Vertical Federated Feature Screening
Authors: Huajun Yin
Liyuan Wang
Yingqiu Zhu
Liping Zhu
Danyang Huang
Venue: NeurIPS 2025 poster ()
Keywords: Vertical federated learning
Feature screening
Ultrahigh dimensionality
Imbalance data
Subsampling
Primary area: general_machine_learning
TL;DR: We propose the vertical federated feature screening (VFS) algorithm, which effectively reduces computational, communication, and encryption costs.

Abstract: With the rapid development of the big data era, Vertical Federated Learning (VFL) has been widely applied to enable data collaboration while ensuring privacy protection. However, the ultrahigh dimensionality of features and the sparse data structures inherent in large-scale datasets introduce significant computational complexity. In this paper, we propose the Vertical Federated Feature Screening (VFS) algorithm, which effectively reduces computational, communication, and encryption costs. VFS is a two-stage feature screening procedure that proceeds from coarse to fine: the first stage quickly filters out irrelevant feature groups, followed by a more refined screening of individual features. It significantly reduces the resource demands of downstream tasks such as secure joint modeling or federated feature selection. This efficiency is particularly beneficial in scenarios with ultrahigh feature dimensionality or severe class imbalance in the response variable. The statistical and computational properties of VFS are rigorously established. Numerical simulations and real-world applications demonstrate its superior performance.

Vertical Federated Feature Screening
Huajun Yin Yin!.2 Liyuan Wang!.2 Yingqiu Zhu?, Liping Zhu', Danyang Huang! 1.2
'Center for Applied Statisties and School of Statistics,
Renmin University of China, Beijing, China
Bigdata and Responsible Artificial Intelligence for National Goverannce, Renmin University of China, Beijing. China
School of Statistics, University of International Business and Economics, Beijing, China Institute of Statistics and Big Data, Renmin University of China, Beijing, China
Abstract
With the rapid development of the big data era, Vertical Federated Learning (VFL.) has been widely applied to enable data collaboration while ensuring privacy pro- tection. However, the ultrahigh dimensionality of features and the sparse data structures inherent in large-scale datasets introduce significant computational com- plexity. In this paper, we propose the Vertical Federated Feature Screening (VFS) algorithm, which effectively reduces computational, communication, and encryp- tion costs. VFS two-stage feature screening procedure that proceeds from coarse to fine: the first stage quickly filters out irrelevant feature groups, followed by a more refined screening of individual features. significantly reduces the resource demands of downstream tasks such as secure joint modeling or federated feature selection. This efficiency particularly beneficial in scenarios with ultra- high feature dimensionality or severe class imbalance in the response variable. The statistical and computational properties of VFS are rigorously established. Numeri- cal simulations and real-world applications demonstrate its superior performance.
Introduction
In the era of big data, the exponential growth of privacy have highlighted
data and concerns Learning (FL). Among its the need for Federated variants, Vertical Federated Learning (VFL) enables collaborative model training using the shared infor- mation from overlapping users, without exchang- ing raw data, thereby ensuring privacy and security A typical example, shown in Figure occurs when an e-commerce company and bank collaborate train model predicting mutual interest, such as individual credit score. To collaboratively train predictive model, institutions need to utilize secure multi-party party computation for data transmission and computation 3
Credit
Dehavioral
0
I 38
49
74
76].
Users
Encryption
Features
data
40 53 677.
Bank
E-commerce platform
Following existing literature, the party that possesses the labels (e.g., credit defaults) referred to as the active party, while the other parties are designated
Active party
Passive party
Figure 1: An example of VFL.
as passive parties [27 53 73].
Coorespondencce to: Liyuan Wang <wangly2023@rum www.:002220m..uu..c.... and Danyang Huang <dyhuang edc.ccu.cn.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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However, the increasing feature dimension presents significant challenges for VFL analysis. Numer- ous real-world cases and theoretical studies indicate that, in
data, only a small proportion of features are truly useful for prediction [23 56 80] To address this, researchers have proposed several feature selection methods in VFL to reduce the cost of building joint predictive However, due to iterative parameter estimation and information communication across multiple parties, methods based on joint modeling using all features still incurs high costs when the feature dimension (p) and sample size (r) are ultrahigh. Therefore, improving the efficiency of these feature selection or joint modeling methods remains a crucial issue VFL.
models 7 [37 45).
Another challenging issue in real-world data is the rare event problem, which represents an extreme case of class imbalance. Following prior literature, observations in the minority class are referred to as cases, while those in the majority class are called controls [30 44]. Although controls often
dominate the dataset, they typically contribute limited information gain compared to cases 39 55 Moreover, the rare event setting implies that cases are extremely scarce, which may invalidate the statistical properties of standard model estimates 9 33 existing federated learning methods, the rare event problem should also be taken into account.
70 71]. Therefore, when improving
This study aims address the challenges of feature selection and joint modeling in VFL, considering both the ultrahigh dimensionality with large and the occurrence of rare event with large n. We propose a subsampling- -ased model-free screening statistic, along with a novel two-stage VFL screening algorithm, called the Vertical Federated Feature Screening (VFS) algorithm. In the first stage, the tttrahigh-dimensiie features are deleted based on VFS statistics. In the second stage, we conduct a more precise screening process for the remaining features. Once fine-grained screening results are obtained, the downstream federated learning techniques can be applied to the condensed dataset with significantly reduced computational complexity. We have established the statistical properties of the VFS algorithm to ensure its reliability. Extensive experiments using simulated and real-world datasets show that our method achieves promising performance with low costs of encryption, decryption, computation, and communication in VFL applications. The main contributions of this paper are threefold.
features are partitioned into multiple groups, irrelevant groups of
Unified model-free screening statistic. We propose the VFS statistic which allows for rare events. The VFS statistic is general framework, compatible with various commonly used statistics but newly designed for rare event scenarios and capable of detecting complex nonlinear relationships.
Coarse-to-fne screening algorithm. We propose a two-stage screening algorithm, which first quickly removes large number of irrelevant features and then performs This approach balances the performance and efficiency of the screening process, further reduce the cost of encryption, decryption, computation, and communication in VFL system.
refined sereening.
Theoretical guarantee. We establish the statistical properties of the proposed screening statistics and algorithm, providing a rigorous theoretical foundation in VFL scenarios allowing for rare event features.
and
2 Related works
Federated feature selection. Existing literature has investigated how to identify important features during the collaborative training of models in a VFL system [31 For instance, Feng 224 proposed a VFL-based feature selection method that uses deep learning models and complementary information from different parties. Castiglia et al. introduced LESS-VFL which optimizes communication efficiency by using a short pre-training period and local feature selection techniques.
Li et al. [45] presented FedSDG-FS, a secure feature selection method that uses a Gaussian stochastic dual-gate and partially homomorphic encryption. Ji et al. (37] proposed MUSE, vertical federated feature selection framework based on mutual information. However, when the feature dimension is ultrahigh (i.e.. P grows exponentially with 1t), existing VFL methods incur high communication and encryption costs, highlighting the need for an efficient preprocessing step to reduce dimensionality.
Non-FL feature screening and independence test. Unlike feature selection which precisely identifies true features, the primary goal of feature screening is to address
data challenges by eliminating irrelevant features through the marginal correlation [5 22 In particular, model-free screening methods aim to identify irrelevant variables without relying on specific model
assumptions [2 15 51
72 78). One effective strategy for model-free screening involves using 65

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statistics derived from independence tests, which focus on the asymptotic distribution of test statistics 77 79]. In the theoretical analysis of such statistics, generalized U-statistics provide 62 However, there is limited discussion of related methods in VFL. settings. VFI. with rare events, direct application of existing methods incurs high computational costs and invalidates asymptotic properties due to the extreme sparsity of cases. This calls for more efficient screening methods and revised theoretical foundations.
[8 61 75
classical and powerful tool
43
Imbalanced data. Undersampling controls 19] [22 and oversampling cases 9 18] are commonly
used approaches for handling imbalanced data. Of the two approaches. undersampling can enhance
model performance and reliability without further increasing computational costs 63], Through
undersampling, the statistical properties of methods such as logistic regression [26] 71, decision trees [50], support vector machines (2]. and high- -igeh-immnnional linear discriminant analysis [57] have been re-examined, leading to refined theoretical results. Nevertheless, rare event and its impact on
general feature screening statistics are seldom considered in the existing VFL. literature.
Cost in VFL. Existing studies have examined various costs in VFL. To ensure data privacy, encryption and decryption procedures introduce substantial resource consumption [10 54). Communication cost arises from the transmission of encrypted data and model parameters
Computational
cost is dominated by local model training on encrypted data S [3]. There is need for a feature
screening approach designed for VFL with minimum costs of encryption, decryption, computation, and communication, which significantly accelerates downstream tasks.
3 Methodology
3.1 Notations and background
In VFL, multiple parties collaboratively build a model using a sample dataset D = (X,,, Y.) where X; = ..., Xp.;) the series of labels, and X E RB"p represents n observations of features. The observations of the j-th feature are contained in the j-th column of X, denoted as Xy. For simplicity, we assume a two-party VFL setting with a binary response. The extension to multi-party multi-class scenarios straightforward and will be discussed later. The active party owns the label column Y, where each Y; takes value between O and 1, while the passive party owns the features, represented by the columns X1,...: Xp. We next present the problem settings, which reflect the two key advantages of our method.
is a p-dimensional feature vector. Let Y (Yi,..., Yn) E R2 denote
The first one is that we allow for extreme imbalance (i.e., rare event) in the response variable. Following previous literature, we refer to observations with Y; = 1 as cases with sample size mi, those with Y; = 0 as controls with sample size ni, and the total sample size is 14g + n1. Rare events imply that grows much slower than n. resulting in n1/n + O. Such settings frequently arise in collaborative modeling scenarios, including fraud detection
and disease prediction (14).
The second advantage is that we allow ultrahigh- dimeeniionn features, where the number of features p can grow at an exponential rate relative to n1, e.g., logp = O(ni) with & E (0, 1). For real- world altrahigh-dimensinnal datasets, among the p features, only a few, say m, are truly informative 23 56 80). Thus, a screening procedure can dramatically reduce these costs by identifying a feature set F to estimate the true feature set F, thereby reducing the costs of subsequent downstream tasks.
However, in VFL system, encrypting Y directly would make many model-free statistics infeasible to compute, particularly rank- statistics. This limitation arises because simple and fast homo- morphic encryption schemes, including RSA [59], Paillier (58), and CKKS (13, do not support direct comparisons of magnitude. To this end, we propose a paired encryption procedure for feature screening. Specifically, the active party first generates label matrix Y
(Viik))nxx from Y, where ii. indicates whether the i1-th and ig-th observations belong to the same class. The matrix Y is then encrypted as [Y] and transmitted to the passive party, where [c] denotes the eiphertext of c. Using YY] and its features X, the passive party computes the encrypted screening statistic [T] and returns it to the active party. Finally, the active party decrypts [7] to obtain 7 for further analysis. For example, in the case of the distance covariance used in our subsequent implementation and CKKS
3

