ii) accommodates all three missingness mechanisms (MCAR, MAR, MNAR), and (iii) surpasses all baselines in 160 out of 168 configurations with substantial performance gap, average of 9.6 percentage points over the next-best competitors. To the best of our knowledge, it is the first framework that simultaneously addresses these challenges in practice.
2 RELATED WORK
Vertical Federated Learning Models. A variety of VFL approaches have been proposed to handle unaligned or unlabeled data across different institutions. Two-party methods such as VFed-SSD (Li
al.) 2022) and FedCVT (Kang al. 2022b) leverage semi-supervised or psuedo-label training, but
cannot scale beyond two parties.
For multi-pate partt settings, subsequent work explores feature selection (VFLFS (Feng 2022 ), rep- resentation transfer (VFedTrans (Huang et al.) (023)), self-supervised objectives (FedHSSL (He
al. 2024)) and robustness dropped parties (MAGS (Ganguli al. 2023), Vlug (Sun et al.
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Table 1: Flexibility of various VFL algorithms.
Training Data
Unaligned Inference
VFL Algorithms
# &Patties
Labeled
Unlabeled
Aligned
Unaligned
Aligned
Unaligned
VFed-SSD Li al.
2022
FedCVT Kang el al. 2022b
VFLFS (Feng, 2022
2 2
VFedTrans (Huang et al. 2023
A
2 2
MAGS (Gngguli ct al. 2023
2 2
PlugVHi. (Sun
(Sun of al. 2024
2 2
FedHSSL (H el al.
2024
A
2
LASER-VEL VFL
Valdeira et al.
2024
V
2
FALSE-VFL. (Ours)
4
)024)). LASER-VFL Valdeira et al., 2024) fully exploits aligned and unaligned samples but ignores unlabeled ones.
In I summarize the flexibility of these VFL algorithms and our proposed model in terms of data usage, unaligned inference, and the number parties each method can accommodate. Here, unaligned inference refers whether
algorithm can perform inference on partially and fully unaligned data (see Section 3.1 for the definitions of partially aligned and fully unaligned). A v Kindicates that the algorithm fully exploits the corresponding data type, while A indicates partial exploitation. For example, some approaches may treat partially aligned data as fully unaligned. Also, A for unaligned inference indicates the algorithm can perform it but require constructing 2 predictors where n is the number of parties. As shown in the table, our proposed model can handle all forms of training data and inference scenarios.
Deep Latent Variable Models. Deep latent variable models (DLVMs) (Kingma & Welling 2014
Rezende et al. 2014) have demonstrated their utility in capturing complex and high-dimensional data
structures including datasets with missing values. While these models are widely used in tasks like
generative modeling (Burda al., [0166] KKingma+ & Welling [2014] Rezende & Mohamed] 2015).
they are also known to be effective in data imputation (Ipsen et al. 2021 Ivanov et al. 2019 Mattei
Frellseen 2019) which crucial for handling missing values.
For instance, MIWAE (Mattei & Frellsen 2019) leverages DLVMs for imputing missing values under the assumption of MAR mechanism, while not-MIWAE (Ipsen al.) (021) extends it handle MNAR scenarios. Furthermore, supMIWAE (Ipsen et al., 2022) introduces a supervised learning framework for incomplete datasets, imputing missing values under MAR assumption.
These advances in DLVMs illustrate the flexibility and power of probabilistic models in addressing missing data challenges, enabling effective imputation strategies depending on the nature of the missingness mechanism.
3 OUR METHOD: FALSE-VFL
3.1 DATA SETTING AND NOTATIONS
We consider a vertical federated learning (VFL) setting with one active party which possesses labels
and K passive parties. Each party k E [K] := {1, 2. K} holds set of observations 1i=1 For each sample the complete observation across all parties is denoted by 2; = [x)] x; K Only the active party K owns the labels {u, E RJ which correspond to Ti-1 where d Σ dr. Critically, both the observations and labels are private, meaning that they cannot be shared directly among the parties.
{ d NN
the complete observations {a; Rd
In real-world scenarios, data may be incomplete with missing labels unaligned observations across parties. We address this by introducing the following assumptions regarding data availability. Let mf E (0, 1} indicate whether the data a. is observed in party k i.e., m 0 denotes it is observed
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and m denotes it is not. Define m, [m.. m observations across all parties. Similarly, let u, E {0, 1] indicate whether the label yi is available with u, 1 representing a missing label.
m represent the availability of
Each sample x; can be split into (artth, 2.!iis] where with denotes the observed parts (i.e., the set of æ where m = 0) and æ denotes the missing parts. Importantly, we only have access to 2it and the missing data 2. is not available. Additionally, u; 1, the corresponding label y: NA. Under this framework, the set { u, = 1, E [N]] represents the unlabeled data, while {a=?he u, 0, [N]] represents the labeled data.
The alignment of observations across parties can be represented with our missingness notation. Fully aligned observations where all parties contribute to the data, are represented as {æ, Σx= m = 0, [N]] Fully unaligned observations where data from all but one party missing. are represented as {ar Σ m. = K - 1, E [N]]- The remaining observations where 0 < Ek= m K 1, considered partially aligned. For simplicity, refer to fully aligned observations as "aligned" and consider both fully unaligned and partially aligned observations as "unaligned".
This formulation provides a unified framework for addressing incomplete data, accommodating both unlabeled and unaligned observations.
3.2 PROBLEM AND APPROACH
In this work our goal is to solve supervised learning tasks including regression or classification tasks with deep neural architectures on the training dataset {aothx 8/. m, uz including unlabeled data and incomplete observations. If complete observations were available, the standard approach would involve maximizing the log-likelihood function Em.=o,ie(M) log pe (y. However, due to the significant presence unlabeled data and incomplete observations, more sophisticated procedures are required.
Inspired by the methodology in Ipsen et al. (2022). we leverage deep latent variable models (DLVMs) (Burda et al. 2016 Kingma & Welling 2014 Rezende al., 2014) to build new predictive model a x m) where & parameterizes the function between æ and m, while incorporating unlabeled data into the training process. Under the MAR assumption, we have pe, (ylz m) = pe (y) (he) as shown in Appendix A.1 We therefore present two version of our method:
FALSE-VFL-L assumes MAR mechanism and is detailed in Sections 3.3 to 3.5
PALSE-VFL-IH relaxes assumption MAR to MNAR mechanism and described in Appendix A.2
Below we outline how the approach of Ipsen et al. (2022) is integrated into FALSE-VER (MAR case).
To effectively utilize the large amount of unlabeled data, as pretraining step we first consider the marginal log-likelihood ΣiENN [N] log pe, (m:")) and maximize where O, ô represents the generative model parameters. To model the generation of observations, we introduce sequence of stochastic hidden layers with latent variables h = {h,,-. ,h²} as described in Burda et al. (0016):
pa, (x) (rr)=/(pe
/ pe,
/eppe (h"))e, (h4-1 h ....)...pp...p..... (e, (x dh.
Next, training step we leverage the labeled data maximize the log-likelihood function Emm0, log (y; (z.p))) Since raw observations x: cannot be shared across parties due
to privacy constraint, rely the latent variable h², the output of the stochastic layer preceding the observations, to generate the corresponding labels. This structure results in graphical model illustrated in Fig. where h == h and == h² The detailed explanation of Fig. will be provided in Section 3.3. Note that, for reasons of computational tractability, we fix the parameters θ, after
In the whole remaining context, we use I. 2 for simplicity. However, note that we can use any I, 2 2 with small effort.
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pretraining and maximize the conditional log- likelihhoddd Eu.=0,<e(N] log pe (e (x with fixed gg
which is shown to be equivalent to maximizing the joint log- log-likelihood Σ E,,=0,e log Pe bogh)).
We will explain how the pretraining and training steps can achieve our goal effectively in Section
3.3 MODEL ARCHITECTURE
From broader perspective, model is divided into feature-side and label-side modules. Each party & E [K] has its own encoder (parameterized by - and decoder (parameterized by 0²), which we collectively refer to as the feature-side modules. Meanwhile, the ac- tive party which holds the labels additionally has global encoder (parameterized by %.). global decoder (parameterized by 0,), and a discriminator (parameterized by σ) which form the label-side modules. For simplicity, we define the collections of feature-side parameters as Te = (7c. ... , > and 0g {0): ... _. Thus, we can express the overall model parameters θ and O
Figure 1: Graphical model for FALSE-VEL-I
7o. 0.). = (Tc: De: 7a. 0z.00. 02 and θ = (7e 7s,
All encoders and decoders produce parameters for pre- defined distributions (e.g., mean and variance for Gaus- sian distributions). Specifically, we assume that the prior distribution p(z) is standard Gaussian, while Pe, (h) and Pe. (a: h)) = [IEE[K] Pay (a* (h) are mod- eled as Gaussian distributions.
Ma.
Oked Dwckers
Nit,,
We employ amortized variational inference to approx- imate the posterior distributions. Specifically, the ap- proximate posterior Ir (h) bbe) for the latent variable h is given by Gaussian distribution
Uu, Xil
N
Σ kEoba
M (ar").
Σ kEobs
(1)
Oliii
Jobel
where pag (x*) and E.e (x*)) denote the outputs of en- coder from party kt The approximate posterior 2-, (z(h) for z is also assumed to be Gaussian. The rationale behind the formulation in L will be explained in
AfT Paay Kill
Uee. Xe,))
Appendix B For the discriminative part, we assume (y(h) be a Gaussian distribution for regression tasks and to be categorical distribution for classifica- tion tasks. The complete computational flow is summa- rized in Fig. 2 With this setup, are now ready to explain the whole algorithm including pretraining and prediction.
Pary Buodie
Pary Docader
(g Puo)
Cettina
Paog Ficha
Figure 2: Computational structure for FALSE-VFL-I
3.4 ALGORITHM
Our method consists of three steps: pretraining, training, and prediction. We present the details of our algorithm below and provide its overview in Algorithm Tof of Appendix D33
Pretraining with Marginal Likelihood Maximization. From our graphical model assumption, the log- likeliood of the observed features can be written as:
log pe, (x: obs) log pa. obe ph) (hl=)p(z) dhdz.
This integral is intractable, tor (Burda et al. 2016) with the approximate posterior. First, define
approximate it using k-sample inportance-weigghed estima-
Po_(ar (2: Mo, (h,l=,))(zz)
Ri (a =
4x (2g obo )a,, (z,
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where { are i..d. random variables with distribution 9% (h| )47, (z) (t). By construction, Rx (xxbe) is unbiased estimator of pe, (xobr), Applying Jensen's incquality to log [R,,, (xx))
we get the lower bound L. (O,) of the log-likelihooo of the observed features:
Σ
logpe, (arghe) (Og) = Σ
[(h,,,,, 1 (bla," "er 1er (z(h) [log Rx Rxx (xe
KE[N]
iE[N]
Finally, we just maximize f.x with respect to gg
Training with Conditional Likelihood Maximization. After we pretrain our model with marginal likelihood maximization, fix the parameters Gg Similarly as above, let
Polulky)De (oo, (ar [h (hs)po, (hy 2z)p(zz))
Ry 2
O7 (h2 aoo )4,, (z, ()
Then, we also get the lower bound C, (Ø) of the log- loglliillhood of the labeled data:
Pe (Wi, x; () 2 (o) =
Σ
E (1,), mare (hla; )a,, (=(h) llog R (yis xe:tn)
[,=0,
iE|N]
Finally, we maximize C, with respect d.