---

eneryption method, we denote
{ [4]. [-2]]
Yi + Y Yi
= Yi
It is worth noting that the same plaintext correspond to different ciphertexts, and thus the ciphertexts are no longer binary and cannot be easily decrypted. The encrypted statistic [T,] for the j-th feature
X ||2 where X3, denotes the j-th feature the 1h user. An illustration of this process is provided in Figure
computed as [T,] (nomn) Ei, Eig [Yi i
Active Party
Label
-2
-2
1
2
Compute
-2
Enerypt upper triangular matrix
Transmit to Passive Party
label matrix
-2
-2
3
-2
5
-2
Figure 2: Paired encryption procedure.
Remark 1 (Effect to Privacy-Preserving Mechanisms). VFS is compatible with any suitable homo- morphic encryption (HE) scheme. While our focus on the statistical and computational aspects of feature screening in VFL, for reference, the privacy guarantees of HE are well established by Gentry Although certain inference attacks may exist, more advanced HE variants can provide stronger protection 21], and combining HE with differential privacy (DP) offers an additional layer of privacy assurance
29).
3.2 Vertical Federated Feature Screening algorithm
To address the challenges posed by the sttrahigh-dimensional and extremely imbalanced data, propose the Vertical Federated Feature Screening (VFS) and shown in Algorithm [ Guided by the sparsity of true features, we adopt coarse-to-fine strategy identify useful features through two-stage process. The overall procedure is illustrated in Figure 3
Passive Party
Active Party
Encrypcion Tranemission
Feature Group
Feature Group
F Fenture Group k
Subsampliing
Stage Group-by-Group
Discard
Retain
0
Retain
Transmission Decrypion
Course Screening
f
Spiit
I Split
Tannmission Decypioon
:
Transmission Decrypion
One-by-4hrc
{
Fine Screening
Feature Selection
Figure 3: Framework Vertical Federated Feature Screening (VFS).
Preliminary Step (Subsampling): Generate binary indicator variable ni (1 < i < no) from Bernuulii /no). indicating whether the i-th control is selected, where is predefined constant representing the subsample ratio.
Stage (Group-by-Group Coarse Screening): Features are partitioned into groups, and coarse screening process rapidly eliminates irrelevant groups, significantly reducing dimensionality. Specifi- cally, for each group Xx, screening statistic T. computed and hypothesis testing is performed. If the null hypothesis is rejected, the feature group X, is considered to contain true features and is retained; otherwise, it is discarded. The coarse screening is repeatedly performed until the number of remaining features becomes computationally acceptable.

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Algorithm Vertical Federated Feature Screening (VFS)
Input:
Feature matrix X E R=X,, label series Y R", subsample ratio s, initial group size so, decay coefficient p. # features of stage switching p', screeming threshold 8.
Output: Retained feature matrix X' Raxd
for i to rap do
Sample the i-th control with probability sm1/nu.
Active party: Generate Y, enerypt Y into [Y) send [Y] to passive party.
Po P.r
while p--1 p' do
Passive party: Divide pr-1 features into 9- [pr- 1/ groups.
for k to g. do
Passive party: Compute [T&] for k-th group of features, send [Tx] to active party.
Active party: Decrypt [Tx) into Ta perform hypothesis test to determine whether to discard k--th
group of features.
Pr +- # retained features, <- p8r-1, -
p' + Pr-1 for j= p' do
Passive party: Compute [Ty] ] for j-th feature, send [Ty] to active party.
Active party: Decrypt [T]] into Ty, use threshold to determine whether discard j-th feature.
Stage (One-by-One Fine Screening): The remaining features are subjected a refined screening process, ensuring both efficiency and precision. The screening statistic Tý computed for each feature X, T2 exceeds the threshold the feature Xj is retained; otherwise, it discarded. This stage enables the accurate identification of true features, For convenience, let d denote the number of features retained.
Compared to classical methods that rely solely on fine screening, coarse screening substantially lowers the time required for encryption, decryption, computation, and communication, further enhancing the overall efficiency of the VFS algorithm. The comparison of the time costs is shown in Table [ with detailed derivation provided in Appendix A particular, the VFS statistics only depend on (s + 1)m1 observations with n < n, enabling feature screening with minimal costs, and facilitating downstream tasks. Moreover, to enhance the generalizability our method, essential to utilize a model-free statistic for the screening process. We propose VFS screening statistic whose definition and theoretical properties are provided in Section
Table 1: Comparison of costs between classical screening procedure with VFS based on group partition. The costs for encrypting, sending, and decrypting number are Le, Le and Eg, respectively, and the cost of multiplying a plaintext number with a ciphertext number is La. Without loss of generality, assume that the time complexity of the screening statistic is (
Cost
Classical Screening
VFS
Eneryption
(nr(t))
( + fe
Computation
Onnppl.)
(s + 1) ( 12 + 8o)
Communication
(pl.))
0 - + t0)
Decryption
(pfla)
0
P + s0)es
4 Theoretical analysis of VFS
In this section, we provide theoretical support for the effectiveness of VFS by illustrating its asymp- totic properties. We establish the screening consistency of the model-free VFS statistic and demon- strate its suitability for group screening in VFL with uirrahigh-dimensionaal features.
4.1
VES statistics
Before introducing the proposed VFS statistic, we briefly review generalized U -tssttissics (3). Con-
sider K independent collections of observations {x("), ,... x.(1 ,,,,,. ,* x(K) }. the
(K) ((K)

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generalized U U-statistic could be defined as
K U U=
T ma
Σ
h
x(2 F1,my
x(K) 1
x-(K 1K
11,1
where ny denotes the number of observations in the k-th collection, my denotes the num- ber of observations used in kernel function from the k-th collection, Σ. denotes the summa- tion over all possible combinations of distinct elements {iR,1..... d, J from {I.... , and x(() xx(K) is symmetric within each block of arguments. This <x.mm
f1,1" M(x.)) . x.!) ft.m, formulation generalizes the classical U-statistic by allowing the kernel to depend observations from different populations.
In our setting, we adopt the generalized U-statistic framework instead of the conventional one because we are particularly interested in scenarios where the class-O and class-I observations are imbalanced. By explicitly separating the two classes within the kernel function, this formulation provides greater flexibility in modeling the asymmetric contributions of each class. Inspired by the generalized U-statistics, we consider a sampled statistic to deal with the rare event scenario. Specifically, let no Ei= T/, where 7)2 denotes whether i-th control is selected. We define the model-free VFS statistic for j-th feature X as
ř, =
ñu
n1 m
Σ
1I
71o h
Moo.
(x
J.0.1
X
J01.
x
(I)
mg
where x and x
denote observations from controls and cases, respectively. The ker-
j.51
nel function h(-)) is symmetric within each set {x(1) "Ai,,"".... x! k for 1 0, Denote
0g = E(M(X
x
(1) :.
x j.h.im }}. so T3 serves as an unbiased estimator of
J.f0,1
j.
0y. For the k-th group of features X the VFS group statistic is similarly defined as Tx except
that X, in (I) is replaced by Xt. Note that ( is not a classical generalized U-statistic because
additionally introduce the sampling indicators 7):
4.2 Asymptotic properties of VFS statistics
To investigate the theoretical properties of the VFS statistic, we consider the following conditions. (C1) (Feature Distribution) There exist constants ě1. 02 > such that for all > 0,
sup max Pr (IX,) > 1 < ē exp(- p EEjEP
(C2) (Feature Dimension) There exists 0 < 1/2 such that log p o(n) 1-)).
(C3) (Imbalance Restriction) Case sample size ni allowed be n. 22(og logp).
(C4) (Signal Separation) There is signal gap between true feature set F and irrelevant feature set Fe. Formally, there exists < < 1/2 - such that minjez | maxjer= 10,l > 2n
Condition (C1) holds naturally if
Xp) is uniformly bounded or follows multivariate normal distribution. This condition prevents features from taking extreme values with high probability data analysis 22 [69] Condition (C2) is same
and is widely used in classical assumption in screening methods. However, the effectiveness depends on assumptions related to n2 instead of due to sample imbalance. In the balanced case, where n; is proportional to n, it reduces to the classical screening assumption. Condition (C3) specifies the growth rate n1, preventing it from growing too slowly, thereby ensuring the convergence of T, [60. Condition (C4) requires sufficiently strong signals for true features to distinguish them from numerous irrelevant ones [2 48). Under Conditions (CI)-(C4), we can prove the following theorem.
Theorem (Asymptotic Properties of VFS Statisties). Assume the screening threshold in Stage of VFS is 6. Then we have the following conclusions.
(1) Convergence Rate:
(exp Pr Pr - 0,1 0,1200 0,]2mT*] m < exp ( -2 1-27-20 2x nexp { - (C22n)))

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where 0 < < 1/2 C1 and O2 are constants.
(2) Screening Consistency:
Fr(F < exp { -
1-27--2 n exp { {-Can
(3) Model Size Bound:
Pr 1 15 2 (0 (gpgnnn 1-22 20] + nexp (-anpp
2
[expp
where O3 and C4s are constants,
denotes the number of elements in the set.
Theorem 1) demonstrates that T converges to its expected value fj and specifies the convergence rate. Theorem 2 shows that the probability of failing to screen true features is negligibly small, thereby confirming the screening consistency of y.. Theorem (3) states that, for any threshold 6, we can derive an upper bound, which constrains the number of screened features with high probability. In summary, subsampling in VFS significantly reduces computational costs while maintaining its effectiveness for feature screening, even for rare event. This guarantees the statistical property of Stage 2 in VFS algorithm.
For Stage of VFS, noting that we employ hypothesis testing, rather than threshold screening, to determine whether each group of features should be retained. Its validity is ensured by Theorem 2 with the definitions of ha.b and Sa,b provided in Appendix B.4 Define the null hypothesis as {HE X₁ IL Y}. which means all features within the k-th group are independent of the corresponding response variable Y.
Theorem (Asymptotic Normality of Group- Based VFS Statistics). Under the null hypothesis and ni/n + if 6o.1 or Êτ.0 and (x()) oc, then the group- group-based VFS
statistics T, satisfies mi 1/2r, 4 (0, m2fo.1 + mbL),o/s).
When Eo.1 $100 0, fo.2 > 0 and s + co, further denote
G( = 2x (1)
, 20) (y))
ho,2(,)) depending on n, satisfying
E G x2) + ni ' h (x
(1)
x
k.1
+0
(2)
E 2
x k.1 x!1
where ( and ((1) indepnedent copy of x then n, T T2/ S N(0,mf(m, - 11)2/2 as M1 - 0c.
The proof of Theorem can be found in 35 Theorem motivates to group all features and conduct hypothesis testing on each group. Rejecting the null hypothesis implies that the features in the group are independent of Y, suggesting they may contain true feature(s). It is worth noting that Theorem 2 does not require specific grouping strategy, which implies that the grouping strategy does not have significant impact on the screening results. This guarantees the statistical property of Stage in VFS algorithm.
In summary, the VFS statistic effectively addresses data imbalance challenges while maintaining robust performance under balanced scenarios. By selecting appropriate kernel functions, it generalizes existing test statistics-such event scenarios. It captures complex nonlinear dependencies often overlooked by linear methods. Developed within general framework and supported by solid theoretical guarantees, offers reliable foundation for wide range of applications. These strengths make the VFS statistic a adaptable and robust tool for feature screening.
distance correlation (28 61] and projection correlation [77] 79 to rare
Remark (Generalization Multi Party and Multi-Clas Scenarios). The properties of VFS discussed in this section can be naturally extended to both multi-partyy and multi-class settings. In the multi- party
7