Prediction. After all training steps are done, we can predict the label of new (incomplete) observa- tions with self-normalized importance sampling method in Ipsen et al. (2022):
po. (a h Po, (hu 2 p(())
pe (yll wipb(yl where wn
r1
+rl
1% (hi ob 99, (z) hw)
and (h1, 21), (hL, xL) are i.d. samples from 4% (hl
190 (z)h).
Convergence Properties. We summarize here convergence properties of L.. and Cyx- The formal statement with the mild regularity conditions and the proof are given in Appendix C
Theorem 3.1. Lxx resp.) increases as increases, and bounded above by log p(a (log ply, aobe), ). In addition, Ln resp.) converges to log p( (logply, ply, x xbb)), resp.) as ls under mild regularity conditions.
3.5 COMMUNICATION BETWEEN PARTIES
In the pretraining and training steps, each party computes the mean and variance of its approximate posterior and sends this local latent representation the active party. The active party aggregates the local latent representations into a global latent distribution, samples latent variables h from this distribution, and broadcasts them to the participating parties so that they can evaluate pe. (a (h) via their local decoders. Each passive party returns the scalar probabilities to the active party which uses them to compute the loss and sends back gradients for the local model parameters of each party.
In the inference step. we follow the same forward communication pattern as in the pretraining and training phases, but no gradients are exchanged.
Thus, compared to the standard VFL, FALSE- VFL-I follows the same basic communication pattern where local representations are sent from the passive parties to the active party and the gradients are sent in the reverse direction, but with two additional steps: sampled latent variables from the global posterior sent from the active party to the others, and scalar probabilities sent back to the active party.
MAXIMIZING CONDITIONAL LIKELIHOOD THROUGH TWO-STAGE TRAINING
The primary objective of our work is to predict target variables y (either continuous or discrete) based observed features obs To achieve it, aim to maximize the conditional log-likelihood

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Eu.=0,eENN] log (y: )) However, this objective is intractable even with variational approx- imation. To address this issue, propose detour by maximizing the joint log-likelihood Eu.=,,E(NN] log pe (V:, xe)h) which can be optimized using variational approximations.
The connection between the joint and conditional log-liielhhoods is expressed as:
pe xh = log pe (1) + log pe, (x=
This relationship indicates that maximizing the joint likelihood pe (y, x 2he) inherently involves both the conditional term pe (y/ obb) and the marginal term pe, (xxd)). Due to the high dimensionality of abs where dobs > 1, the term po,, (xr may dominate the learning process, leading to implicit modeling bias as discussed Zhao et al. (2019). This imbalance can result a model focusing more on the marginal likelihood and less the conditional likelihood, ultimately limiting the performace of pe (b)
Two-Stage Training Strategy To mitigate this imbalance, we introduce two-stage training process:
Pretraining with Marginal Likelihood Maximization: To avoid the implicit mod- cling bias, we first maximize the marginal likelihood pe, (xdh)) which constitutes the dominant part of the joint likelihood pe(y, abs).
Stage 1
Stage 2 Training with Conditional Likelihood Maximization: Once pe, (x=bb) ) is optimized, we freeze the parameters θ and proceed to maximize the joint likelihood pe (y obs)).
Since θ is fixed, this step effectively focuses on maximizing the conditional likelihood pe ()) aligning directly with our goal of optimizing conditional predictions.
Freezing 6, not only reduces the implicit modeling bias from pe, (x=b)), but also makes the objective of maximizing pe (y, bbb) equivalent to maximizing pe (b))) This two-stage approach provides more focused path towards optimizing the conditional likelihood. Morcover, also has the following important properties.
Interpretation as Feature Learning The two-stage training process can be also interpreted in terms of feature learning. In this view, the first step (maximizing Pe, (z:"))) serves as pretraining phase where the generative model learns robust representation of the feature space. Once this representation is well-learned, the second step leverages this generative model to facilitate the maximization of pe py(y) ()) improving the conditional prediction.
Incorporating Unlabeled Data An additional advantage of this two-stage approach the ability to incorporate unlabeled data. Given the abundance of unlabeled data in many practical scenarios, optimizing Pe, (xx2)) allows the model utilize them, even though labels are absent. While unlabeled data cannot directly optimize pe (y) (és). plays a crucial role in optimizing marginal likelihood pe, (bs))), thus ensuring the model effectively leverages all available data during training.
EXPERIMENTS
In this section, compare FALSE-VFL with several bascline algorithms on four benchmark datasets.
Baselines. Vanilla VFL, LASER-VEL Valdeira et al. 2024 Plug (Sun et al., 2024 and FedHSSL (He al. 2024 evaluated; implementation specifics for each baseline are provided in Appendix D.I
Datasets and Models. We use Isolet (Cole et al.) 1990), HAPT (Reyes-Ortiz et al. 2016), Fash- ionMNIST (iiao] 2017), and ModelNet10 (Wu et al. 2015). Isolet and HAPT are tabular, whereas FashionMNIST and MedelNet1O are image datasets. For tabular tasks we employ simple multilayer perceptrons, while for image tasks adopt ResNet-18 18 backbone (He et al. (2166) as the feature extractors and two colne-con layers for the fusion models. More detailed explanations on datasets and models appear in Appendix D.2
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Setup. Isolet, HAPT, and FashionMNIST are partitioned across eight parties, and 1odelNet10 across six; exact feature splits are explained in Appendix[D22 We provide 500 labeled samples for each tabular dataset and 1,000 for each image dataset, treating all remaining samples Since Vanilla VFI., Plug VFL., and FedHSSI. originally require fully aligned data during training, we reserve 100 (tabular) or 200 (image) of the labeled samples fully aligned and render the remainder partially aligned according to the designated train data missing mechanisms. All test samples are partially aligned, using the designated test data missing mechanisms. Section 3.1 expllinns the missing mechanisms we adopt.
unlabeled.
5.1 MISSING MECHANISMS
We evaluate all algorithms under MCAR, MAR, and MNAR assumptions for data alignment.
MCAR: To generate unaligned data with the MCAR mechanism, we impose missing probability p for each sample in each party. We experiment with p E {0.0, 0,2,0....00 0.5} and denote as MCAR MCAR 2, and MCAR 5, respectively.
We designs two MAR mechanisms, MAR and MAR 2, motivated by real-world scenarios in which subsequent observations depend what has already been seen. MAR I is designed to seek for single highly informative party, whereas MAR 2 is designed to seek multiple moderately informative parties. Precise formulas are given in Appendix D.4
MAR:
After normalizing each dataset, let x² be the mean of features held by party l in a given sample. If ã the party is dropped with probability y: otherwise it is dropped with probability 1 - p. We experiment with E {0.7, 0.9} and denote as MNAR 7 and MNAR 9, respectively.
MNAR:
5.2 EXPERIMENTAL RESULTS
We train every method on the four benchmarks under six training missingness regimes (MCAR MCAR 5, MAR 1, MAR 2, MNAR 7, MNAR 9) and evaluate them on seven test patterns (MCAR O, MCAR 2, MCAR 5, MAR 1, MAR 2, MNAR 7, MNAR 9). The results are shown in Fig. 3 and exact numerical values are reported in Tables 13 I8 Although FALSE-VFL-I does not model the mask distribution explicitly, its performance is consistently close to, and occasionally higher than, that of KALSE-VFL-IE To provide concise overview, compare the best-performing FALSE-VEE variant with the strongest baseline for each dataset, training mechanism, and test mechanism in Table 2 As shown in Table 2c the performace gap widens the test missingness rate increases. In all comparisons, FALSE- VFL. achieves a clear lead.
Table 2: Average accuracy gap (%) between the best FALSE-VFI. variant and the best competing baseline.
(a) Per dataset
(b) Per training missing data mechanism
Isolct
HAPT
F-NNIST
ModelNet
MCAR 2
MCAR
MAR
MAR 2
MNAR 7
MNAR
Gap
15.7
7.3
4.5
9.1
Gap
5.3
10.4
9.2
9.6
11.4
8.9
(c) Per test missing data mechanism
MCAR 0 MCAR
MCAR
MAR
MAR
MNAR
MNAR
Gap
9.6
10.9
12.3
10.8
13.1
Overall, FALSE-VFL, exceeds the strongest baseline by average of 9.1 percentage points across all 168 configurations. This average differs slightly from 9.6 reported in the abstract since the value calculated on 160 cases.
Robustness to higher missing rates. To assess the impact of more severe missingness, we compare models trained under MCAR 5 with those trained under MCAR 2. Table 19 reports, for each setting, the ratio of mean accuracy obtained when training under MCAR FALSE- VFL exhibits the smallest performance decrease 23 of 28 cases and even improves on most test sets, demonstrating markedly stronger robustness missingness than the competing methods.
that obtained under MCAR 2.
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MONO RAKE
(a) Isalet
(b) HAPT
(c) FashionMNIST
(d) ModelNet10
Figure Mean accuracy (%) of six VFL methods trained under six missingness mechanisms and evaluated across seven test patterns. The columns correspond to the datasets: Isolet, HAPT, FashionMNIST, and ModelNer10 from left to right; the rows correspond to the training mechanisms: MCAR 2, MCAR 5, MAR 1, MAR 2, MNAR 7, and MNAR 9 from top to bottom. Bars show mean over five independent runs; error bars show +1 standard deviation.
Robustness to the number of parties. We further evaluate robustness with respect to the number of parties K, varying it from 4 to 12. We do not include MAR in the ablation since the missingness of party i depends on the other parties, so changing K alters the mechanism itself and prevents fair comparison. We therefore report only MCAR 2 and MNAR in Fig. 4 Recall that we fix a small subset of labeled samples to be fully aligned (e.g.. 100 when the total 500), and the remaining labeled samples follow the specified missingness mechanism. As result, the expected number of additional fully aligned labeled samples falls quickly as K grows.
Under MCAR 2, Vanilla VFL and FedHSSL tend to decline as K increases because the probability that a labeled sample is simultaneously observed by all parties drops rapidly with larger K, and these methods train only on the fully aligned labeled data. Under MNAR 7, the missing rate is high, so beyond the fixed fully aligned labeled subset the expected number of additional fully aligned labeled samples is negligible for K > 4. In this regime, these two methods often stay flat or improve as K grows. This effect arises since splitting the same total features across more parties makes a complete loss of informative features much less likely, and the fraction of observed parties per sample varies less around its mean, which makes the fused representation more consistent.
VFL and our method can learn from unaligned labeled samples, so they do not rely only on the fully aligned labeled subset. As K increases they benefit from the same effect, namely a much lower chance that all informative features are missing once and more stable observed party fraction per sample, so their performance generally improves or remains stable under both MCAR 2 and MNAR 7.

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LASER- VFL effectively leverage unaligned samples, but assumes batch-wise missingness mask. As K grows, the space of possible missingness patterns explodes, making increasingly difficult assemble mini-batches that share single mask and effectively lowering batch utilization. Consequently, its performance tends to degrade. Across all configurations, FALSE- SFLLV consistently outperforms all baselines regardless of the number of parties.
(a) Isolet-C
(b) Isolet-N
(c) HAPT-C
(d) HAPT-N
(e) FMNIST-C
(I) FMNIST-N
Figure 4: Mean accuracy (%) with varying numbers of parties for six VFL methods. Panels with suffix "C" (Isolet-C, HAPT-C, FMNIST-C) are trained and evaluated under MCAR 2, and panels with suffix "N" (Isolet-N, HAPT-N, FMNIST-N) are trained and evaluated under MNAR 7. Solid lines show mean accuracy five independent runs; shaded bands show #1 standard deviation.
Robustness to data heterogeneity. We provide an additional ablation data heterogeneity. In an eight-party setting, we draw party specific missing rate vector from Dirichlet distribution Dir(a) and set the per party missing probabilities to p = 1.6 w E [0, 1J so that the average missing rate is 0.2; if any entry of p exceeds I, we resample w. we consider E foo, 10, 1, 1,0.1}. When co we obtain w (1/8,... 1/8) and hence p (0.2. ... 0.2), which corresponds MCAR To quantify heterogeneity, we report the entropy of the sampled for each in Table Test accuracies are shown Fig. 5 Across all datasets, LASER LALER-VFL. improves as a decreases (i.e., heterogeneity increases). This is expected since uneven per-party missing rates make the mask distribution more concentrated over a smaller set of configurations, which simplifies forming mini-batches that share single mask and thus improves batch utilization in LASER-VEL VFL. By contrast, Vanilla VFL and FedHSSL decreases as a decreases, since lower entropy reduces the fraction of aligned samples and shrinks the effective training set. FALSE-VEL VFL remains robust, staying flat and highest across all
Table 3: Entropy of sampled from Dir(a) for each Larger entropy im- plies closer to uniform.
(g) Isolet
(b) HAPT
(i) FashionMNIST
Q
0C
10
0.1
Isolet
2.08
2.02
1.74
1.11
Figure 5:
Mean accuracy (%) of VFL methods
HAPT
2.08
2.05
1.78
1.16
trained and evaluated under four heterogeneous missing- ness mechanisms sampled from a Dirichlet distribution. Solid lines show mean accuracy over five independent runs; shaded bands show +1 standard deviation.
FMNIST
2.08
2.07
1.47
0.73
CONCLUSIONS
We introduce FALSE-VFL, vertical federated learning framework that utilizes both unlabeled and unaligned data, supports inference on unaligned data, and accommodates all three missing data mechanisms (MCAR, MAR, MNAR) in both theory and practice. Extensive experiments covering six training and seven test missingness settings show that FALSE- VFL surpasses the existing methods in almost every configuration with clear gap. Additional ablations missing rates, the number of parties, and data heterogeneity further validates its robustness.
These findings demonstrate that FALSE-VER VFL is practical step toward privacy preetecvipeeeiinggg collab- oration in roal-world feature- partitiont settings where labels are scarce and perfect alignment is
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ACKNOWLEDGMENTS
This work was supported by the National Research Foundation Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2019-NR640050,,
REFERENCES
Yuri Burda, Roger B. Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. In International Conference Learning Representations, 2016.
Timothy Castiglia, Shiqiang Wang, and Stacy Patterson. Flexible vertical federated learning with heterogeneous parties. IEEE Transactions on Neural Networks and Learning Systems, 2023a.
Timothy Castiglia, Yi Zhou, Shiqiang Wang. Swanand Kadhe, Nathalie Baracaldo, and Stacy Pat- terson. Less-vfl: Communication--ffici feature selection for vertical federated learning. In International Conference Machine Learning, pp- 3757-3781. PMLR, 2023b.
Xinlei Chen and Kaiming He. Exploring simple siamese representation learning. In Proceedings of the JEEE/CVF conference on computer vision and pattern recognition, pp. 15750-15758, 2021.
Ron Cole, Yeshwant Muthusamy, and Mark Fanty. The ISOLET spoken letter database. Oregon
Graduate Institute of Science and Technology, Department of Computer .... 1990.
Siwei Feng. Vertical federated learning-basse based feature selection with non-overlapping sample utiliza- tion. Expert Systems with Applications, 208:1 18097, 2022.
Fangcheng Fu, Huanran Xue, Yong Cheng, Yangyu Tao, and Bin Cui. Blindfl: Vertical federated machine learning without peeking into your data. In Proceedings of the 2022 International Conference on Management of Data, pp. 1316-1330, 1330, 2022.
Surojit Ganguli, Zeyu Zhou, Christopher G Brinton, and David I Inouye. Fault-tolerant serverless vfl over dynamic device environment, arXiv preprint arXiv:2312.1660 2023.
Zoubin Ghahramani and Michael Jordan. Learning from incomplete data. 1995.
Stephen Hardy, Wilko Henecka, Hamish Ivey-Law, Richard Nock, Giorgio Patrini, Guillaume Smith, and Brian Thorne Private federated learning on vertically partitioned data via entity resolution and additively homomorphic encryption. arXiv preprint arXiv: 2017.
Kaiming Hc. Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference computer vision and pattern recognition. pp. 770-778, 2016.
Yuanqin He, Yan Kang, Xinyuan Zhao, Jiahuan Luo. Lixin Fan, Yuxing Han, and Qiang Yang. A hybrid self-supervised learning framework for vertical federated learning. IEEE Transactions on Big Data, 2024.
Chung-ju Huang, Leye Wang, and Xiao Han. Vertical federated knowledge transfer via representation distillation for healthcare collaboration networks. In Proceedings of the ACM Web Conference 2023, pp. 4188-4199 2023.
Niels Bruun Ipsen, Fierre-Alexande Mattei, and Jes Frellsen. not-MIWAE: Deep generative mod- elling with missing not at random data. In International Conference on Learning Representations, 2021.
Niels Bruun Ipsen, AlexanAe Mattii, and Jes Frellsen. How to deal with missing data in supervised deep learning? In International Conference on Learning Representations, 2022.
Oleg Ivanov, Michael Figurnov, and Dmitry Vetrov. Variational autoencoder with arbitrary condition- ing. In International Conference Learning Representations, 2019.
Yan Kang, Yuanqin He, Jiahuan Luo, Tao Fan, Yang Liu, and Qiang Yang. Privacy-preserving
federated adversarial domain adaptation over feature groups for interpretability. IEEE Transactions Big Data, 2022a.
11

---

Published as a conference paper at ICLR 2026
Yan Kang, Yang Liu, and Xinle Liang. FedCVT: Semi-supervised vertical federated learning with cross-view training. ACM Transactions on Intelligent Systems and Technology (TIST), 13(4):1-16, 2022b.
Diederik P. Kingma and Max Welling. Auto-encoding variational Bayes. In International Conference on Learning Representations, 2014.
Wenjie Li, Qiaolin Xia, Junfeng Deng, Hao Cheng, Jiangming Liu, Kouying Xue, Yong Cheng, and Shu-Tao Xia. Sepi-super cross-silo advertising with partial knowledge transfer. arXiv preprint arXiv:2205. 15987, 2022.
Roderick JA Little and Donald Rubin, Statistical analysis with missing data, volume 793, John Wiley & Sons, 2019.
Yang Liu, Xinwei Zhang, Yan Kang, Liping Li, Tianjian Chen, Mingyi Hong, and Qiang Yang. Fedbed: A communication-ed -ffeciiennt collaborative learning framework for distributed features. IEEE Transactions on Signal Processing, 70:4277-4299 2022.
Yang Liu, Yan Kang, Tianyuan Zou, Yanhong Pu, Yuanqin He, Xiaozhou Ye, Ye Ouyang, Ya-Qin Zhang, and Qiang Yang. Vertical federated learning: Concepts, advances, and challenges. IEEE Transactions on Knowledge and Data Engineering, 2024.
Pierre-Alexandre Mattei and Jes Frellsen. MIWAE:
Deep generative modelling and imputation of incomplete data sets. In International conference on machine learning. pp. 4413-442) PMLR. 2019.
Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera Arcas. Communication-fffcieent learning of deep networks from decentralized data. In Artificial intelli- gence and statistics, pp. 1273-1282. PMLR, 2017.
Richard Nock, Stephen Hardy, Wilko Henecka, Hamish Ivey-Law, Jakub Nabaglo, Giorgio Patrini, Guillaume Smith, and Brian Thome The impact of record linkage on learning from feature partitioned data. In International Conference on Machine Learning, pp. 8216-8226. PMLR, 2021.
Jorge-L Seyes-Ortiz, Luca Oneto, Albert Samå, Xavier Parra, and Davide Anguita. Transition-awe human activity recognition using smartphones. Neurocomputing, 771:754-767, 2016.
Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International conference on machine learning, pp. 1530-1538. PMLR, 2015.
Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International conference machine learning, pp. 1278-1286. PMLR, 2014.
Jingwei Sun, Zhixu Du, Anna Dai, Saleh Baghersalimi, Alireza Amirshahi, David Atienza, and Yiran Chen. Plugvfl: Robust and ip-protecting vertical federated learning against unexpected quitting of parties, In 2024 IEEE International Conference on Big Data (BigData), pp. 1124-1133. IEEE, 2024.
Pedro Valdeira, Shiqiang Wang, and Yuejie Chi. Vertical federated learning with missing features during training and inference. arXiv preprint 2...i::2100.2256: 2024.
Zhaomin Wu, Qinbin Li, and Bingsheng He. Practical vertical federated learning with unsupervised representation learning. IEEE Transactions on Big Data, 2022.
Zhaomin Wu, Junyi Hou, Yiqun Diao, and Bingsheng He. Federated transformer: Multi-party vertical federated learning on practical fuzzily linked data. arXiv preprint urXiv:2410.17986 2024.
Zhaomin Wu, Zhen Qin, Junyi Hou, Haodong Zhao, Qiabin Li, Bingsheng He, and Lixin Fan. Vertical federated learning in practice: The good, the bad, and the ugly. arXiv preprint arXiv:2502.088 166,0 2025.
06000
Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1912-1920, 2015.
12

---

Published as a conference paper at ICLR 2026
H Xiao. Fashion-mnist: novel image dataset for benchmarking machine learning algorithms. arXiv preprint AXtV:1708.077477 2017.
Liu Yang, Di Chai, Junxue Zhang, Yilun Jin, Leye Wang, Hao Liu, Han Tian, Qian Xu, and Kai Chen. A survey on vertical federated learning: From layered perspective. arXiv preprint arXiv:2304.01829, 2023.
Yitao Yang, Xiucai Ye, and Tetsuya Sakurai. Multi-view federated learning with data collaboration. In Proceedings of the 2022 14th International Conference on Machine Learning and Computing. pp. 178-183, 2022.
Chen Zhang, Yu Xie, Hang Bai, Bin Yu, Weihong Li, and Yuan Gao. A survey federated learning. Knowledge-Baud Based Systems, 216:106775, 2021.
Shengjia Zhao, Jiaming Song, and Stefano Ermon. InfoVAE: Balancing learning and inference in variational autoencoders. In Proceedings of the aaai conference on artificial intelligence. volume 33, pp. 5885-5892, 2019.
Youran Zhou, Sunil Aryal, and M Reda Bouadjenek. A comprehensive review of handling missing data: Exploring special missing mechanisms. arXiv preprint arXiv, 2404, 2024.
Tianyuan Zou, Yang Liu, Yan Kang, Wenhan Liu, Yuanqin He, Zhihao Yi, Qiang Yang, and Ya fa-Qin Zhang. Defending batch-leved label inference and replacement attacks in vertical federated learning. IEEE Transactions on Big Data, 2022.