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setting, each passive party only needs to compute the VFS statistics for its own features, while the active party handles decryption: no interaction between features held by different parties required. In the multi-class setting, the kernel function h in the VFS statistics only needs to be modified to accommodate differences among multiple classes. The properties and derivations of this extension are analogous to the theorems presented in this section, with further details provided in Appendix
Notably, the VFS framework differs from classical multiple testing in that it does not require controlling the type I error; moreover; without introducing any additional mechanisms, can still guarantee that the cumulative type II error across iterations converges to zero under certain assumptions. addition, Stage plays an essential role in refining feature screening and should not be skipped. Detailed discussions are also provided in Appendix
5 Simulation studies
In this section, we adopt numerical results on simulated datasets to demonstrate the effectiveness of VFS. In the following parts, the kernel function similar to the distance covariance but designed for rare events, whose definition and properties are presented in the Appendix
5.1 Simulation settings
Data generation. The simulated dataset is created with the following steps:
Step 1. Covariance matrix. We randomly generate X E TR from a multivariate normal distribution with zero-mean and covariance matrix (o pxpp where jid22 0.5l
real-world scenarios, true features often tend to appear groups Therefore, select true features in groups of 5 with the coefficients for each group predefined as (81, B2, B3, 34, (b5) = (3, 2, -2, -3).
Step 2. True features.
Step 3. Response. For 1 < i & the binary label variable Y; is drawn from Bernoulli distribution with success probability Y," We generate Y* from the following models:
Model 1: Y = L Σ m
3 - -7)
Model 2: Y = L exp プ 83 83,1s}
Model 3: Y = L Σ - By X,: + ビソ= B, exp(X -,)
Here, L(t) denotes the Logistic function, y is positive constant ensuring that n1/n is small enough.
Encryption settings. To conduct feature screening in a VFL system, different encryption methods required depending on the selected statistic. In these experiments,
consider both the Paillier [58] and CKKS 13 homomorphic encryption schemes, following the paired encryption procedure
described in Section 3.1 Since the results are almost the same, we only present the result for Paillier.
Performance measurement. We repeated each experiment M = 200 times on machine equipped with an Intel(R) Xcon(R) Gold 5320 CPU @ 2.20GHz and 72 GB of RAM, and evaluated the screening performance from two perspectives: (1) statistical effectiveness, measured by the Positive Selection Rate (PSR = I n Fi F)F and the False Discovery Rate (FDR = j Fl and (2) computational efficiency, evaluated by the average total computation time denoted Time.
Hyperparameter tuning. To demonstrate the robustness of VFS, we design comparative ex- periments of hyperparameter selection. Specifically, investigate the impact of hyperparam- eter on the performance VFS, including subsampling ratios s = 1, 3, 4, 5, initial group size So 10, 20, 50, 100, 200, and number of retained features after screening d = 50, 100, 150, 200, 250.
Comparison with other methods. It is remarkable that most existing literature focuses on feature selection and few screening methods have been developed for VFL. Since VFS includes distance covariance and improved projection covariance designed for rare events as special cases, we compare the screening results with these corresponding classical statistics in simulations. Additionally, we also include comparisons with mutual information (7 and Chi-squared statisties encompassed within the VFS framework.
34 which are not
8

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5.2 Simulation results
subset of the experimental results presented Figure while the complete simulation results can be found in Ap- pendix Compared to classical screening methods that evaluate each feature individually, VFS significantly reduces computational time while achieving comparable or even su- perior performance. This advantage becomes greater the feature dimension p increases, because VFS can quickly filter out a large number of irrelevant features. This is con- sistent with the theoretical results and strongly demonstrates the role of VFS in reducing feature dimensionality.
5000
1000
0000
2000
0.-
300
1020
2000
3020
4004)
35041
Moreover, as shown in Figure VFS robust to hyperpa- rameter settings. Increasing the subsampling ratio raises computational costs but provides little performance improve- ments. Choosing = 1 is sufficient for handling our sim- ulated data. larger initial group size 8p leads to greater time savings, but excessively large So may hinder the identification of true features. The number of retained features after screening d involves a trade-off between PSR and computational time.
Figure 4: Performance of VFS and classical screening.
1290
0.5
250
(a) Subsampling ratio
(b) Initial group size so-
(c) Number of retained features d.
Figure 5: PSR and computational time of different hyperparameter.
Finally, we compare VFS with other feature screening methods. The results presented in Table demonstrate that VFS exhibits significant advantages.
Table 2: PSR standard deviation of different feature screening methods.
Screening Method
Model
Model 2
Model
VFS
0.965 + 0.012
0.985 + 0.013
0.856 + 0.013
Mutual Information
0.789 + 0.017
0.847 + 0.020
0.666 0.019
Chi-squared
0.904 1 0.016
0.949 0.014
0.712 0.017
6 Real data analysis
To illustrate the performance of VFS in real data applications, we first consider four publicly available
real-world datasets: Activity [1. Gina (32), p53 Mutants [42), and RNA-Seq [25]. We adopt LESS-
VFL. 7 and MUSE (37 as feature selection methods. The following two approaches are considered:
(1) VFS+Selection (VFS+): Feature screening is first performed using VFS algorithm, followed by
further feature selection using LESS-VER VFL or MUSE; (2) Selection: Directly applying LESS-VE VFL or
MUSE for selection all features.
Since the true feature set F is unknown real-world datasets, PSR and FDR cannot be directly computed. Therefore, we compare model performance using Accuracy and AUC of predictive models, and also compare the computational cost (i.e., Time) of the two methods. As shown in Table 3 VFS+Selection achieves comparable accuracy while significantly reducing computational costs, with this advantage becoming more pronounced as the feature dimension p increases. For instance, in RNA-Seq dataset, VFS+Selection requires less than 5% of the computation time compared to using either LESSS-FFL or MUSE alone. Table [ Appendix provides the complete results for all datasets, which also demonstrates the robustness of VFS in real-world applications.

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Table 3: Feature selection performance. Accuracy and AUC evaluated the test set, Time includes the time for feature screening (if performed) and feature selection.
LESS. VFL
MUSE
Dataset
Method
Accuracy
AUC
Time
Accuracy
AUC
Time
Activity
VFS+
1.000
1.000
678.5
1.000
1,000
23.0
p =56
Selection
0.998
1.000
953.4
1.000
1.000
44.8
Gina
VFS+
0.836
0.915
201.8
0.842
0.842
21.8
p=968
Selection
0.843
0.924
467.0
0.843
0.842
61.5
p53
VFS+
0.835
0.830
248.8
0.839
0.845
53.0
p 5408
Selection
0.843
0.859
1493.3
0.849
0.862
279.3
RNA-Seq
VFS+
0.994
1,000
74.9
1.000
1,000
30.9
= 20531
Selection
0.944
1,000
1791.1
1.000
1,000
1062.3
To further demonstrate the robustness of VFS in actual VFL pipelines, we conduct an application based evaluation motivated by real-world credit modeling task in collaboration with a leading third-party payment platform in China. Our goal is to build a model using merchant risk labels as the response and large number of transaction- bbased features inputs. The dataset contains sample size of n = 10,000 merchants and a feature dimension of = 189,2% 223,, covering key features in different groups and their interactions. Those feature groups are, respectively, merchant attributes, transaction activity, revenue scale, business growth, customer structure, payment channel distribution, transaction stability, and temporal transaction patterns. The model outputs are further utilized to inform financial product recommendation strategies for merchants.
After removing redundant features with the VFS algorithm, validate its efficiency gains in downstream VFL tasks using SplitNN [66]. a neural network-based VFL method, as an example. The results in Table show that VFS achieves substantial reduction in computational cost, with the runtime reduced to less than 10% of that required by the original method. Furthermore, VFS achieves improved out-of-sample AUC while using significantly fewer features. This application- based evaluation further demonstrates the practical relevance, scalability, and robustness of VFS, especially in scenarios involving ultrahigh-dimensione features.
Table Performance VFS on the real-worke credit modeling dataset using SplitNN.
# Features
AUC
Screening Time
Modeling Time
Total Time
100
0,937
29.1
100.3
129.4
500
0.941
30.9
125.3
156.2
1000
0,936
31.7
151.9
183.6
All
0.901
1911.8
1911.8
Conclusion
In this paper, introduce Vertical Federated Feature Screening (VFS), two-stage feature screening algorithm based on model-free statistics and designed for accelerating feature selection process in VFL, especially when handling with ultraahih-dimensionee dimensional and event datasets. Theoretical analysis demonstrates that by combining subsampling and group screening techniques, VFS effec- tively eliminates irrelevant features in Stage 1, while providing more precise screening in Stage 2. This approach reduces the overall costs existing feature selection methods. Experiments on both synthetic and real-wor wal-worldd datasets show that VFS not only lowers computational costs but also retains strong screening performance, underscoring its potential for real-world VFL applications.
While VFS is effective in practice, it also has certain limitations and offers several avenues for future improvement. First, although VFS a flexible framework that accommodates wide range of screening statistics, its computational efficiency may depend the specific statistic used. Second, as VFS conducts marginal screening, the retained feature set may still contain redundancies. Finally, an interesting direction for future research lies in integrating homomorphic encryption with differential privacy to strengthen resilience against potential attacks targeting homomorphic encryption.
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Acknowledgments and Disclosure of Funding
This research supported by the National Natural Science Foundation of China (NSFC 72471230, 12225113 and 12171477), the National Key R&D Program of China (2023YFA 1008702 and 2023YFC3304701) the Beijing Social Science Fund(3GLA000)) the MOE Project of Key Re- search Institute of Humanities and Social Sciences (grant 22JJD110001) and Public Computing Cloud, Renmin University of China.
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Broader impacts
Question: Does the paper discuss both potential positive socictal impacts and negative socictal impacts of the work performed?
Answer: [Yes]
Justification: This paper's proposed Vertical Federated Feature Screening (VFS) algorithm has significant applications in fields like healthcare and finance, enabling institutions to collaborate on sensitive data while preserving privacy, and enhancing the efficiency of collaborative modeling in privacy-preserving frameworks.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g.. disinformation. generating fake profiles, surveillance), fairness considerations (e.g.. deploymentt of technologics that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Answer: [NA]
Justification: After thorough consideration, conclude that the VFS algorithm presented in this paper does not present any apparent risk of misuse.
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The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
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Question: Are the creators or original owners of assets (e.g.. code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
The experiments in this paper utilize publicly available datasets, and the original papers corresponding these datasets are duly cited.
Justification:
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The answer NA means that the paper does not existing assets.
The authors should cite the original paper that produced the code package dataset.
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Question: Are assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
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Appendix
The appendix consists of the following parts. Appesettt the derivation of the computational costs reported in T AppendixB|pret provides the proofs of the main theorems. Appendix [opendix(Cdiseuusess several technical details and extensions of the VFS framework. Appendix Definnes the screening statistic used in the experiments. Finally, Appendix D
reports additional experimental results.
A Computational complexity analysis of algorithm
First, we discuss the number of encrypted computations of VFS statistics required for feature screening. For classical screening, we need to compute 7, for each of the features Xj, so the number of computations can be written as O(p). For VFS, the number computations depends on the distribution of m true features. We consider two extreme cases.
Best Case. All m true features divided in one group. After one round of group screening, only this group is retained. Assuming 80 < p', these 8o features need to be screened individually. Thus, the total number of computations is O + 50
Worst Case. In each round of group screening, all m true features are placed in different groups Assuming the group screening process involves r rounds, is obvious that the first round will require p computations, and the subsequent r - rounds will each require to computations, where eo
satisfies mpp"11p 8g < p'. Noting that p' features need be screened individually, the total number of men +p/).
computations is O 2 log,
90
To summarize, we combine the results discussed above Table 5
Table 5: Comparison of encrypted computation counts for Ty across different methods.
Method Classical Screening VFS (Best Case)
VFS (Worst Case)
Computation Counts
(pp)
P + 80
m
log,
p
+ p
Next, will outline the cost for each part of feature screening in VFL. Recall that the costs for encrypting, sending, and decrypting a number are denoted as le, l, and &a, respectively, while the cost of multiplying plaintext number with ciphertext number is L.. In general, let P represent the number of encrypted computations of VFS statistics required for feature screening. and let N denote the sample size.
Encryption. The label matrix E F N needs to encrypted, so the encryption cost is O(N2L.)
Computation. Since P encrypted computations of VFS statistics are required, and each computation has time complexity proportional the square of the sample size, i.e. N2, the computation cost can be written as O PP.).
Communication. After each encrypted VFS statistic is computed, the result needs to be sent, so the communication cost ((P(.).
Decryption. After the active party receives encrypted VFS statistics, decryption is required for each, so the decryption cost ((P()).
Clearly, for classical screening. N = n, and for VFS, N = (s+1)n By substituting the computation counts P from Table 5 into these expressions, obtain the results presented in the following table.
B Proof of theorem
B.1
Useful lemmas
Before presenting the proof details of the theorem, we introduce two useful lemmas.
Lemma 1 (Hoeffding's Inequality for Two Sample U-Statisties). Let XI,..., Xm. Yi.... Y, be (m + n) independent random variables. For 2 and consider random variable of the
23

---

Table 6: Comparison of costs for feature screening across different methods in appropriate group partition. Without loss of generality, assume that the time complexity of the chosen statistic is O(n²).
Cost
Classical Screening
VFS (Best Case)
VES (Worst Case)
Eneryption
((r
Oi(s 1)4nft))
Oi(s L)
Computation
O(n²p(.)
(s + 1) Z to)le)
0
((s 1)
ma log,
Ex)
Communication
O(pl,)
0 Gu t006,)
m log
t p
E
mine
Decryption
O(pla)
0 (R 80)
m log
p
form
-1
U
11
Σ M((i ((Xi...... X Yi...., Yi.);
8
where the summation Ee is taken over all r-tuples (f1,...,;y) 1r) of distinct positive integers & m and all s-tuples (fi,..., :s) of distinct positive integers &
If a & h(X........ X Yn...... Yz.) & b, then for any t O, and m, 8 & n, have
Pr (IU - E(U) 2 1) & 2exp { {- (b - a)²
2\12
(
where X = min[[m/r], [n/s]]. [n/s]]
Lemma (Multiplicative Chernoff Bound). Let X1, X Xn, be independent Bernoulli random variables, and let Sn - X; denote their sum. Let μ (( be the expected value of Sn- Then, for any 0 < X < 1, the following bound holds:
2² Pr[S, & (1 -))m & exp
2
B.2 Asymptotic properties of generalized U-statistics
We first focus on the scenario where subsampling is not considered, i.c., when using the full sample. To perform feature screening, it is sufficient to compute the marginal correlation between each X; and Y, such as Kendall's т, Spearman's 's rank correlation, distance correlation, and improved projection correlation [6). All of these correlation measures be unified within the framework of two sample takes values between and 1, we can define a generalized U-statistic for the j-th feature Xj and Y follows, using the notation in the main text:
U-statistics. Specifically,
T₃
x
mo
ma
m
Similarly, we propose the following condition in order to study the theoretical properties of Ty:
(C1') (Feature Distribution) There exist constants C1, > 0 such that for all t > 0,
sup max Pr f > t < exp(-cat).
P EUj
(C2') (Feature Dimension) There exists < 7 < 1/2 such that logg = a(n 1-1))
(C3') (Class Imbalance) Case sample size n, satisfing n7 22log n + log p).
(C4') (Signal Strength) For all true features X, with j E F, there exists
1/2 such that
min
JE 10,1 > 2n
Under Conditions (CI')-(C4'), we can obtain a result similar to Theorem Theorem 3 (Asymptotic Properties of Generalized U-Statisties in VFS). Assume the screening threshold in Stage 2 is 8. Then we have the following conclusions.
24

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(1) Convergence Rate.
Pr [ < <0 exp [-nnn 2 n nexp C-2n
where K 1/2. < < 1/2 - K, C₁ and C2 are constants.
(2) Screening Consistency.
Pr Fr{FeF} <0 m exp - 12 n nexp (-xn(-C
(3) Model Size Bound.
Pr F
2
21
Σ
031
- 20] + n nee (CCan]])
where C3 C4 constants.
(0) Proof. To simplify, we denote h 1. the subsequent analysis, we truncate h at M, then the kernel function can written as
X (0)
x ) as hy. For the needs of
d. 3.d
Jihi.my
def h4.1 + h322
hg hy 1[/b & M) + h2 1[b,] > M)
and denote f, = E Ty, = (m)) -1 ( -e h311. 1 = 1,2.
E.
For the first part 1,,1, since -M h2.1 M, according Lemma for all e 0, we have
Aea
Pr Pr 011 > of < 2 exx
2M2
where A = min [m]] [m/mm]]}
Suppose that mn is exactly an integer multiple of my and that no is sufficiently large. In this case, we have À = n2 n and
{
c
Pr[[T,,1 0,11 > e) < 2exp
2m, M
It's noteworthy to point out that only consider the case where is divisible by mi; otherwise, we can discard redundant positive observations: this approach loses some observations, but makes our analysis simpler.
For the second part 22 let M n7.0 <ソ< K, then f3.2 + when n > oc(in that case, both n1 and M tend to infinity, and |hy] < M holds with high probability), thus
Pr[[T,,2 0 c) < Pr 21 c/2).
If we assume that |hy] - oo only if there exists (1) + xo. {0. 1). E (1...., m.), we can find constant >> 0 such that [hy] < M if X,, JäH,
a.M for each observation & In this case, h = 0 and T3 = 0, so we have the inclusion relationship of events:
[7,22 c/2) S (3i (1....., n) s.t. |X,al aM).
Under Condition (C1'), we have
Pr( TT22 0322 c) Pr[[T,,2] c/2)
< n max Pr X > aM}
18jEp
< ncn -
where C1, 02 > 0 are constants.
25

---

In summary, 7 converges to its parameter function f and we have
Pr(IT) 0yl > 2c} & Pr([TI.
011 o)
Pr(1T2, - 022 e)
E (exx {
2
(3)
+ +nexp(-c2aM())
2m1 2m1M² M2
At last, we l = /2 and M = n,. denote C1, C2 > 0 constants, and get the following inequality
Pr[[; 05l n < O(exp( 2: + nexpl
Condition (C3') ensures that - guaranteeing the convergence of T to 0z.
tends to as and n, approach infinity, thereby
The proof of Theorem 3 (1) is complete.
Noting that f = {ji |7 > 8}. From condition (C4') and the Theorem 311 we know that for any j & F, f = 0 and [Ty < n2 with high probability; for any F, 10 2n and [Ty > ni with high probability. Specifically, if we let å = nTT", we have
Pr(i & Fljj EF <
2 + - (2n
Since we have true features, the probability that there exists an important feature that is not screened can be expressed as
Pr[ F) & O(plexp(-Cnn 1-22--2 nexp(-C2n
The proof of Theorem is complete.
Let E = 8/4 and M = n 33 can written as
O(exp { - 1-27 nexpl-
Pr T 0,l
(4)
where C3, C4 are constants.
define event = {maxies<r IT, 0,1 < 8/2). When A holds, the number of [j [T,1 > 8}
(i.e. IFD cannot exceed the number of { 10,1 > 6/2}, which is bounded by (8/2) Σ, |0 Together with we find that the model size (i.e. number of selected feature by our method) is bounded.
Pr
1051
10
O (-22) nexp(-Can]H)
> 11(
The proof of Theorem
3) is complete.
B.3
Proof of Theorem
Proof. Recall that the VFS statistic is obtained by considering subsampling based generalized U-statistics. Therefore, the proof strategy for Theorem is essentially similar to that of Thcorem 3 showed below.
First of all, we need define the event (io > n1). We want to hold, otherwise we cannot use all positive observations.
Since 7o random variable with mean of sm from Lemma for any 0 À < we have
Pr(ño & (1 - A)sn,} < exp {
X²
2 88
Let À = 1 1/s, the probability of {īo & (i.e. B does not hold) can be obtained as {
(s
1
{
1--1)2n
Pr(ño & ni & exp
< exp
(5)
2s
2s
26

---

It is clear that n1 - oc as - co. Thus, the probability of no & tends to 0, i.e., B holds with high probability.
Similar to the method used in the previous section, we truncate the inner part of summation at M. Thus, have
100
Σ
nio
n1 m=
Σ c9
13
def T3 + T32,
(6)
mi
m_
mo
where es denotes all possible combinations satisfying di such combinations (ma) (m;). so T₃ in (6) remains a generalized U-statistic.
for I...., ma- The number of
For the first part T as when using full data, we have
PH[[T,, 0,,1 > +) < 2exp {
2m,
M2
Since holds with high probability, we can be assured that this operation is feasible. For the second part 22 we can also follow similar approach show that satisfies
Pr(
G.2l 2 c) < Pr[1T,,2| > c/2)
Pr[ aM}
(8)
16j
náj exp(-
It's important to note that, since we do not know the sampling method 6, in advance, we should ensure that all observations (instead of only the observations drawn from subsampling) satisfy |Xyl < aM to guarantee the correctness our proof.
Combining inequality
[), and (8) we have
T
+ ncxp(-caaM) + exp {
(s - 1-])
Pr[1ī, - 8,1 > 2c} < (expp
2m, Mm,M2 M
2s
Let M = n7.00 <
K, the above inequality can be written as
en 1-2
+ nexp(-(zam?) + exp { (s - 1)²m
Pr[ - 0,1 > 2 < (eep
). (9)
2m1
2s
At last, we let c = ni" /2, denote Ö₁, Ō2 0 constants, and get the following inequality.
Pr 0,| > n m & O(exp(-Cx 1-2; 22-22) + nexp(-C2n7)).
The proof Theorem (1) is complete.
Noting that f { : [Ty] 6}. From condition (C4) and the Theorem 1 ()+ we know that for any j & F, 0, = and |ł,1 < ni with high probability; for any j E F, 10,1 > 2n and [T, ni with high probability. Specifically, if we let = ni we have
Pr[j & Fli F) < O(expf - 1-1-17 22n } +
Since we have m true features, the probability that there exists an important feature that is not screened can be expressed
Pr[F of F) < O(plexp(-C1n) 1-27- nexp(-C2nn]1
The proof of Theorem 2 is complete.
Furthermore, if we let = 8/4, inequality
9 can be written as
Pr [T 0,1 > _ < O(exp -088 + nexp(-C,nf])
(10)
27