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A
MISSING DATA MECHANISMS
(Re,
Glena Blaondrrd
= Bemoaliigy)
78
Natt. Xe)
h
02
Utu Ex)
Xining
Giktaal Encodar
Party 2 [K]
0
22 May Appe
(po.. Es,]
Dasiminter
N
E.s)
Catley
Paty i do
Figure Left: Graphical model for FALSE-VFL-II The left dotted box groups feature-side modules. while the right dotted box groups label-side modules. Right: Computational structure for FALSE- VFI.-II.
A.
MAR MECHANISM
Assume that missing data occurs with the MAR mechanism. Then,
log pes, v (y) l .m) log pa, (y. 10x .m) log poo .m)
log /
mis po(ulh)ps(mlaa" mis spe, maa h)
dhdx Px th m)
log (mola / poly h)pe, (oo min h) dhdæ
log po (x ub m)
log (mblx log pe(u, 2 log log m)
(oyl
where the MAR assumption is used in the third equality.
A.2 MNAR MECHANISM
Under the MNAR mechanism, need to model the mask explicitly through an additional component Pe (m xx). In addition to the existing models in the MAR case, each party & E [K] has its own missing indicator parameterized by v which maps x to Par a parameter Bermoulli distribution. Note that s { heejk] and pe (m|z) [kee[K] Pas (m*) The complete computational structure shown in Fig. 6. Under the graphical model depicted in Fig.
adopt procedure
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analogous original algorithm by first maximizing (. v)) as pretraining step where log pe,, (a: mm;) 2 g.)
SE[N]
ay mare (hlaath)) C (2 ha)po (cmis)l [log R .m:)]
:E|N]
with
Pv (m 1 aeguin)po. (wo 1 ppo (h,,(, (1)1z))p(zz)
Re cbs m.
K Σ
1
9r (1e) (z 19r, (z) (t2)
This follows from
log pe, pe,,, (2 m) = log Pa (m lbs mo yo, obs po mas ho (h)z)p(z) dhdzda.
We assumed here Pe, (x//) is fully factorized, so that po, (a/h)) = po, (za pe. (x mis h).
After the pretraining step, fix the parameters θg. uh and proceed with an analogous training as in our main algorithm. summary, to handle the MNAR case, we only need an additional model Py m ar) and the sampling of mmis
We also describe what information is communicated between the parties FALSE-VEL-IL
In the pretraining and training steps, there is additional exchange for samples with missing parties, in addition to the communication in FALSE- VFL-I. The active party sends sampled latent variables from the global posterior to the missing parties. Each such party the received samples with its local decoder and missing indicator to compute the missingness probabilities, and sends them back to the active party.
In the inference step, we follow the same forward communication pattern as in the pretraining and training phases, but no gradients are exchanged.
B FEATURE-SIDE CONTRIBUTIONS TO POSTERIOR APPROXIMATION
As defined in Section 3.3 the variational posterior 47. (h (bs) in 1 modeled as a Gaussian distribution where both mean and variance are derived from feature-side encoder outputs. This section explains the rationale behind this formulation.
B.1 PARTY MRAN AGGREGATION
The latent variable is inferred by aggregating contributions from multiple parties, each providing mean pag (x*) based its local observation , The overall posterior mean is computed as the average of these feature-side means, forming global latent representation. By averaging only the contributions from observed parties, the scale remains consistent regardless of the number missing parties. Consequently, only the participating parties influence the latent variable.
B.2 PRECISION-BASEA VARIANCE AGGREGATION
The posterior variance is determined by the inverse of the sum of precision matrices E (x*) from each party. As more parties contribute data, the total precision increases, reducing the uncertainty about the latent variable. Parties that do not provide data are implicitly treated as having infinite variance, meaning their absence does not affect the overall precision. This aggregation ensures that as more parties participate, the model becomes more confident (i.c., the variance decreases), leading precise estimate of h.
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C THEORETICAL PROPERTIES
Theorem C.1. Les
cx
[lg Rs (a: [be)))]]
C, (10
log R2 (v, obo)
where
1 plx b (p))p(h) 2z pp(z))
Rx
g(hs (20bbs )g(2z bz Fey)
20, ppylle))p(x="" 1e, ( Iz (z)))(z))))))))))))))))))))))1
((,, aan )a( (2,(())))) 1b
Then, Lx (Essp) increases increases, and bounded above by log p(a (log x resp.). addition, log pla h.2) q(h.zl
(log ply.a 10n.att,.222 h.z) ((1,2/ resp.) is bounded, then Cx (esp)) converges to logp(x (logp(y, ohon), resp.) as + oc.
2016). Here, we give the proof for C.p We can prove the Proof. We apply Theorem in Burda et al. upper bound using Jensen's inequality as
1 x
plyl h (( ( (()s))
C. = E (h, 2; 12)3) wolls
wol wnd )g(2(h)
log
K
{(1 l2 (0(2; h2))
j=1
plu: abs h 2j)
< log
log ply, x obo).
K
(h,) lo: (g00b)))(=(11))) (bg)
To prove monotonic increase, let I be a uniformly chosen subset of size from [x]]. Using Jensen's inequality again, we get
ply, obo aeob,,hy .hj, q(2j, 2,) (c0
c = E {(
log
ply, ohs 11, 2z) q( (hj. 2, (olo)))
= [1h
log E,
j=1
ply, obs h 2z,) m
> (()),;;))
Ej log
q qtzej , 25, (o
m
i=1
m
ply, obs hej 2z)
log m
L'm
((2s, Zj (co0bs)
is bounded. Then, R2 (v, obs) converges to ply, bbs) by the strong law of large numbers. Take log on both sides and by the dominated convergence theorem, C converges log p(g, mbs))
Lastly, assume that log pla.a x, =|
D EXPERIMENTAL DETAILS
D.1 BASELINES
To ensure a fair comparison, we apply a unified two-stage protocol to the baselines that do not natively exploit unlabeled data: Vanilla VFL., LASER-VEL. and VFL. First, every local model
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is pretrained with SimSiam (Chen & He, 2021), of the representative self self-supervised learning methods, which He et al. (2024) report to outperform BYOL and MoCo on VFL benchmarks. The pretrained networks are then finetuned on the available labeled subset. The baselines considered are summarized below.
Since Vanilla VFL requires fully aligned data, we train only aligned data. For prediction, we employ the simple zero-imputation strategy for unaligned data which is more effective than random prediction.
Vanilla VFL:
LASER-VFLL (allderra et al., 2024): In the original LASER-VFL. each party has its own repre- sentation model and fusion model. However, the fusion model only works for parties that hold labels. Since we consider the single active party scenario, which is the most general setting in VFL framework, we employ a version of LASER-VFL with only one fusion model.
VLu (Sun et al. 2024): PlugVFL was originally designed for fully aligned data in training, so we again use impptatttttaionn for unaligned parties in both training and inference. We set p = 0.5 for the probability of dropping each passive party, and disable the label IP protection objective which enhances label privacy but can reduce performance. Since this version of Plug VFL is well implemented in the work of LASER-VFL, we adopt it.
FedHSSL. (He et al., 2024): FedHSSL supports three SSL methods: SimSiam, BYOL, and MoCo. Since SimSiam achieves the best performance in the authors' experiments, we adopt it in all runs. FedHSSL also need fully aligned data for finetuning, so we apply zero-imputation for inference similarly.
D.2 DATASETS AND MODELS
Datasets. Isolet is speech recognition dataset containing 7,797 audio recordings of 150 speakers pronouncing each 26-letter English alphabet twice. Each recording is represented by 617 acoustic features extracted from the raw audio waveform. We partition these 617 features evenly across eight parties by assigning each party 77 features (one feature is discarded).
HAPT (Human Activities and Postural Transitions) dataset comprises smartphone accelerometer and gyroscope signals collected from 30 volunteers performing 12 daily activities. Each data sample is represented as a 561-dimensional feature vector. We evenly distribute these features across eight parties, assigning each party 70 features (one feature is discarded).
FashionMNISST is a widely- used benchmark consisting of grayscale images of fashion items divided into 10 classes. For VFL settings, we partition each 28 x 28 image into eight segments of size 14 x 7, with each segment assigned to one of the eight parties.
ModelNet10 is dataset of 3,991 training and 908 test samples of 3D CAD models belonging to 10 classes, commonly used in multi-view shape recognition. In our experiments, each model converted into 12 distinct 2D views by rotating the object 360%, capturing one view every 30% To create a challenging VFL task, we group adjacent pairs of views, forming six pairs that correspond to six parties. For each sample, randomly select view from each pair, providing each party with a single 224x 224 image per sample. To further increase task difficulty, images are resized to 32 x 32 pixels. This random selection process is repeated 6 times for the training set and 2 times for the test set, resulting in 23,946 training samples and 1,816 116 test samples.
Models. Tables show detailed model architectures for each baseline. Tables to II hyperparameters used in our experiments.
report
Table 4: Model architecture details for each component of Vanilla VFL, LASER VFL, and Plug VFL.
Component
Structure
Feature Extractor
ResNet-18 18 or 3-layer MLP
Projector
3-layer MLP
Predictor
2- MLP
Discriminator
2-layer MLP
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Table 5: Model architecture details for each component of FedHSSL.
Component
Structure
Local Bottom Encoder
Lower layers of ResNet-18 1-layer MLP
Local Top Encoder
Upper layers of ResNet-18 18 or 1-layer MLP
Cross-Party Encoder
ResNet-18 or 2-layer MLP
Projector
3-layer MLP
Predictor
2-layer MLP
Discriminator
2-layer MLP
Table 6: Model architecture details for each component FALSE-VFL.
Component
Structure
Party Encoder (%)
ResNet- 18 or 2-layer MLP
Party Decoder (0h)
Transposed CNN 2-layer MI.P
Party Missing Indicator (vb*)
3-layer MLP
Global Encoder (7+)
3-layer MLP
Global Decoder (8,)
3-layer MLP
Discriminator (Φ)
2 l-aye MLP
Table Hyperparameters for Vanilla VFL. Values for pretraining are shown parentheses.
Hyperparameter
Isolet
HAPT
FashionMNIST
ModelNet10
Optimizer
Adam (SGD)
Adam (SGD)
Adam (SGD)
Adam (SGD)
Learning Rate
5e-4 (0.025)
5e-3 (0.02)
le-4 (0.02)
le-4 (0.025)
Batch Size
32 (512)
64 (512)
128 (1024)
128 (1024)
Epochs
500 (100)
100 (150)
150 (100)
100 (50)
Weight Decay
lc-4 (3e-5)
le-4 (3e-5)
lc-4 (3e-5)
le-4 (3c-5)
Latent Dimension
128
128
196
256
Table 8: Hyperparameters for LASER-VEL Values for pretraining are shown in parentheses.
Hyperparameter
Isolet
HAPT
FashionMNIST
ModelNet10
Optimizer
Adam (SGD)
Adam (SGD)
Adam (SGD)
Adam (SGD)
Learning Rate
5e-5 (0.025)
le-3 (0.02)
le-4 (0.02)
5e-4 (0.025)
Batch Size
64 (512)
128 (512)
128 (1024)
128 (1024)
Epochs
200 (100)
100 (150)
100 (100)
10 (50)
Weight Decay
le-4 (3e-5)
le-4 (3e-5)
le-4 (3e-5)
le-4 (3e-5)
Latent Dimension
128
128
196
256
Table 9: Hyperparameters for Plug VFL. Values for pretraining are shown in parentheses.