---

We define event A = {maxi IT, 8,1 < 6/2}. When A holds, the number of { [T]] 8}
(i.e. (FI cannot exceed the number of {j |0,1 > 8/2), which is bounded by (6/2) (6/2)~E,; 10 Together with inequality (10), find that the model size (i.e. number of selected feature by method) is bounded.
Pr
031
Pr(A}>1- 16ppex
- nexpl-
The proof of Theorem 3) is complete.
B.4 Preliminaries for Theorem 2
To facilitate the discussion of Theorem for 0 & a < my and 0 < b < mi, we define (a. b)- observation projection of
ha.b
X
(2)
k,
k1
=E h
(0) 2,1
(
(() k.m=
x
(0)
x x (1)
k.ma
k.1
k,1
Based on this, we define La,b as the variance of ha,b, that is,
La.b Var (ooob (x (
x
k. (_.)...00))))
(0) (1)
C Additional discussions on VFS framework
This section provides supplementary discussions extending the main text. We elaborate on three aspects of the VFS framework: (i) its extension to multi-party and multi-class settings; (ii) its conceptual distinction from classical multiple testing and the associated cascading error analysis; and (iii) the necessity of Stage 2 for refined feature selection.
Extension multi-party and multi-class settings. In the main text, we only considered the simplest case involving two parties and two classes. In fact, VFS can be naturally extended to multi-party and multi-class settings.
First, it is important to note that VFS is performed marginally on individual features and does not involve interactions between features. In the multi-party setting, the active party only needs to encrypt the pairwise label matrix and send the result to each passive party. Ciphertext computations are carried out locally by each passive party, while the active party is only required to perform decryption. This process is identical to that in the two-party scenario.
The extension multi-class settings is also feasible. Define the screening statistic as
-1
K
Tk
ñoo mg
n m
Σ
II Too
(x
(K)
x(K)
mg
j=1
The asymptotic distribution of Tk can be referred to Corollary in 351 The cut-off value can be selected accordingly, and then theoretical guarantees in Theorems T and 2 hold in the multi-class
case. This establishes the theoretical foundation for extending VFS to multi-clase settings.
Difference with multiple testing and cascading error analysis. In VFS framework, our focus differs from classical multiple testing, where the goal is typically to control the Type error. For instance, the Bonferroni correction reduces false positives by using a/m, but at the cost of increased contrast, we a regardless of the number of groups to maintain statistical power, thereby focusing on reducing Type Il error rather than controlling Type error. Theoretical guarantees on the Type error are established in Theorem 2). Furthermore, the potential accumula- tion of cascading errors in Stage can be explicitly quantified. Suppose that each iteration involves
Type error [16 20
28

---

I groups and that total of r iterations are conducted with Ir S p. Under similar conditions as in Theorem 22, the probability that any relevant feature is mistakenly removed after Stage satisfies
P( [ Ir (ex (- exp (-Ō2n) +0,
ensuring that errors do not accumulate uncontrollably across iterations.
Necessity of Stage 2 in VFS. Theorem 2 shows that the screening statistic is asymptotically normal when condition (2) is satisfied, which requires that each feature group in Stage contains sufficiently large number of features (empirically more than five (77). Consequently, Stage alone not sufficiently fine-grained, the retained groups may still contain considerable number of irrelevant features. This highlights the necessity of Stage 2 for further refinement.
D Definition and asymptotic properties of the VFS statistic adjusted from distance covariance
For simplicity, omit the subscript j of the VFS statistic T3 here. The VFS statistic used in Section and is defined as
Tdenw
110
111
Neccorr (x,0,, x( x(1))
(II)
2
i=1
j>i
k= 1>k
with kernel function
hacov X, (0),
(x(") x
x = [x(0) - -x(1) + | - -x,"1 + | - x
1
| -x,")
x²
-2 x - xx - -2 xx,"
(12)
Furthermore, corollary of Theorem
2 can obtain the asymptotic distribution of Tacov-
Corollary (Asymptotic Normality of Group-based de Under the null hypothesis, ni/n and condition in (2) the VFS statistic defined in (77) satisfies
NaTowo 1/2 4 N(0,2)
E Additional experimental results
In numerical simulations, the default simulation setting is as follows: sample size n = 10000, number of features 2000, number of true features 20, initial group size So 100, decay coefficient and number selected features after screening d 100. While keeping other settings fixed, Tables 7 [O] investigate the effects of total number of features p, number of true features m, initial group size So, and number of retained features after screening d, respectively. In these tables, Time reports the computational time for encrypted feature screening in seconds, and Time Ratio compares the average computational time of VFS with that of classical screening. Results from the ablation studies suggest that method is robust and not sensitive hyperparameter choices.
-= 0.5 0.5
For real-world datasets, Table reports the results under different numbers of retained features after screening (i.c., d), which further demonstrates the robustness of VFS in real-world applications.
29

---

Table 7: Performance of VFS and classical screening under different p.
VFS
Classical Screening
PSR
FDR
Time
PSR
FDR
Time
Time
P
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Ratio
Model
1000
0.962
0.950
0.808
0.810
316.2
318.7
0,927
0.950
0.815
0.810
1519.0
1472.6
0.208
2000
0.959
0,950
0.808
0.810
507.5
502.
0.894
0.900
0.821
0.820
3023.1
3029.5
0.168
5000
0.945
0.810
852.3
866.7
0.846
0.850
0.831
0.830
7634.5
7463.1
0.112
Model 2
1000
0.985
1.000
0.803
0.800
382.6
382.2
0.964
0.950
0.807
0.810
1490.8
1522.5
0,257
2000
0.983
1.000
0.804
0.800
506.8
513.9
0.943
0.950
0.811
0.810
2988.0
3047.6
0.170
5000
0.978
1,000
0.804
0.800
1085.2
1127.2
0.910
0.900
0.818
0.820
7385.2
7534.5
0.147
Model
3
1000
0.860
0.850
0.828
0.830
470.5
474.8
0.786
0,800
0.843
0.840
1526.4
1491.2
0.308
2000
0,869
0.850
0.826
0.830
530.0
529.
0.743
0.750
0.852
0.850
3056.6
2997.0
0.173
5000
0.836
0.850
0.833
0.830
825.7
836.5
0.684
0.700
0.863
0.860
7432.0
7356.0
0.111
Table 8: Performance of VFS and classical screening under different m.
VFS
Classical Screening
PSR
FDR
Time
PSR
FDR
Time
Time
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Ratio
Model
T
0.992
1.000
0.901
0.900
363.7
368.3
0
1.000
0.903
0,900
3004.6
3032.4
0.121
20
0,957
0.950
0.809
0.810
429.6
436.8
0.896
0.900
0.821
0.820
3019.2
2950.1
0.142
50
0.836
0,840
0.582
0,580
658.6
659.3
0.733
0.720
0.633
0.640
2996.8
2992.
0.220
Model 2
0,999
1.000
0,900
0,900
500.0
490.1
0.991
1.000
0.901
0,900
3007.0
2999.9
0.166
20
0,981
1,000
0.804
0.800
455.8
459.9
0,940
0.950
0.812
0,810
2970.6
2976.1
0.153
50
0.864
0,860
0,568
0,570
748.9
744.1
0.761
0.760
0.619
0.620
2999.9
2972.9
0,250
Model 3
0,956
1.000
0.904
0,900
406.2
398.7
0.886
0.900
0.911
0,910
2952.8
3039.8
0.138
20
0.858
0.850
0.828
0.830
546.5
543.5
0.738
0.750
0.853
0.850
3058.4
3007.6
0.179
50
0,487
0.480
0.756
0,760
592.9
611.0
0.451
0.460
0.775
0.770
2940.8
2952.0
0.202
Table 9: Performance of VFS and classical screening under different 8p-
VFS
Classical Screening
PSR
FDR
Time
PSR
FDR
Time
Time Ratio
80
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Mean
Med.
Model
20
0.946
0.950
0.811
0,810
831.2
841.9
0.279
50
0.958
0.950
0,808
0,810
528.8
528.6
0.891
0.900
0.822
0.820
2978.8
2979.0
0,178
100
0.958
0.950
0.808
0.810
473.7
474.9
0.159
Model 2
20
0.978
1.000
0.805
0,800
633.3
640.4
0.208
50
0
1.000
0.804
0.800
623.6
623.2
0.943
0.950
0.811
0.810
3042.8
2956.6
0,205
100
0.984
1.000
0.803
0,800
522.0
512.1
0.172
Model 3
20
0.843
0.850
0.832
0.830
665.8
685.6
0,220
50
0.852
0.850
0.830
0.830
565.9
562.3
0.744
0.750
0.851
0.850
3021,2
2982.6
0.187
100
0.858
0.850
0.828
0.830
383,2
389.4
0.127
Table 10: Performance of VFS and classical screening under different d.
Classical Screening
PSR
FDR
Time Mean
PSR
FDR
Time
Time
d
Mean
Med.
Mean
Med.
Med.
Mean
Med.
Mean
Med.
Mcan
Med.
Ratio
Model
I
50
0.927
0.950
0.629
0.620
455.6
446.6
0.841
0.850
0.664
0.660
3036.8
2955.6
0.150
100
0.957
0.950
0.809
0.810
486.9
500.7
0.894
0.900
0.821
0.820
2984.3
2984.0
0.163
200
0.968
1.000
0.903
0.900
818.9
832.0
0.933
0.950
0.907
0.905
3020.8
3026.7
0.271
Model 2
50
0.966
0.950
0.614
0.620
4
404.7
0.901
0.900
0.640
0.640
3031.4
3007.3
0.133
100
0.984
1.000
0.803
0.800
583.0
590.3
0.944
0.950
0.811
0.810
3000.9
2982. 1
0.194
200
0.985
1.000
0.901
0.900
752.2
736.4
0.965
0.904
0.905
2978.9
29977.
0.253
Model 3
50
0.808
0.800
0.677
0.680
378.3
368.7
0.683
0.700
0.727
0.720
2978.8
3051.6
0.127
100
0.860
0.850
0.828
0.830
424.8
409.0
0.741
0.750
0.852
0.850
3029.6
2979.8 8
0,140
200
0.877
0.900
0.912
0.910
864.7
873.3
0.799
0.800
0.920
0.920
3040.9
3028.1
0.284
30

---

Table 11: Performance of Selection v.s. VFS+Selection. 80% of the data is randomly split for training and 20% for testing. Accuracy and AUC both evaluated the test set.
LESS VFL
MUSE
Screening
Selection
Inial Time
Selection Time
Total
Dataset
Method
Accuracy
AUC
Accuracy
AUC
100
6.1
0.810
0.885
138.0
144.1
0.823
0.823
5.1
11.2
Gina
VFS+
200
9.9
0.836
0.915
191
201.8
0.842
0.842
11.8
21.8
500
21.3
0.859
0.933
295.8
317.0
0.843
0.842
31.4
52.6
Selection
All
0.843
0.924
467.0
467.0
0.843
0.842
61.5
61.5
50
0.999
1,000
325.7
329.1
1.000
1.000
Activity
VFS+
100
4.9
0.999
1,000
469.3
474.2
1.000
1.000
6.4
11.3
200
8.5
1,000
L.000
670.0
678.5
1.000
1.000
14.6
23.0
Selection
All
1,000
953.4
44.8
44.8
200
21.6
0.994
1,000
53.3
74,9
1,000
1,000
9.3
30.9
VFS+
1000
17.4
1,000
1.000
123.3
140.7
1.000
51.1
RNA-Seq
2000
18.0
1,000
1.000
207.1
225.1
1.000
1.000
103,9
121.9
Selection
All
0.944
1.000
1791.
1791.1
1.000
1.000
1062.3
1062.3
77.7
VFS+
500
2.7
0.835
0.802
137.8
140.5
0.808
0.813
24.0
26.7
1000
0.835
0.830
245.3
248.8
0.839
0.845
49.6
53.0
Selection
All
0.843
0.859
1493.3
1493.3
0.849
0.862
279.3
279.3
31


Reviews:
[{"review_id": "YJl9FiBwpg", "reviewer": "Reviewer_ekrS", "summary": "The proposed method attempts to address the ultrahigh dimensionality of features and the sparse data structures inherent in large-scale datasets in a VFL context. The gist of the method lies in a two-stage approach by which features are selected in a coarse-to-fine manner: first by groups then by individual features.", "questions": "1. how different options in the grouping method of stage one would affect the overall performance?\n\n2. in terms of scalbility, what will happen if apply the method to features more than 100K?\n\n3. how would privacy-preserving mechanisms e.g. DP or HE affect the proposed method?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Stength: \n+ The idea, which is based on Generalized U-statistic, seems straightforward and effective. \n+ Theoretically analysis elucidate the effectiveness of the method. It also provides kind of performance estimation which seems agree with the empirical results. \n\nWeakness: \n- empirical results on real datasets are limited, in terms of the number of features tested (50K features are only considered moderated in real-life applciations)\n- feature selection methods are not necessarily limited to VFL. It would be good to compare the proposed with more baseliens e.g. those referred in Sect.2.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank Reviewer ekrS for recognizing the strengths of our paper, specifically highlighting the straightforward and effective idea based on Generalized U-statistics and our rigorous theoretical analysis. We highly appreciate these insightful comments and constructive suggestions, which help clarify and enhance our contribution. Below we address every question and comment raised by the reviewer in detail.\n\n**1. Empirical Results with More Features.** We would like to kindly note that this work was originally inspired by **a real-world credit modeling task**. While privacy concerns initially prevented us from including the real data results, we have now incorporated them following your suggestion. The data provider has approved their inclusion under confidentiality agreements, and the results are demonstrated through a secure sandbox environment (Intel(R) Xeon(R) Gold 5320 CPU @ 2.20GHz and  62 GB of RAM). The goal is to build a model using merchant risk labels as the response and a large number of transaction-based features as inputs. The model outputs are further utilized to inform financial product recommendation strategies for merchants. Specifically, the total sample size is  $n=10,000$ and the feature dimension $p=189,236$ (**far larger than 100K**). The features are generated from transaction-level payment records to describe each merchant's business behaviors, including but not limited to merchant attributes, transaction activity, revenue scale, business growth, customer structure, payment channels' distribution, transaction stability, temporal transaction patterns. Applying the VFS method, we remove redundant features to efficiently detect risky merchants. We validated the efficiency gains of VFS in downstream VFL tasks , **taking SplitNN (Vepakomma et al., 2018) as an example**. This is a neural network VFL method, as suggested by Reviewer xnKO.  As shown in the table below, **VFS achieves a substantial reduction in computational cost**, with the runtime reduced to less than 10% of that required by the original method. Furthermore, **VFS achieves improved out-of-sample AUC while using significantly fewer features**. Other competing methods are also considered in the revision, and the results are similar. This further demonstrates the scalability of our method, as you kindly asked, highlighting its superior performance even in scenarios involving more than 100K features.\n\n| \\#   Features |  AUC  | Screening Time | Modeling Time | Total Time |\n| :-----------: | :---: | :------------: | :-----------: | :--------: |\n|      100      | 0.937 |     29.098     |    100.313    |  129.411   |\n|      500      | 0.941 |     30.876     |    125.343    |  156.219   |\n|     1000      | 0.936 |     31.699     |    151.941    |  183.640   |\n|      All      | 0.901 |       -        |   1911.829    |  1911.829  |\n\n**2. Additional Baseline Comparisons.** Thank you very much for your insightful comment regarding the baselines. Indeed, previous studies have already discussed feature selection within federated learning (FL), but feature screening in FL has not yet been explored. Hence, in Section 2.1, we have only reviewed feature selection methods in VFL as well as feature screening methods in non-FL scenarios. In response to your suggestion about the performance of non-FL feature selection methods, **we have now additionally included classical feature selection methods, such as Lasso (Tibshirani, 1996), SCAD (Fan and Li, 2001), and Elastic Net (Zou and Hastie, 2005), in subsequent analyses**. Using the p53 dataset as an example, the following table demonstrates that VFS quickly removes irrelevant noise, thereby enhancing both the efficiency and performance of feature selection, as evidenced by our theoretical analyses.\n\n|      Method      |  AUC  | Total Time |\n| :--------------: | :---: | :--------: |\n|      Lasso       | 0.824 |   498.0    |\n|   VFS + Lasso    | 0.890 |    17.4    |\n|    ElasticNet    | 0.824 |   522.8    |\n| VFS + ElasticNet | 0.890 |    17.5    |\n|       SCAD       | 0.885 |   1227.6   |\n|    VFS + SCAD    | 0.923 |    42.2    |\n\n**3. Grouping Strategy.** The grouping strategy in Stage 1 has minimal impact on the overall performance. Theoretically, we have established the asymptotic normality of the group-based VFS statistics in Theorem 2. Empirically, to address your concern, we conducted **an additional experiment in which the feature rankings were randomly permuted in each replication to create a new grouping strategy**. The results yielded a PSR of $0.967\\pm0.033$, which is very similar to the original result of $0.965\\pm0.042$, with **no statistically significant difference**. We have made this point clear in the revision per your kind advice.\n\n**4. Effect of Privacy-Preserving Mechanisms.** Our method is compatible with any suitable homomorphic encryption (HE) scheme, requiring only the availability of one appropriate HE method. We would like to kindly remark that **the primary focus of this work is on the statistical and computational aspects of feature screening in VFL, rather than the cryptographic guarantees of HE**. However, per your kind advice, we have added discussions about the effect of privacy-preserving mechanism in the revision with a newly added remark. Specifically, the privacy-preserving property of HE could be secured by Gentry’s foundational work (Gentry, 2009). There is also a possibility that HE could be vulnerable to certain inference attacks, more advanced variants of HE could offer stronger protection (Bost et al., 2014; Falcetta and Roveri, 2022). Furthermore, combining HE with differential privacy (DP) is a promising direction for enhancing privacy guarantees,  as suggested in recent studies (Li et al., 2022; Wang et al., 2023), which we now have cited in the revised manuscript. Thank you again for this excellent point.\n\n**References**\n\n[1] Vepakomma, P., Gupta, O., Swedish, T., & Raskar, R. (2018). Split learning for health: Distributed deep learning without sharing raw patient data. arxiv preprint arxiv:1812.00564.\n\n[2] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology, 58(1), 267-288.\n\n[3] Fan, J., & Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association, 96(456), 1348-1360.\n\n[4] Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society Series B: Statistical Methodology, 67(2), 301-320.\n\n[5] Gentry, C. (2009). A fully homomorphic encryption scheme. Stanford university.\n\n[6] Bost, R., Popa, R. A., Tu, S., & Goldwasser, S. (2014). Machine learning classification over encrypted data. Cryptology ePrint Archive.\n\n[7] Falcetta, A., & Roveri, M. (2022). Privacy-preserving deep learning with homomorphic encryption: An introduction. IEEE Computational Intelligence Magazine, 17(3), 14-25.\n\n[8] Li, B., Micciancio, D., Schultz-Wu, M., & Sorrell, J. (2022). Securing approximate homomorphic encryption using differential privacy. In Annual International Cryptology Conference (pp. 560-589). Cham: Springer Nature Switzerland.\n\n[9] Wang, B., Chen, Y., Jiang, H., & Zhao, Z. (2023). Ppefl: Privacy-preserving edge federated learning with local differential privacy. IEEE Internet of Things Journal, 10(17), 15488-15500."}, {"review_id": "0g41TWWeAC", "reviewer": "Reviewer_yBdV", "summary": "This paper proposes a Vertical Federated Feature Screening (VFS) algorithm designed to filter out spurious features that may arise in Vertical Federated Learning (VFL) settings. The authors support the effectiveness of their method through both theoretical results and empirical evaluation.", "questions": "- In Section 6, is there a communication cost difference between using LESS-VFL alone and using VFS + LESS-VFL?\n- Please clearly define m_0 and m_1 in Equation (1).\n- The statement of Theorem 1.1 does not appear sufficient to justify the claim in line 203. In particular, tilde{C_2}>1/C_1 is required where C_1 is defined as C_1 = liminf n_1^gamma/log n > 0 (the positivity comes from assumption (C3)). Such a result would be needed to ensure that the second term on the left-hand side of Theorem 1.1 tends to zero as n_1\\to\\infty.\n- Are the datasets used in Section 6 class imbalanced?", "limitations": "No. I would encourage the authors to include a dedicated section that discusses the limitations of their approach.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The proposed VFS algorithm can be used as a preprocessing step before existing VFL algorithms such as LESS-VFL to improve overall efficiency. Notably, VFS appears to be more efficient than classical screening methods under class imbalance condition.\n- The authors propose a model-free VFS statistic inspired by generalized U-statistics, and they prove theoretical properties of the proposed method.\n(Due to time constraints, I was unable to fully verify the theoretical proofs provided in the appendix. I assume that the proofs are correct and base my review on the theoretical results presented in the main text.)\n\nWeaknesses\n- While feature screening based on generalized U-statistics has been studied in the statistics literature, it may be unfamiliar to a general machine learning audience, including myself. I suggest providing a brief background on this topic in the main text or appendix, which would help clarify the novelty of the proposed method.\n- Given recent trends in the ML community, more extensive experiments on real-world datasets are expected.\n- In the experiments, the authors still apply an additional federated feature selection method such as LESS-VFL. Is it difficult for the proposed method to perform sufficiently accurate feature screening to enable the use of a vanilla VFL algorithm? If possible, it would be helpful to include results of applying vanilla VFL directly after VFS.", "was_revised": true, "final_justification": "The authors have addressed most of the issues I raised through adequate explanations and additional experiments. Accordingly, I will maintain my positive score.", "rebuttal": "We thank Reviewer yBdV for the positive feedback and for recognizing the theoretical contributions of our work. We appreciate your comment that VFS improves efficiency especially under class imbalance and serves well as a preprocessing step for VFL feature selection algorithms. Below, we address your comments and questions in detail.\n\n**1. Background on Feature Screening Based on Generalized U-Statistics.** We appreciate the reviewer’s suggestion. While feature screening is an active research topic in statistics and generalized U-statistics serve as a classical powerful tool in theoretical analysis, **the adoption of generalized U-statistics dealing with data imbalance and sampling technique has not been well explored**. We would like to kindly remark that this is one of the key contributions of our work. Thank you for pointing this out. Per your kind advice, we have **added more background on generalized U-statistics in Section 2.1 in the revision**. Furthermore, we have clarified the novelty of the proposed VFS method by emphasizing how generalized U-statistics are leveraged to construct new screening statistics tailored for VFL scenarios, **enhancing both statistical efficiency and computational feasibility**. Thank you for helping clarify our contribution.\n\n**2. More Experiments on Real-World Datasets with Vanilla VFL algorithm.** We thank the reviewer for emphasizing the importance of real-world validation. Our study is indeed directly motivated by a practical credit risk modeling task subject to privacy constraints. Following your suggestion, we now include this case study based on collaboration with a third-party payment platform. The data contain 189,236 features across 10,000 merchants (financial product users), covering diverse aspects such as merchant profiles, transaction behaviors, and temporal dynamics. See also our reply to Reviewer ekrS. Using the proposed VFS method, we have filtered out redundant features and have **employed SplitNN (Vepakomma et al., 2018) as an example of Vanilla VFL algorithm**. The results below confirm that **VFS reduces runtime** by over 90% while **improving prediction accuracy**. This further supports the scalability and practical value of our method, no matter whether the downstream tasks are trained by federated feature selection algorithms or Vanilla VFL algorithms.