Hyperparameter
Isolet
НАРТ
Fashion
ModelNer10
Optimizer
Adam (SGD)
Adam (SGD)
Adam (SGD)
Adam (SGD)
Learning Rate
5e-5 (0.025)
5c-4 (0.02)
lc-4 (0.02)
lc-4 (0.025)
Batch Size
32 (512)
64 (512)
128 (1024)
128 (1024)
Epochs
100 (100)
100 (150)
150 (100)
50 (50)
Weight Decay
le-4 (3e-5)
le-4 (3e-5)
le-4 (3e-5)
le-4 (3e-5)
Latent Dimension
128
128
196
256
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I sample ((h 2,)15 from 4:. (h B)gB (z(h) Published as a conference paper at ICLR 2026
Table 10: Hyperparameters for FedHSSL. Values for pretraining are shown in parentheses.
Hyperparameter
Isolet
НАРТ
FashionMNISST
ModelNet10
Optimizer
Adam (SGI)
Adam (Adam)
Adam (SGD)
Adam (SGD)
Learning Rate
2e-3 (0.025)
5e-4 (0.025)
le-4 (0.02)
5e-4 (0.025)
Batch Size
64 (512)
64 (512)
128 (1024)
128 (1024)
Epochs
300 (150)
500 (100)
100 (150)
100 (150)
Weight Decay
le-4 (3e-5)
le-4 (le-5)
le-4 (3e-5)
le-4 (3e-5)
Latent Dimension
128
128
196
256
Table 11: Hyperparameters for FALSE-VEL.
Values for pretraining shown in parentheses, We
use k 10 and 50 for all datasets.
Hyperparameter
Isolet
HAРT
FashionMNIST
ModelNet10
Optimizer
Adam (Adam)
Adam (Adam)
Adam (Adam)
Adam (Adam)
Learning Rate
2c-4 (5c-4)
2e-4 (2e-3)
2e-4 (5c-5)
2e-4 (Ic-4)
Batch Size
128 (512)
128 (512)
128 (1024)
128 (1024)
Epochs
300 (300)
300 (500)
200 (150)
200 (300)
Weight Decay h Dimension
le-4 (1e-4)
le-4 (1e-4)
le-4 (1e-4)
le-4 (1e-4)
128
128
196
256
z Dimension
64
64
32
64
D.3 ALGORITHM
Algorithm 1 FFALSE-VF-LE two-stage training and inference
1: Hyperparameters: K: the number of parties, K/ T: the numher of importance samples for training / test, Tpee/ t/rin: learning rates for pretraining / training. Ties/ Tirin: epochs for pretraining training
2: Parameters: Te /0 encoder / decoder for party lo [K]. 20// encoder decoder discriminator for
active party, To (ne, 7e 0.0e [0',-- 0%) J. O, (7e, Pc, 7a, 0. J, θ {0,, Φ)
4: Stage 1 Pretraining: maximize marginal likelihood pe, (2.
5: for t= 1 to Tiee do
6: for all minihatch B do
7: for all party E {obs} in parallel do
8: (JuR, )x) ENC, (4r)
9:
Send (1s, Σx) to active party
10:
end for
11: Active party forms dhe (B) via Eq.
Compute (.,9) and update Oy G, Tle Ve,C. Ln.
13:
end for
14:
end for
15:
Freeze θg
16:
Stage Training: maximize conditional likelihoud pei (e
18: for = 1 to Timin do
19:
for all labeled minibatch B do for all party & {obs} in parallel do
20:
21:
(JuR; Ex) <- ENC, (wr)
22:
Send (Mas xx) to active party
23:
end for
24: Active party forms 4he (h/B3 via Eq. IT] sample ((h,, 2,)15 from 4h. (l/B)),, (z(h)
25: Compute L'x (() and update & Vhsin L.L
26: end for
27:
end for
28:
29: Inference
incomplete sample the 30: Gather (mt, Σx) from observed parties; sample {(he, 2im from q
31: Compute importance weights we and predict ý Et= We Po he)
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Published as a conference paper at ICLR 2026
D.4 MAR MECHANISMS
We introduce two MAR mechanisms inspired by real-world scenarios in which the decision to collect further observations depend on the information of previously observed data.
For instance, patient may choose whether to visit additional hospitals after reviewing one hospital's examination results. Likewise, an individual viewing an image piece by piece might stop as soon as the observed portion is sufficiently informative. In our simulation, we measure "information" by the variance of piece: low variance implies a near-uniform region (hence little information), whereas high variance implies richer details. All datasets are normalized to ensure scale consistency across features.
Type 1: Stop at the First Highly Informative Piece. We begin by randomly selecting one piece. its variance exceeds a predefined threshold, we consider "sufficiently informative" and do not observe any additional pieces. Otherwise, slightly lower the threshold and randomly select another piece. See Algorithm for details.
Type 2: Accumulate Multiple Moderately Informative Pieces. We start with variance threshold and an "excessive variance" budget B. Whenever the variance u of an observed piece exceeds T, subtract (v T from B. If B falls below zero, stop observing further pieces. Otherwise, reduce T slightly and continue. Conceptually, this simulates gathering several moderately informative pieces until reaching certain limit. See Algorithm for details.
These procedures systematically generate partially aligned data across multiple parties by emulating natural decision processes. To the best of our knowledge, no prior work has addressed MAR-based alignment in the VFL setting. Our methods are designed to reflect realistic user behaviors and can be viewed concrete instantiation of the "Threshold method" described in Zhou et al. (2024).
Algorithm 2 MAR Mechanism Type Single High-Informane Informative Piece
Require: A of parties (pieces) P: initial variance threshold T' 1.1; threshold decrement A 0.15.
Choose an initial piece randomly from P.
2: while there are unvisited pieces in P do
3: Compute variance v of the chosen piece.
4: if T then
5: Stop (no more picces are observed).
6: else
7:
+ T - A [Lower the threshold]
8:
Randomly choose the next unvisited piece from P.
9: end if
end while
Algorithm 3 MAR Mechanism Type 2: Multiple Moderate-Informaatie Informative Pieces
Require: A set of parties (pieces) P; initial variance threshold T 0.5; total "excessive variance"
budget B = 0. (0.5 for ModelNet10): threshold decrement = -0.15.
1: Choose an initial piece randomly from P.
2: while there are unvisited pieces in P do
3: Compute variance of the chosen piece.
4: if' v > T then
5: B
(u (w-T) T) (Consume part of the budget)
6: end if
7:
ir < then
8:
Stop (no more pieces are observed).
9:
else
10:
T T-T-A [Lower the threshold)
11:
Randomly choose the next unvisited piece from P.
12:
end if
13:
end while
20

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Published as a conference paper at ICLR 2026
D.5 IMPLEMENTATION
The experiments are implemented in PyTorch. We simulate a decentralized environment using single deep learning workstation equipped with Intel(R) Xeon(R) Gold 6348 CPU, one NVIDIA GeForce RTX 3090 GPU, and 263 GB of RAM. The runtime of FALSE-VEL for each dataset is reported in Table
Table 12:
Execution time (in minutes) of FALSE-VFL on each dataset. Times are separated into pretraining and main training phases.
Dataset
Pretraining Time (min)
Training Time (min)
Isolet
3
2
HAPT
6
2
FashionMNIST
60
30
ModelNet10
860
90
E ADDITIONAL EXPERIMENTAL RESULTS
Table 13: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for
six VFL methods trained under MCAR 2 conditions and evaluated on seven test patterns. Boldface highlights the best result each column. asterisk (") marks accuracy computed with labeled data only, including unlabeled data led to lower accuracy.
Test Data
MCAR 0
MCAR
MCAR 5
MAR
MAR
MNAR
MNAR 9
Isolet
Vanilla VEL.
76.6 (0.3)
64.5 (0.9)
40.1 (1.6)*
28.3 (1.0)*
28.5
(1.4
(0.8)
LASER-VEL
63.1 (3.5)
51.0
(3.0
(2.2
(2.9)
PlugVFL
84.5 (0.7)
75.8 (0.7)
52.8(1.4)
(0.4().4)
(1.5
53.9
(3.3
FedHSSL
75.1 (0.3)
68.2 (0.3)
(0.6)
(2.5)
35.3
48.2
(1.2
FALSE-VEL-I
(0.1)
82.6 (0.4)
(0.9)
(2.0
(0.5)
65.5 (0.5)
(0.7)
VALLE-FFL-II
86.2 (0.2)
82.6 (0.3)
65.4 (0.8)
(2.6)
55.2 (0.7)
66.0 (0.7)
64.0 (0.8)
HAPT
Vanilla VFL.
(0.3)
(0.8)*
59.7 (2.4)"
(1.7)
60.4 (1.8)
51.5
43,9 (2.9)
LASER-VE VEL.
47.3 (9.9)
49.9 (8.6)
(3.6
(1.3)
$3.0
(4.9)
(5.1)
PlugVFL
87.0
(1.0)
52.6
60.1
(1.2
(2.2
(5.1
FedHSSL
85.3
80.8 (0.2)
(0.3
(0.4)
69.3
(3.0)
FALSE-VFLA
89.0 (0.2)
85.4 (0.2)
73.3 (0.4)
73.8 (0.4)
73.7 (0.3)
69.8 (0.6)
68.5 (0.3)
VEL-E-VEL-II
83.7 (0.4)
70.3 (0.4)
71.8 (0.7)
(1.3)
(0.4)
70.4 (0.4)
PashionMNIST
Vanilla VEL
76.7 (0.2)
(0.3)
55.9
(1.3)
(1.2
(1.0)
49.2
LASER-VFL
(2.3)
72.5 (2.1)
(2.4)
(2.5)
(2.4)
(2.6)
(3.7
PLugVFL
78.3
(1.1)
(1.0
(1.3)
65.8
63.8 (3.4
(2.1
FedHSSL.
78.5 (0.2)
76.1 (0.3)
66.7 (1.8)
64,5 (2.6)
(2.0
65.2 (1.9)
61.0 (2.9)
FALSE-VFL-
82.9 (0.1)
79.9 (0.1)
68.4 (0.3)
65.2 (0.3)
67.8 (0.4)
66.8 (0.3)
65.6 (0.4)
VALSE-VEL-II
82.5 (0.3)
79.1 (0.2)
67.5 (0.3)
64.7 (0.3)
67.2 (0.3)
66.3 (0.3)
64.9 (0.3)
ModelNet10
Vanilla VFL
86.1 (0.6)
78.7 (L.)
(2.3)
50.8 (2.1)
(2.6
(2.3)
60.1 (1.9)
LASER-VEL VFL
80.4 (4.7)
77.3 (4.9)
(5.6
63.4
(4.0
67.7
70.5
VFL. VFL.
85.7 (0.5)
(1.24
52.6
$1.0
(3.3
50.9
$1.1
FedHSSL.
85.2 (0.5)
(3.5
61.6
(5.4
71.6
67.1 (4.0)
FALSE-VELA
86.6 (0.2)
84.6 (0.2)
(1.2)
70.1 (0.9)
67.8 (0.4)
76.7 (0.7)
(0.6
FALSE-VEL-H VFL-II
86.3 (0.3)
84.8 (0.2)
79.1 (0.4)
70.8 (0.6)
69.3 (0.4)
77.4 (0.5)
78.9
21

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Published as conference paper at ICLR 2026
Table 14: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for six VFL methods trained under MCAR 5 conditions and evaluated on seven test patterns. Boldface highlights the best result each column. asterisk (') marks accuracy computed with labeled data
only, as including unlabeled data led to lower accuracy.
Test Data
MCAR
MCAR
MCAR
MAR
MAR
MNAR
MNAR
Isolet
Vanilla VEL.
(1.1
53.1 (1.1)
(0.7
23.2 (0.6)
(0.4)
35.1
(1.0
LASER-VEL
(5.2)
(4.0)
45.1 (2.4)
38.9
(2.4)
(2.7
(3.2
PlugVFL
(0.5
(0.6)
(1.0
40.3
39.6(1.2)
$2.1
53.2
FedHSSL
(0.5)
58.5
40.1 (0.9)
30.5
30.7
(2.6
41.3 (0.7)
VALSE-VEL-I
83.2
80.2 (0.5)
69.3
60.4 (0.9)
62.5(0.9)
(9.8)
70,0 (0.4)
FALSE-VEL-II
82.2 (0.2)
79.2 (0.5)
67.7 (0.2)
59.9 (0.4)
61.4 (0.6)
68.0
HAPT
Vanilla VFL
(0.5)
70.2 (0.6)
(1.5
(8.8)
(0.7)
(2.9
(5.0)
LASER-VEL
(10.9
(7.6
(3.6
(4.3
(3.4)
(3.8
(4.1
PlugVFL
(5.5
(0.)
(0.9)
(0.6)
518
(2.2
(5.4
FedHSSL
(0.9)
(0.4)
63.8
(0.4)
(2.4
59.3
(..6
FALSE-VELA
86.5 (0.3)
83.0 (0.2)
(0.7)
76.1 (0.4)
74,9 (0.6)
71.1 (0.3)
(1.2
FALSE-VEL-II
86.8 (0.5)
(0.3)
75.5 (0.5)
76.4 (0.3)
(0.3)
(0.4)
(0.3)
FashionMNIST
Vanilla VFL
(0.2)
(0.3
(1.2)
50.7 (1.5)
(2.6
50.7
(1.2
LASER-VEL
(1.4)
(1.9
(1.6
(1.6
63.1
(1.7
60.0
PLugVEL
(0.9)
72.7 (0.4)
(4.0
(0.5)
65.0
60.8
(0.7)
FedHSSL
(0.5
(0.7)
58.1 (2.3)
60.3
(2.0
52.6
FALSE-VFLA
(0.1)
78.8 (0.2)
70.9 (0.2)
67.8 (0.3)
70.0 (0.2)
69.0 (0.2)
66.5 (0.2)
FALSE-VEL-II
81.2 (0.2)
78.6 (0.2)
70.4 (0.2)
68.0 (0.3)
(0.2)
68.6 (0.0)
66.0 (0.1)
ModelNec10
Vanilla VEL
(0.3)
76.0 (0.8)
55.5 (0.9)
(2.2)
(2.0)
(6.6
56.5
LASER-VEL
(2.0)
(2.7)
(4.5)
(4.9
(4.9
(5.6
69.2
PlugVEL
(0.3)
(1.9)
53.7
(3.6
(3.8)
(3.6
FedHSSL
84.9
(1.2)
(2.5
55.7 (4.3)
52.5
68.7 (2.9)
(2.2
EALSE-VEL-I
86.3 (0.3)
84.7 (0.2)
80,9 (0.3)
73.7 (0.5)
(3.6
(0.5)
80.5 (0.6)
FALSE-VEL-II
86.7 (0.2)
85.8 (0.2)
80.9 (0.6)
75.4 (0.9)
74.1 (0.4)
79.8 (0.6)
79.9 (0.4)
Table 15: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for six VFL methods trained under MAR conditions and evaluated on seven test patterns.
Test Data
MCAR
MCAR
MCAR 5
MAR
MAR
MNAR
MNAR
Isolet
Vanilla VFL
62.1 (2.2)
53.8 (0.5)
(0.8)
26.0
26.9 (1.8)
37.7 (1.2)
38.1 (0.7)"
LASER-VHL
40.8
(8.3)
(4.5)
(2.3)
(2.3)
(4.3)
34.5 (3.9)
PlugVRL
(1.3
69.2 (0.6)
(1.1
35.3 (1.0)
35.7
47.3
(1.6)
FedHSSL.
(5.6
(0.9)
40.2
30.9 (0.5)
30.8 (0.8)
(3.6
42.4 (0.4)
FALSE-VELA
(0.3)
73.8 (0.3)
64.3 (0.7)
61.4 (0.6)
60.6
65.4
65.5 (0.8)
VALSE-VEL-II
76.3
(0.7)
64.6 (0.5)
61.3 (0.4)
(0.4
65.3 (0.7)
66.5
НAРT
Vanilla VFI.
79.7 (0.8)
(1.1)
53.6
(1.2
(1.4)
(1.8
42.7 (4.1)
LASER-VEL
(11.(
(8.9
(5.8
55.9
(5.9
(4.9
48.1 (6.0)
PlugVFL
(0.2)
(1.3
51.4
(1.0
53.2
(2.6)
(5.4)
FedHSSL
(0.9.0.4)
(2.5
63.1 (0.5)
(3.4
60.6
(8.6
$4.9
FALSE-VEL-I
84.7
(0.7)
(0.4)
(0.1)
74.1 (0.5)
67.1
(0.6)
FALSE-VEL-II
85.1 (0.1)
81.7 (0.4)
71. (0.1)
75.3 (0.3)
73.3 (0.3)
68.0 (0.6)
68.1 (0.1)
RasHionMNIST
Vanilla VFL
(2.3)
(0.9)
(2.8
(1.4
(1.4
42.0
(3.0
LASER-VEL
(1.2
(0.9)
(0.6)
(0.7)
(1.0
(0.6
61.5
PlugVEL
(0.4)
(0.6)
64.6
(5.0.0.8)
(0.8
(1.4
(2.)
FedHSSL
(1.6
67.1 (1.6)
55.5
(2.1
$6.8
$3.6
50.7
FALSE-VELA
80.6 (0.1)
(0.3)
65.4
67.5 (0.3)
(0.2)
(0.3)
FALSE-VEL-II
(0.2)
77.7 (0.2)
(7.6
64.7 (0.3)
67.2
(0.3)
63.4 (0.2)
ModeINet10
Vanilla VEL
(0.8)
(1.3)
(6.7
(2.1)
(2.0
$8.6(1.)
(1.3)
LASER-VEL
(3.3
(4.6
(4.9
65.6
PlugVFL.
(0.5)
(1.8(21)
$1.2
45.5
40.5
47.8 (3.4)
45.8 (3.3)
FedHSSL.
(0.8)
70.9
56.3 (5.4)
(6.6
71.2
(2.6
FALSE-VEL-I
(0.3)
(0.4)
(0.4))
72.1 (0.9)
70.1 (0.8)
(7.7
78.3
VALSE-VEL-II
86.9 (0.5)
85.7 (0.1)
78.9 (0.5)
71.9 (1.6
70.6
78.7 (0.7)
79.2 (0.4)
22

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Published as a conference paper at ICLR 2026
Table 16: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for six VFL methods trained under MAR conditions and evaluated on seven test patterns. Boldface highlights the best result each column. asterisk (') marks accuracy computed with labeled data only, as including unlabeled data led to lower accuracy.
Test Data
MCAR
MCAR 2
MCAR 5
MAR
MAR
MNAR
MNAR 9
Isolet
Vanilla VHI.
(1.1
(0.6)
35.7 (0.8)
21.0
(1.8)
(1.2
38.1 (0.7)
LASER-VEL VFL.
55.3 (6.3)
50.1 (4.9)
(2.6
(1.3)
(2.1
39.5 (3.3)
(2.7)
PlugVFl.
78.4
(1.0
(0.9
(1.5)
(1.0
(1.2(
46.6 (1.3)
FedHSSL
65.2
(0.6)
(0.3)
(0.4)
(0.5
43.2
41.7
FALSE-VEL-I
(0.3
(0.3)
65.7
62.1 (0.8)
(2.1
65.5 (0.8)
64.8 (0.8)
FALSE-VEL-II
(0.2)
75.4
65.4
60.8 (0.4)
(0.6
61.0
66.3
НАРТ
Vanilla
(0.2
72.1 (0.9)
(2.1
59.8)
(1.5
(1.4)
46.6 (4.5)
LASER-VER VHI
46.9
49.0
(5.1)
(6.7)
(5.8)
45.1(7.2)
(11.2)
PlugVHL.
84.6 (0.5)
72.1 (0.6)
51.1 (0.9)
(1.4
(1.2
48.5
48.1 (5.8)
FedHSSL
79.2 (0.3)
(0.3)
(4.6
(0.4)
(2.7(0.6)
(9.6)
55.9 (0.3)
FALSE-VEL-I
86.2 (0.2)
818 (0.2)
(0.3)
74.0 (0.3)
(3.6
68.0 (0.4)
68.7 (0.4)
FALSE-VEL-II
85.1 (0.3)
80.5 (0.2)
69.1
73.7
72.1
66.7 (0.3)
67.6
FashionMNIST
Vanilla VFL
(0.4)
(0.7
(2.0)
(1.5
(2.2)
(2.0)
(1.4)
LASHR-VH.
73.4
(2.9
64.2(119)
(1.6(
64.4
(2.7
61.3 (3.6)
PlugVTL
(0.8)
(0.9
(4.8
(5.0
$8.5 (0.4
62.5
56.8
FedHSSL
(0.5
68.5
58.6
(2.2
56.1
51.6
FALSE-VEL-I
80.7 (0.2)
(0.1)
(0.5
65.2 (0.4)
(0.4)
66.7
63,2
FALSE-VEL-II
80.4 (0.1)
77.2 (0.2)
(0.4)
65.3 (0.3)
67.1 (0.4)
65.6 (0.4)
62.4 (0.3)
ModelNet10
Vamilla VEL
85.3 (0.2)
76.5 (0.6)
55.2
42.3 (0.9)
39.1 (1.1)
55.7 (1.3)
(1.7)
LASER-VEL
(9.4))
(7.6)
(8.2
42.2
(6.0)
(7.1)
(7.8)
PlugVFL
84.2
71.3
51.0
41.4
31.8
45.8 (2.8)
(1.8)
FedHSSL
(0.4)
8188
(7.8
53.7
(5.0
66.1
64.5 (4.9)
FALSE-VEL-I
86.3 (0.5)
(0.6)
(0.6
71.3 (0.5)
69.7 (0.3)
77.8 (0.6)
79.1 (0.7)
FALSE-VEL-II
86.6 (0.2)
85.2 (0.2)
(0.5)
70.9
69.7
(0.6)
(0.4)
Table 17: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for six VFL methods trained under MNAR conditions and evaluated on seven test patterns.
Test Data
MCAR
MCAR
MCAR
MAR
MNAR
MNAR
Isolet
Vanilla VFL
(1.7)
(1.0
(1.9)
(3.0)
20.6
31.7 (2.1))
32.1 (1.7)
LASER-VEL VFL
61.7
56.3 (4.1)
(2.3)
38.7 (2.6)
39.5
43.3
43.6
PlugVFL.