\n\n| #   Features |  AUC  | Screening Time | Modeling Time | Total Time |\n| :----------: | :---: | :------------: | :-----------: | :--------: |\n|     100      | 0.937 |     29.098     |    100.313    |  129.411   |\n|     500      | 0.941 |     30.876     |    125.343    |  156.219   |\n|     1000     | 0.936 |     31.699     |    151.941    |  183.640   |\n|     All      | 0.901 |       -        |   1911.829    |  1911.829  |\n\n**3. Discussion on Communication Cost.** Thank you for raising this important point. **We have already provided the expression for the communication cost order in Table 1 of the submitted version**. Indeed, the VFS procedure introduces an initial communication cost, which is inevitable for most federated screening or selection methods. However, **the communication cost accounts for only a small fraction of the overall cost**, which leads to no further influence on the iteration of joint modeling. For example, under the default simulation settings in Appendix E, the time spent on communication accounted for an average of 4.26% of the total runtime. In real-world applications, the actual communication cost can be further influenced by various factors such as network conditions and hardware configurations (McMahan et al., 2017).  Thus, **the communication overhead introduced by VFS is relatively small when weighed against the significant reduction in per-party computation cost.**\n\n**4. Limitation of VFS.** Per your kind advice, we have included a discussion on the limitation of VFS in the revision. First, although VFS is a flexible framework that accommodates a wide range of screening statistics, its computational efficiency may depend on the specific statistic used. Second, as VFS performs marginal screening, the selected feature set may include redundant ones as well. In some scenarios, a subsequent feature selection step may still be required for identifying the important features rather than removing the redundant ones. This is also why we originally adopted feature selection methods as downstream tasks to mitigate feature redundancy and correlation. However, for downstream models that are robust to correlated and partially redundant features, such as SplitNN, VFS can be directly integrated without additional selection steps.\n\n**5. Other Issues.** \n\n(5.1) **Notations for $m_0$ and $m_1$.** We sincerely apologize for the confusion caused by the notations. In the VFS statistic, **$m_0$ and $m_1$ denote the numbers of class 0 and class 1 samples used by the kernel function**, respectively. We have clarified these definitions in Section 4.1 of the revised manuscript for better readability.\n\n(5.2) **Condition clarity.** According to Condition (C3),  we have $n_1^{\\gamma} \\gg \\log n + \\log p$, which implies that the quantity you mentioned as $C_1 = \\liminf_{n_1 \\to \\infty} \\frac{n_1^\\gamma}{\\log n} \\to \\infty$, and hence $1/C_1 = 0$. Therefore, we would like to kindly remark that **(C3) in the submitted version is sufficient** to ensure that the second term on the left-hand side of Theorem 1.1 tends to zero as $n_1\\rightarrow\\infty$.\n\n(5.3) **Class Imbalance in Datasets.** The real-world datasets **include both balanced and imbalanced ones**, with case ratios of 45.36% (Activity), 49.16% (Gina), 37.45% (RNA-Seq), 0.48% (p53), and 5.57% (newly added credit dataset). As shown in Table 3 in our submitted version and supported by the newly added experiments, VFS consistently demonstrates robust performance across a wide range of case ratios.\n\n**References**\n\n[1] Vepakomma, P., Gupta, O., Swedish, T., & Raskar, R. (2018). Split learning for health: Distributed deep learning without sharing raw patient data. arxiv preprint arxiv:1812.00564.\n\n[2] McMahan, B., Moore, E., Ramage, D., Hampson, S., & y Arcas, B. A. (2017). Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics (pp. 1273-1282). PMLR."}, {"review_id": "o5RjoYqpF9", "reviewer": "Reviewer_xnKo", "summary": "The authors address the problem of high computational, communication, and encryption costs in Vertical Federated Learning (VFL) under ultra-high dimensional feature spaces and imbalanced (rare event) response variables. Existing federated feature selection methods struggle to scale in these settings due to the enormous feature dimension and the resource-intensive secure multi-party computations involved. To solve this, the authors propose the Vertical Federated Feature Screening (VFS) algorithm, a two-stage coarse to fine screening procedure that first eliminates irrelevant feature groups and then performs refined screening on individual features, all under secure encryption. They combined subsampling technique to address class imbalance with a model-free statistic compatible with encrypted data, they significantly reduce the resource burden for downstream federated learning tasks. The authors achieve theoretical guarantees of consistency and asymptotic properties for their method, and demonstrate through both simulations and real-world experiments that VFS can maintain high predictive performance while reducing encryption and computation costs by an order of magnitude compared to classical approaches.", "questions": "1. In Stage 2, the fine screening depends on the features retained from Stage 1. Could cascading errors accumulate if Stage 1 removes too many features? Did the authors investigate how robust the method is to Stage 1 mistakes? Since the group-level screening uses a hypothesis-testing framework, how are the type I and type II errors controlled across potentially many groups, and is there a multiple testing correction strategy built in? \n\n2.  Would it be feasible to skip Stage 2 altogether if the initial grouping was fine-grained enough? In other words, is there a principled trade-off between group granularity and the necessity of a second stage?", "limitations": "1. One limitation of the work is that authors have not tested their method in a VFL setup with multiple clients involved with a diversed amount of features and classes (multi-class). Therefore, it is difficult to say at this moment the applicability of the algorithm in a real scenario.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The authors present a strongly motivated problem. In practice, datasets with very high feature dimensionality and severe class imbalance can create serious challenges for federated learning, especially when participating devices have limited computational resources.\n\n2. The application of U-statistics to define a model-free VFS statistic is an elegant choice, providing an asymptotically consistent way to measure associations between features and rare-event labels without relying on restrictive parametric assumptions.\n\n3. The paper offers a solid theoretical analysis, rigorously establishing convergence, screening consistency, and bounded model size properties of the proposed method.\n\n\nWeakness:\n\n1. The authors did not implement a full vertical federated learning algorithm (such as SplitNN or SecureBoost) to directly verify how their proposed feature screening would impact downstream VFL model training.\n\n2. The paper does not adequately investigate how robust the feature selection is when coupled with an actual VFL training pipeline, leaving open questions about its real-world effectiveness.\n\n3. The paper is not studied the scalability of their algorithm across a VFL framework. With N number of clients. \n\n4. The work does not analyze potential privacy leakage risks from the intermediate screening statistics themselves, which might expose sensitive label information to adversaries under certain attack models.", "was_revised": true, "final_justification": "The authors gave justifications for my queries, and the answers are satisfactory. \n1. The authors empirically tested their VFS approach on a VFL framework. \n2. Error analysis and the necessity of stage 2 in VFS are addressed. \n\nTherefore, I think it is a good paper. Consequently, I raised the score.", "rebuttal": "We sincerely thank Reviewer xnKo for the thoughtful and detailed comments. We truly appreciate your recognition of the practical challenges addressed by our work, especially the computational burden in VFL under ultra-high dimensional and imbalanced settings. Your acknowledgment of the novelty of using model-free U-statistics for rare-event feature screening and the rigor of our theoretical analysis is highly appreciated. We have carefully addressed your suggestions and revised the manuscript accordingly, as detailed below.\n\n**1. Comparison with Full VFL Algorithms.** Thank you for bringing us these excellent works. We would like to kindly emphasize that **the examples SplitNN (Vepakomma et al., 2018) and SecureBoost (Cheng et al., 2021) considered are low dimension cases**, and the maximum of feature dimension in these two works is $p=32\n\\times 32 \\times 3 = 3{,}072$, which is much smaller than the corresponding sample size $n=60{,}000$. This is because there could be prohibitive computational cost when $p \\gg n$ (Ceballos et al., 2020) and also leads to non-robust estimation (Li et al., 2017). More specifically, experiments show that SplitNN is infeasible for training on a machine with 16GB GPU and 62GB RAM when $p > 200{,}000$, even with $n=10{,}000$. Therefore, from both empirical and theoretical perspectives, **feature reduction is considered necessary in high-dimensional VFL settings** (Cassará et al., 2022). This is why we **only consider feature selection methods as baselines** to deal with high-dimensional features in the original submission. However, we fully agree that comparisons with full VFL algorithms can more effectively demonstrate the efficacy of VFS.  Following your advice, we have compared with full VFL method (i.e., SplitNN) in an actual VFL training pipeline (described in the next point) based on the dataset with $n=10{,}000$ and $p=189{,}236$. The results below further support the scalability and efficiency of our method, which are similar to the results in the submitted version. The results below demonstrate that **incorporating VFS reduces the computational cost to less than 10% of that incurred by using SplitNN alone, while simultaneously improving the prediction performance**. See also our reply to Reviewer ekrS. In addition, we have carefully compared and cited both of the works you have brought to us in the revision.\n\n| #   Features |  AUC  | Screening Time | Modeling Time | Total Time |\n| :----------: | :---: | :------------: | :-----------: | :--------: |\n|     100      | 0.937 |     29.098     |    100.313    |  129.411   |\n|     500      | 0.941 |     30.876     |    125.343    |  156.219   |\n|     1000     | 0.936 |     31.699     |    151.941    |  183.640   |\n|     All      | 0.901 |       -        |   1911.829    |  1911.829  |\n\n**2. Robustness Demonstration in Actual VFL Training Pipeline.** Thank you for this insightful suggestion. This work is directly motivated by **a real-world VFL application** in collaboration with a leading third-party payment platform in China. Due to privacy constraints, the data cannot be publicly released and was not included in the initial submission. Following your advice, we now report results based on the dataset in a secure sandbox containing 189,236 features for 10,000 merchants, covering key feature groups such as merchant attributes, transaction activity, revenue, and business growth. See also our reply to Reviewer ekrS for more background description of the dataset. We adopt SplitNN as the downstream model per your advice and present the results above. **This application-based evaluation further demonstrates the practical relevance and robustness of VFS**.\n\n**3. Scalability to Multiple Clients and Classes.** As discussed in Remark 1 and Appendix C, **our method can be readily theoretically extended to the multi-party setting**. Empirically, we have added simulation results in Appendix E of this revision. Specifically, we investigate the performances of the proposed method in scenarios with 1, 2, and 5 passive parties. The resulting PSRs are $0.965\\pm0.042$, $0.959\\pm0.037$, and $0.961\\pm0.046$, respectively, indicating no statistically significant difference. In the newly added real-world case, merchant profile data and transaction data are naturally stored across different institutions, which are treated as separate clients. The results remain stable, further supporting the scalability of VFS to multiple clients. **For the scalability of multi-class scenario, the theoretical property has already been guaranteed by Corollary 1**. Empirically, we have now validated the correctness of our proposed theory by additional simulations in the revision. Taking Model 1 as an example, consider the three-class label variable. When the number of features is set to $p = 1000, 2000, 5000$, the resulting PSRs are $0.943 \\pm 0.039$, $0.926 \\pm 0.050$, and $0.923 \\pm 0.056$, respectively, which are very close to the results obtained in our binary classification simulations (see Table 6 in our submitted version). We have made this point clear in the revision.\n\n**4. Privacy Leakage from Intermediate Statistics.