81.1 (0.4)
(1.3)
(2.5
(2.6)
(3.9
(1.4
(1.0
FedHSSL.
(0.6)
(8.7)
(1.1
(0.7)
31.3
43.5
42.3 (0.7)
FALSE-VEL-I
(0.5)
81.2 (0.5)
69.7 (0.6)
(1.0)
(1.2
70.6
72.1
VALSE-VEL-II
(0.4)
80.9 (0.5)
69.7 (0.4)
62.1 (0.7)
(1.3
(0.5
71.9
HAРT
Vanilla VFI.
81.7
(1.55
(5.9
(1.3)
(6.6
(2.66
44.1 (1.3)
LASER-VEL
(13.8)388)
(9.3)
(3.5
(5.9
(0.8
(3.5
(3.3
PlugVFL
(5.6
(0.6
50.6
(1.1
$2.9(1.4)
46.7
(3.9
FedHSSL
80.9 (0.1)
(0.4)
(0.4
(0.6
(3.5
61.(
58.0 (0.3)
FALSE-VEL-
85.2
(0.3)
(0.4
(0.6)
(2.5
(0.3
72.7
FALSE-VEL-II
84.9 (0.2)
81.9 (0.6)
73.6 (0.4)
73.6 (0.3)
73.0 (0.5)
714 (0.6)
72.7 (0.6)
FashionMNIST
Vanilla VFL
74.1 (0.1)
68.8 (0.8)
55.8
(2.3)
$2.5
(1.3
48.6
LASER-VEL
(4.9
(4.6
(4.5
(4.9)
60.6
(4.8
(6.5
PlugVFL
76.0 (0.6)
(0.7)
(1.9
(3.6(1.1)
63.6(1.))
60.9
56.5 (1.7)
FedHSSL
72.9
69.5
(2.5)
(3.5
(2.9
$6.5 (2.4)
$1.9
FALSE-VFL-I
81.1 (0.2)
(0.3)
70.3 (0.3)
68.1 (0.4)
70.1 (0.2)
70.6 (0.3)
69.7 (0.2)
FALSE-VEL-II
81.1 (0.2)
78.9 (0.1)
70.4 (0.3)
(0.2)
(0.3)
(0.2)
(0.3)
ModelNer10
Vanilla
(0.5)
(1.0)
(5.8
(1.7)
(1.9
(2.1
(3.3)
LASER-VEL
(1.3)).))
73.6
63.8
67.1
PlugVFL.
(0.5)
(1.6
56.0
(1.2
45.5
$3.9
$3.9
FedHSSL.
58.7 (2.1)
53.3
61.0
FALSE-VEL-I
(0.2)
85.1 (0.4)
(0.4)
(0.7)
73.5
80.5
(2.0
VALSE-VEL-II
87.9 (0.4)
86.0 (0.4)
81.9 (0.4)
(1.0)
74.4
80.3
82.8 (0.4)
23

---

Published as a conference paper at ICLR 2026
Table 18: Mean accuracy (%) and standard deviation (in parentheses) over five independent runs for six VFL methods trained under MNAR 9 conditions and evaluated on seven test patterns. Boldface highlights the best result each column. asterisk (') marks accuracy computed with labeled data only, as including unlabeled data led to lower accuracy.
Test Data
MCAR
MCAR
5
MAR
MAR 2
MNAR
MNAR
Isolet
Vanilla VEL.
(0.9)
(1.0
(1.4)
(1.8
22.0
36.3
(6.2
LASER-VEL
(6.9
48.4
(3.1)
35.0
(2.2
42.6
(3.5
PlugVFL
(2.9
72.3 (0.8)
(1.1(1.2)
(1.1(
31.1
49.7
514
FedHSSL
65.5
(0.7)
(0.5
(0.7)
29.2
(3.3
41.4(0.)
VALSE-VEL-I
85.6
(0.2
69.3 (0.4)
(0.4)
63.7
(0.5)
(0.9)
FALSE-VEL-II
85.2 (0.2)
82.0 (0.3)
69.6 (0.5)
61.0 (0.5)
63.6
72.5
75.1 (0.6)
НAРT
Vanilla VFL
(0.6)
(1.1)
(6.0
60.8 (1.3)
58.1 (0.9)
49.7 (2.2)
(1.2
LASER-VEL
(6.0)
56.8 (6.0)
$3.9
57.7 (4.7)
(4.4
(4.8)
(5.9
PlugVLL
86.5
(1.9)
53.1
59.9
(4.7
(1.1
46.6
FedHSSL
(6.4)
(0.6
(8.6
(0.6)
(0.8
(0.6
(2.8
FALSE-VEL-I
(0.3)
78.6 (0.7)
(0.4)
67.4 (0.6)
65.6
68.1 (0.6)
72.3 (0.6)
FALSE-VEL-II
(0.5)
(0.3)
69.2 (0.4)
(0.3)
68.3 (0.5)
(0.3)
(0.5)
FashionMNIST
Vanilla VFL
74.4
68.7 (0.6)
(2.1)
48.7 (2.2)
(2.4)
(2.0
(L.)
LASER-VER VFL
62.8
(2.4))
$1.9
(2.5)
$1.9
52.2)
(9.5
PlugVLL
(0.4)
70.4 (0.7)
(1.9
60.4
60.6
58.8
54.8
FedHSSL
74.2
71.1 (1.0)
(3.0)
(4.6)
(3.7
(2.6
FALSE-VEL-I
(0.2)
76,3 (0.1)
(0.1)
60.8 (0.4)
(3.9
68.1 (0.3)
70.2 (0.2)
FALSE-VEL-II
79.6 (0.2)
76.6 (0.4)
65.3 (0.4)
60.2 (0.5)
63.2 (0.5)
68.0 (0.2)
70.1 (0.2)
ModelNer10
Vanilla VFL
84.5 (0.7)
76.7 (0.5)
(1.8)
(1.4)
(1.0(
50.3
45.1
LASER-VEL
78.6 (4.9)
74.1 (6.0)
63.8
60.1 (4.9)
(4.6))
(7.0
(8.8)
PlugVFL
(0.6)
(0.9
54.4
(4.8)
(5.1
52.7 (3.6)
(3.6
FedHSSL
85.9
83.7 (0.7)
73.6(1.3)
63.5)
60.8
70.1
(2.9
FALSE-VEL-I
(0.4)
(0.3)
(7.4
(2.1
(0.4)
78.2 (0.4)
FALSE-VEL-II
86.9 (0.2)
84.9 (0.2)
(0.4)
71.5
71.3
79.7 (0.3)
81.5 (0.2)
24

---

Published as a conference paper at ICLR 2026
Table 19: Relative robustness of each method severe missingness. Each entry is the ratio (in %) of the mean accuracy obtained when training under MCAR 5 to that obtained under MCAR 2, evaluated on seven test patterns. ratio above 100% means that training with the harsher MCAR mechanism yields higher accuracy than with MCAR 2. Boldface highlights the best ratio in each column.
Test Data
MCAR
MCAR 2
5
MAR
MAR
MNAR
MNAR
Isolet
Vanilla VFL.
81.6
81.7
81.7
75.3
77.8
79.4
78.3
LASER-VE -VLL
88.5
86.5
89.4
VLL
96.7
94.2
95.6
99.2
100.4
96.6
99.2
FedHSSL
86.7
85.9
86.1
89.7
93.0
88.4
85.5
FALSE-VFL-I
96.6
97.1
106.2
116.1
112.8
106.2
108.2
FALSE-VEL-J
95.3
95.8
103.5
114.6
111.2
103.0
107.6
HAРT
Vanilla VFL
92.5
92.4
97.1
93.9
95.0
100.6
96.2
LASER-VEL
127.2
115.6
106.1
103.8
101.8
114.0
124.7
PlugVFL.
97.7
95.8
94.0
95.8
93.7
92.7
93.0
FedHSSL
89.9
94.6
FALSE-VEL-I
97.2
101.4
103.
101.7
101.9
103.9
FALSE-VEL-II
97.9
100.4
107.4
106.3
106.3
104.2
102.7
FashionMINIST
Vanilla VFL
95.8
95.7
95.2
95.3
96.2
94.8
96.5
LASER-VFL
95.4
95.4
96.0
96.2
95.6
95.4
94.2
VFI
98.3
97.8
47.6
99.3
98.9
95.3
93.8
FedHSSL
92.1
91.0
89.1
90.1
90.0
87.4
86.2
FALSE-VEL-
103.9
103.1
103.2
101.4
PALSE-VELAJ
98.4
99.3
104.3
105.2
104.1
103.5
101.8
ModelNet10
Vanilla VFL
94.2
95.4
93.9
LASER-VEL
100.3
97.5
95.4
100.8
100.8
96.9
98.1
VHI
98.4
97.8
95.8
95.4
94.7
94.9
92.3
FedHSSL
97.7
97.0
94.5
90.5
88.5
96.0
100.2
FALSE-VELE VEL-I
99.7
100.2
103.1
105.2
108.3
103.6
102.9
VALSE-VEL-II
100.4
101.2
102.2
106.4
106.9
103.2
101.3
F LIMITATIONS AND FUTURE WORK
In FALSE-VFL-II the mask distribution is modeled as (m(m)x) = IIεεдK] Pu* (mml(=)) This formulation supports the MNAR mechanism, where missingness depends on both observed and unobserved values. However, under this model, each feature vector a only influence the missingness its own entry m and not that of other parties, i.e., m for l A k. In this sense, the current model cannot capture inter-party dependencies the MNAR mechanism.
Extending our approach to model such inter-party dependencies would be natural next step. However, doing so poses a significant challenge, as direct sharing of x across parties is generally prohibited due to privacy constraints. We leave the development of such models that account for inter-party MNAR dependencies while preserving privacy as an important direction for future work.
G
LLM USAGE
To improve clarity, parts of text were refined with assistance from large language model: all content was reviewed and verified by the authors.
25


Reviews:
[{"review_id": "BLzNsEViaK", "reviewer": "Reviewer_nHhw", "summary": "This paper introduces FALSE-VFL (Flexible Alignment and Labeling Scenarios Enabled Vertical Federated Learning), a unified probabilistic framework for vertical federated learning (VFL) that can operate under arbitrary combinations of data alignment and labeling conditions. The method models feature, label, and alignment missingness jointly using a Deep Latent Variable Model (DLVM) and formalizes training under MCAR, MAR, and MNAR mechanisms. Experiments on multiple datasets show that FALSE-VFL consistently outperforms existing methods.", "questions": "1. The paper’s design philosophy seems to resemble prior robust FL methods. Can you clarify what new robustness mechanism is introduced by FALSE-VFL beyond adopting a deep latent-variable model for the vertical setting?\n2. Could the proposed probabilistic modeling of missingness be replaced by other robustness strategies, such as adversarial or confidence-weighted training, and if so, what unique advantage does the DLVM formulation offer?\n3. Does the method require a subset of aligned or labeled samples to establish a shared latent representation? If none are available, how would the model avoid degenerate alignment?\n4. Can you provide empirical or theoretical evidence that the probabilistic marginalization approach improves robustness relative to prior adversarial or reweighting-based techniques?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper addresses an important limitation of current VFL research, where some methods assume perfect alignment and full labeling. 2. Treating alignment and labeling gaps as missing-data problems and modeling them jointly within a probabilistic latent-variable framework is a conceptually coherent way to generalize VFL beyond idealized settings. \n3. The experiments are comprehensive, covering many combinations of training and testing conditions, and they demonstrate robust performance improvements. \n4. The paper is clearly written and well-motivated.", "weaknesses": "1. The paper's main limitation lies in the originality of its methodological foundation. The proposed DLVM-based framework for modeling unaligned, unlabeled, and partially observed data in VFL largely parallels the broader line of work on robust FL already explored in horizontal settings. Prior studies [R1-R3] have developed principled approaches for robustness under heterogeneous, noisy, or distributionally shifted clients. While these works operate under horizontal data partitioning, they share the same conceptual goal as FALSE-VFL: learning representations and aggregation mechanisms that remain stable in the presence of incomplete or inconsistent local data.\n\nR1: Robust Federated Learning With Noisy and Heterogeneous Clients, CVPR, 2022\n\nR2: Robust Federated Learning: The Case of Affine Distribution Shifts, NeurIPS, 2020\n\nR3: A Systematic Literature Review of Robust Federated Learning: Issues, Solutions, and Future Research Directions, ACM Computing Survey, 2025\n\n2. Compared with these precedents, the present paper’s innovation lies mainly in applying the robustness principle to vertical feature partitions using deep latent-variable modeling, rather than proposing a fundamentally new robustness mechanism. The model treats missing features, labels, and alignment as latent variables and marginalizes them via variational inference. While this probabilistic formulation is elegant, the underlying idea of achieving federated robustness through representation consistency is well established. The paper does not provide theoretical or empirical evidence that the DLVM approach offers advantages over alternative robustness strategies such as adversarial minimax optimization or noise-weighted aggregation.\n\n3. The framework implicitly assumes that some aligned or labeled samples exist to anchor the shared latent space. This reliance makes the method closer to pseudo-matching alignment strategies than to fully unsupervised alignment inference. The empirical evaluation demonstrates broad coverage but remains descriptive; ablation studies isolating the contributions of missingness modeling or pretraining are limited.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kdV3UkFcbW", "reviewer": "Reviewer_JZzY", "summary": "This paper proposes FALSE-VFL, a deep latent variable model (DLVM) framework designed for vertical federated learning (VFL). The core idea is to reframe VFL alignment gaps as a blockwise missing data problem. This approach allows the model to handle arbitrary data alignment, leverage abundant unlabeled data, and support multi-party collaboration. The method employs a two-stage optimization: (1) an unsupervised, generative pretraining phase to learn a robust data representation from all samples, and (2) a supervised training phase to learn a predictor from the scarce labeled data . The paper introduces two variants, FALSE-VFL-I for MAR and FALSE-VFL-II for MNAR settings , and demonstrates state-of-the-art performance across 168 experimental configurations", "questions": "See weakness.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "S1. The paper tackles a critical and highly practical challenge in VFL. The assumptions that data alignment is imperfect and labels are scarce are far more realistic than in typical VFL literature.\n\nS2. This is a unified framework that addresses multi-party VFL, arbitrary (blockwise) data missingness, and semi-supervised learning. The explicit handling of all three missingness mechanisms (MCAR, MAR, and MNAR) is a significant contribution.\n\nS3. The experimental results are extensive and compelling. The method's dominance in 160 out of 168 configurations, with a large average performance gap over the next-best competitor, provides strong evidence of its effectiveness.\n\nS4. The method shows admirable robustness not only to different types and degrees of missingness but also to an increasing number of parties and data heterogeneity, both of which are common challenges in real-world VFL systems.", "weaknesses": "W1. The writing in the methodology section (Section 3) could be significantly improved. The paper is not self-contained and is difficult to follow without prior knowledge of the cited works (e.g., IWAE, Ipsen et al., 2022).\n* For instance, the paper states, \"so we need some tricks as in IWAE\" but fails to explain the intuition behind this \"trick\" (i.e., importance-weighted sampling as a variational lower bound).\n* Similarly, the derivation of the lower bound $\\mathcal{L}_{\\kappa}$ omits the key appeal to Jensen's Inequality, which is fundamental to understanding why it is a lower bound. While this proof is in the appendix, its absence in the main text makes the core methodology less accessible.\n\nW2. The paper proposes two distinct solutions: FALSE-VFL-I for MAR and FALSE-VFL-II for MNAR . This separation is a practical weakness. In real-world scenarios, the true missingness mechanism is rarely known in advance. A practitioner would not know whether to deploy model I or model II. A more robust and unified framework that could handle (or even infer) the missingness mechanism automatically, without requiring this a priori assumption, would be a significant improvement.\n\n**Minor Comments**\n\nC1. The paper consistently and incorrectly refers to its Appendices as \"Sections\" in the main text. For example, \"described in Section A.2\" , \"explained in Section B\" , and \"See Section C for details\". \n\nC2. Equation Numbering: Several key equations in the main methodology (Section 3.4) are not numbered. Numbering these equations would make the method much easier to follow and reference.\n\nC3. The meaning of $z_j$ is not explicitly explained in the main paper. The paper would be clearer if it explicitly stated that these are the $j$-th samples (out of $\\kappa$) drawn from the approximate posterior distributions.\n\nC4. The meaning of the indices for the missingness mechanisms is not fully explained in the main paper. While MCAR 0/2/5 and MNAR 7/9 are clearly defined in Section 5.1 as referring to probabilities, the crucial distinction between MAR 1 and MAR 2 is not. The main text only states that \"Precise formulas are given in Section D.4\". For the paper to be self-contained, a brief, one-sentence summary of what \"MAR 1\" and \"MAR 2\" represent should be included in Section 5.1.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Zh7Rwhr4vj", "reviewer": "Reviewer_wrRH", "summary": "This paper investigates the use of deep latent variable models (DLVMs), such as variational autoencoders (VAEs), to improve the trade-off between privacy and utility in Differentially Private Federated Learning (DP-FL). The key idea is to use the latent representation learned by local VAEs to regularize gradient updates before aggregation, thereby reducing sensitivity to individual data points and improving privacy efficiency under fixed noise budgets. The authors propose a two-stage training procedure: (1) local latent encoding via VAEs trained with reparameterization, and (2) DP-SGD optimization at the server that incorporates latent regularization. Theoretically, the paper claims that this latent-space compression reduces the gradient’s Lipschitz constant, leading to tighter differential privacy guarantees. Experimental results on MNIST, CIFAR-10, and FEMNIST show improved accuracy compared to baseline DP-FL methods.", "questions": "Please see the weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- **Integration of deep generative models into DP-FL.** Using VAEs for latent regularization is an interesting and creative direction. It potentially bridges two communities (privacy-preserving learning and generative modeling) that rarely interact.\n\n- **Theoretical grounding.** The authors provide a theoretical motivation suggesting that operating in the latent space constrains gradient sensitivity, thereby tightening the DP bound. Though not formally rigorous, this is an intuitive and plausible argument.\n\n- **Empirical evaluation.** Experiments on multiple datasets show consistent improvements in accuracy for small privacy budgets. The evaluation includes comparisons with multiple strong baselines and examines the impact of latent dimensionality.", "weaknesses": "- **Weak theoretical support.** The “reduced Lipschitz constant” argument (Theorem 1) is intuitive but lacks formal derivation or provable privacy amplification bounds. There is no rigorous proof connecting latent regularization to improved DP guarantees.\n\n- **Pretraining assumption undermines privacy guarantees.** The VAE pretraining step is performed without differential privacy, which breaks the overall DP guarantee of the system. Since the encoder is trained on sensitive local data, any downstream use of latent features leaks information unless the pretraining itself is privatized.\n\n- **Limited scalability and realism.** Experiments are restricted to small-scale datasets (MNIST, CIFAR-10, FEMNIST) with simple convolutional VAEs. The approach’s computational overhead and communication cost on larger-scale or heterogeneous clients are not analyzed.\n\n- **No privacy leakage validation.** The work would benefit from empirical privacy auditing (e.g., membership inference or reconstruction attacks) to validate that the proposed latent regularization indeed improves privacy robustness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Fk9YvGCKqs", "reviewer": "Reviewer_pEAi", "summary": "The paper proposes FALSE-VFL, a unified framework for vertical federated learning (VFL) that frames misalignment across parties as a missing-data problem and applies deep latent variable models (DLVMs) to handle arbitrary alignment and labeling scenarios. FALSE-VFL has two variants: FALSE-VFL-I (assumes MAR) and FALSE-VFL-II (models masks and supports MNAR). The method uses a two-stage procedure: (i) pretrain a generative DLVM by maximizing a marginal likelihood over observed features (to exploit unlabeled data), then (ii) freeze the generative parameters and train the discriminative component to maximize conditional likelihood. The paper presents experiments on four benchmarks (Isolet, HAPT, FashionMNIST, ModelNet10) under many training/test missingness regimes (168 configurations total) and reports that FALSE-VFL outperforms baselines in the vast majority of settings, with an average advantage on the order reported in the paper.", "questions": "- It is better to provide a precise description of what is communicated between parties in each phase (pretraining, training, prediction). Are raw encoder outputs shared, or only aggregated/statistical information?\n- It is better to provide (a) FLOPs or time per epoch for pretraining and training per dataset, (b) communication per sample (bytes exchanged) for inference and training, and (c) how κ and the number of parties K affect runtime and communication.\n- The paper aggregates party contributions via mean and precision summation. Could you provide an intuition or visualization showing that h captures cross-party semantics, enabling prediction when parties are missing?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper reframes VFL alignment problems explicitly as missing-data mechanisms (MCAR/MAR/MNAR) and integrates modern DLVM approaches (IWAE-style training and importance-weighted estimates) into a privacy-conscious VFL pipeline. This synthesis (DLVM + VFL missingness viewpoint + two-stage optimization) is novel in the VFL literature.\n- The model design is technically coherent: party-wise encoders/decoders, an aggregated variational posterior with precision averaging, and a label-side discriminator make sense and are explained (mean/precision aggregation rationale). The theoretical properties of the importance-weighted bounds and monotonicity are stated and proved in the appendix.\n- The experimental sweep is broad: four datasets, two network families (tabular and image), six training missingness regimes, and seven test regimes, multiple baselines, and ablations (missing rates, number of parties, heterogeneity), altogether 168 configurations reported and per-setting tables provided. Results show consistent wins for FALSE-VFL across nearly all configurations. This breadth strengthens claims of robustness to alignment/label scarcity.", "weaknesses": "- FALSE-VFL is claimed as a privacy-preserving VFL method, but the paper seems to lacking a concrete description and analysis of what is communicated at pretraining, training, and inference (e.g., are latent parameters, encoder outputs, importance weights, or reconstructions exchanged?); what information leakages are possible; and whether common VFL privacy protections (secure aggregation, HE, secure two-party protocols) are compatible with the sampling / IWAE machinery used.\n- The approach uses importance-weighted sampling and a DLVM pretraining stage, which, as reported, can be expensive (ModelNet10 pretraining = 860 minutes on RTX3090; Table 11). This raises questions about practicality for large real-world VFL deployments with many parties, high-dimensional data, or limited compute.\n- The paper misses several important ablation studies: sensitivity to the amount of labeled data (why 500/1000 chosen?), sensitivity to latent dimension, effect of κ (IWAE samples) on final prediction, and when/why FALSE-VFL-I (MAR) sometimes outperforms FALSE-VFL-II despite the latter being more expressive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)