** First, we would like to kindly clarify that **the main focus of this work lies in the statistical and computational aspects of feature screening in VFL, rather than in the cryptographic guarantees of homomorphic encryption (HE)**. The privacy of intermediate statistics is protected by adopting appropriate HE schemes, and the proposed VFS framework is compatible with any suitable HE methods. The classical privacy-preserving properties of HE are established by Gentry’s foundational work (Gentry, 2009). While some studies have investigated the potential vulnerability of HE to inference attacks, more advanced variants of HE can offer enhanced protection (Falcetta and Roveri, 2022), and these methods remain fully compatible with VFS. Moreover, combining HE with differential privacy (DP) is a promising direction for strengthening privacy guarantees, as suggested in recent studies (Wang et al., 2023). We have clarified this point and added a corresponding discussion in the revised manuscript. Thank you again for this valuable suggestion.\n\n**5. Difference with Multiple Testing and Cascading Error Analysis.** Thank you for this insightful comment. Our focus differs from classical multiple testing, which typically controls the Type I error (e.g., Bonferroni correction uses $\\alpha/m$ to reduce false positives but at the cost of increased Type II error [Dunn, 1961; Dai et al., 2023]). In contrast, we fix $\\alpha$ regardless of the number of groups to maintain power. This means **we wish to decrease Type II error rather than control Type I error as in a classical multiple testing problem. We have already provide theoretical guarantees on the Type II error in Theorem 1(2) in the submitted version**. Moreover, **we can quantify the cascading error in Stage 1 directly by the result from Theorem 1**. Assume that each iteration involves $l$ groups and a total of $r$ iterations are conducted with $l r \\lesssim p$, then under similar conditions in Theorem 1(2), the probability that any relevant feature is mistakenly removed after Stage 1 satisfies\n$$\n\\\\mathrm{P}\\\\left\\\\{ \\\\mathcal{F} \\\\not\\\\subset \\\\hat{\\\\mathcal{F}} \\\\right\\\\} \\\\le O\\\\left( lr \\\\left[ \\\\exp\\\\left( - \\\\tilde{C}_1 n_1^{1 - 2\\\\gamma - 2\\\\kappa} \\\\right) + n \\\\exp\\\\left( - \\\\tilde{C}_2 n_1^\\\\gamma \\\\right) \\\\right] \\\\right) \\\\to 0 \\\\, ,\n$$\n**ensuring the error does not accumulate uncontrollably across stages**. We have made both of these points clear in the revision with a newly added remark.\n\n**6. Necessity of Stage 2 in VFS.** We would like to kindly remark that we have already pointed out in Theorem 2 in the submitted version that the statistic is asymptotically normally distributed for relatively large $p$. Specifically, as pointed out in Theorem 2,  the asymptotic normality holds only under the condition in line 217, which is satisfied only when each feature group in Stage 1 contains a large number of features (Empirically, this should be larger than 5 [Zhang and Zhu, 2024]). **Therefore, Stage 1 alone is not fine-grained enough, and the retained feature groups may still contain a substantial number of irrelevant features.**  \n\n**References**\n\n[1] Vepakomma, P., Gupta, O., Swedish, T., & Raskar, R. (2018). Split learning for health: Distributed deep learning without sharing raw patient data. arxiv preprint arxiv:1812.00564.\n\n[2] Cheng, K., Fan, T., Jin, Y., Liu, Y., Chen, T., Papadopoulos, D., & Yang, Q. (2021). Secureboost: A lossless federated learning framework. IEEE intelligent systems, 36(6), 87-98.\n\n[3] Ceballos, I., Sharma, V., Mugica, E., Singh, A., Roman, A., Vepakomma, P., & Raskar, R. (2020). Splitnn-driven vertical partitioning. arxiv preprint arxiv:2008.04137.\n\n[4] Li, J., Cheng, K., Wang, S., Morstatter, F., Trevino, R. P., Tang, J., & Liu, H. (2017). Feature selection: A data perspective. ACM computing surveys (CSUR), 50(6), 1-45.\n\n[5] Cassará, P., Gotta, A., & Valerio, L. (2022). Federated feature selection for cyber-physical systems of systems. IEEE Transactions on Vehicular Technology, 71(9), 9937-9950.\n\n[6] Gentry, C. (2009). A fully homomorphic encryption scheme. Stanford university.\n\n[7] Falcetta, A., & Roveri, M. (2022). Privacy-preserving deep learning with homomorphic encryption: An introduction. IEEE Computational Intelligence Magazine, 17(3), 14-25.\n\n[8] Wang, B., Chen, Y., Jiang, H., & Zhao, Z. (2023). Ppefl: Privacy-preserving edge federated learning with local differential privacy. IEEE Internet of Things Journal, 10(17), 15488-15500.\n\n[9] Dunn, O. J. (1961). Multiple comparisons among means. Journal of the American statistical association, 56(293), 52-64.\n\n[10] Dai, C., Lin, B., Xing, X., & Liu, J. S. (2023). False discovery rate control via data splitting. Journal of the American Statistical Association, 118(544), 2503-2520."}, {"review_id": "p5sqB6BHcR", "reviewer": "Reviewer_1Em4", "summary": "This paper introduces a feature selection method to help eliminate unnecessary features in vertical federated learning (VFL). The benefit is to reduce the communication overhead without sacrificing the accuracy of the trained model. It is possible because in ultra-high dimensional settings, only a small number of features truly contribute to the model prediction, and other features can be pruned. The idea is to first ask the passive participants to group the features they own and calculate a group-wise score for the server to remove those surely unimportant ones. Then, a similar process is done on individual features of those remaining groups. Theoretical studies and experiments on simulated and real-world datasets have been conducted to show the effectiveness of the proposed method.", "questions": "1. Could you clarify how to compute the statistic? It is unclear how to make use of Y_{enc} to measure the statistic for pruning.\n2. Could you elaborate on the choice of the real-world datasets used in the experiments? How are those features being partitioned and distributed across different parties. The reviewer is particularly concerned about whether this partitioning is natural or artificial. If it is artificial, whether it can be a representative of real-world VFL problems.", "limitations": "The authors discussed the limitations of existing methods. However, they did not discuss the potential impact of limitations of the method they propose.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths\n+ VFS supports secure VFL and extremely imbalanced datasets. These are practical considerations that are commonly overlooked in the VFL literature.\n+ The hierarchical pruning (i.e., group->individual) is sound. Encryption overhead is often a concern in practical deployment of FL systems. The reduction in such overheads can be of broad impact.\n+ Theoretical studies have been conducted to analyze the properties of VFS.\n\nWeaknesses\n- The paper writing could be improved. Technical details are insufficient. For instance, how Y_{enc} is being used by the clients to compute the encrypted statistic is not clearly articulated.\n- The choice of real-world datasets is not clearly justified. While those datasets are not simulated, it is unclear whether they are for VFL. As shown in prior work [1], VFL datasets are of different characteristics than HFL, and randomly distributing features to different \"clients\" can create an artificial dataset that does not represent VFL.\n\n[1] Wu, Zhaomin, Junyi Hou, and Bingsheng He. \"Vertibench: Advancing feature distribution diversity in vertical federated learning benchmarks.\" ICLR 2024.", "was_revised": true, "final_justification": "The authors addressed my concerns in the rebuttal. I believe that it is a valuable work on VFL. Hence, I maintain my score.", "rebuttal": "We sincerely thank Reviewer 1Em4 for the positive evaluation of our work and thoughtful comments. We are especially grateful for the recognition of the practical value of our method in supporting secure VFL under extreme class imbalance, the effectiveness of the hierarchical pruning strategy, and the theoretical rigor of our analysis. These aspects are indeed the core motivations behind our proposed VFS framework. We have carefully addressed the concerns as detailed in our point-by-point responses below.\n\n**1. Writing improvement and technical details for $\\mathbf{T}_k$  and $T_j$ calculation.**  Per your advice, we have revised the writing to ensure that all notations and algorithm details are clearly defined in the revision. Specifically, we have clearly defined the technical details for $\\mathbf{T}_k$ and $T_j$. In the submitted version (Lines 128–137), we **briefly described how to compute $\\mathbf{T}_k$ and $T_j$, but the specific form may vary depending on the choice of the screening statistic**. For example, in the case of the distance covariance used in our implementation, denote\n$$\n\\left[ Y\\_{i_1,i_2} \\right] = \n\\begin{cases}\n\\left[ 4  \\right] , & \\text{if } Y\\_{i_1} \\ne Y\\_{i_2} \\\\\\\\\n\\left[ -2 \\right] , & \\text{if } Y\\_{i_1} = Y\\_{i_2}\n\\end{cases}\n$$\nwhere $[c]$ denotes the ciphertext of $c$. It is worth noting that the same plaintext may correspond to different ciphertexts after encryption, and therefore the ciphertexts are no longer binary and cannot be easily decrypted. The encrypted statistic $[\\mathbf{T}_k]$ for the $k$-th feature group is computed as \n$[\\mathbf{T}\\_k] = {(n\\_0 n\\_1)^{-1}} \\\\sum\\_{i\\_1}\\\\sum\\_{i\\_2} [Y\\_{i\\_1,i\\_2}] \\\\| X\\_{i\\_1,k} - X\\_{i\\_2,k}\\\\|\\_2$, where $X\\_{i\\_1,k}$ denotes the $k$-th feature group of the $i\\_1$-th user. \n\n**2. Justification of Real-World Datasets.** Thank you for the insightful comment. We have carefully studied the concerns raised by Wu et al. (2024), cited the paper in the revision and agree that preserving real-world VFL characteristics is critical. In the revision, **we have included a new empirical study based on a real VFL scenario from a third-party payment company, which is also the empirical motivation of this work**. The features are naturally vertically partitioned, covering 189,236 attributes for 10,000 merchant (financial product users). This dataset  is analyzed in an actual VFL pipeline in a secure sandbox environment. See also our reply to Reviewer ekrS. Results below show that VFS substantially reduces computational cost to less than 10% of the runtime for baseline methods. Thank you for your suggestion. We believe that the inclusion of this real-world application significantly strengthens the practical relevance of our work.\n\n| #   Features |  AUC  | Screening Time | Modeling Time | Total Time |\n| :----------: | :---: | :------------: | :-----------: | :--------: |\n|     100      | 0.937 |     29.098     |    100.313    |  129.411   |\n|     500      | 0.941 |     30.876     |    125.343    |  156.219   |\n|     1000     | 0.936 |     31.699     |    151.941    |  183.640   |\n|     All      | 0.901 |       -        |   1911.829    |  1911.829  |\n\n**3. Impact and Limitation of VFS.** The VFS framework tackles a key bottleneck in vertical federated learning: performing effective feature screening under the dual challenges of ultrahigh-dimensional data and severe class imbalance, while preserving data privacy. It offers broad adaptability to different federated architectures and screening criteria, and substantially **reduces the computational and communication burden** in downstream tasks, as **supported by both theoretical analysis and empirical validation**. Nevertheless, we recognize certain limitations of the proposed method. First, the overall efficiency of VFS depends on the specific statistic employed for screening. Second, due to its marginal nature, VFS may retain features that are correlated, which could be undesirable in scenarios where independence is required for model interpretability or classical statistical inference. These considerations have been clearly discussed in the revised version.\n\n**4. Elaboration on the Partitioning of Real-World Datasets.** Thank you for raising this important concern. We have carefully studied, cited, and acknowledged the insight of Wu et al. (2024), which emphasizes the unique characteristics of VFL datasets. In real-world VFL applications, features are often naturally partitioned due to their storage across different clients or servers. **In the newly added real-world case study with a third-party payment company, data are naturally stored in different clients and divided into different groups**. Typically, in this dataset, (1) merchant profile attributes and (2) transaction-level behavioral features are stored in separate clients. And they include different groups of features respectively. Together with the active party, this setup constitutes a genuine VFL scenario **without artificial partitioning**. We have clarified this point in the revised manuscript."}]

Author rebuttal:

Meta-review: This paper presents an approach for making vertical federated learning more efficient via a two-stage approach involving a preprocessing step ("screening"). The reviewers praised the applicability of the method and its theoretical analysis but raised strong concerns about the scope and real-world applicability of the empirical results. In the discussion they indicated that these concerns were addressed, but maintained borderline ratings, with remaining concerns devoted mainly to an even broader applicability of the method. Therefore, I tentatively recommend accepting the paper.
Decision: Accept (poster)