FED-$\chi^2$: Secure Federated Correlation Test

1 INTRODUCTION . Correlation test , as the name implies , is the process of examining the correlation between two random variables using observational data . It is a fundamental building block in a wide variety of real-world applications , including feature selection ( Zheng et al. , 2004 ) , cryptanalysis ( Nyberg , 2001 ) , causal graph discovery ( Spirtes et al. , 2000 ) , empirical finance ( Ledoit & Wolf , 2008 ; Kim & Ji , 2015 ) , medical studies ( Kassirer , 1983 ) and genomics ( Wilson et al. , 1999 ; Dudoit et al. , 2003 ) . Because the observational data used in correlation tests may contain sensitive information such as genomic information , centralizing the data collection is risky . To address , we resort to a federated setting in which each client maintains its own data and communicates with a centralized server to calculate a function . Note that the communication should contain as little information as feasible . Otherwise , the server may be able to infer sensitive information from the communication transcript . In the present work , we study a representative correlation test , namely χ2-test , under the federated setting . There are two straightforward methods for conducting χ2-test in such a context . First , clients can upload their raw data to the centralized server and delegate the test to it . While this method is effective in terms of communication , it entirely exposes the clients ’ private information . Second , clients may run secure multiparty computation ( MPC ) under the server ’ s coordination . Thus , clients can jointly run χ2-test without disclosing their data to the server . However , general-purpose MPC imposes significant computation and communication overhead , which is typically intolerable in a federated setting with computationally limited clients , e.g. , mobile devices . To address the dilemma , we present a federated protocol optimized for χ2-test that is computationally and communicationally efficient and discloses limited information to the server . We begin by recasting χ2-test as a second frequency moment estimation problem . To approximate the second frequency moment in a federated setting , each client encodes its raw data into a low-dimensional vector via stable random projection ( Indyk , 2006 ; Vempala , 2005 ; Li , 2008 ) . Such encodings can be aggregated with only summation , allowing clients to leverage secure aggregation ( Bonawitz et al. , 2017 ; Bell et al. , 2020 ) to aggregate the encodings and the server to decode them to approximate the second frequency moment . Because secure aggregation conceals each client ’ s individual update within the aggregated global update , the server learns only limited information about the clients ’ data . Our evaluation on four synthetic datasets and 16 real-world datasets shows that FED-χ2 can replace centralized χ2-test with good accuracy and low computation overhead . Additionally , we analyze FED-χ2 in three real-world use cases : feature selection , cryptanalysis , and online false discovery rate control . The results show that FED-χ2 can achieve comparable performance with centralized χ2-test and can withstand up to 20 % of clients dropping out with minor influence on the accuracy . In summary , we make the following contributions : We propose FED-χ2 , the first secure federated χ2-test protocol . FED-χ2 is computation- and communication-efficient and leaks much less information than trivially deploying secure aggregation . FED-χ2 decomposes χ2-test into frequency moments estimation that can easily be encoded/decoded using stable projection and secure aggregation techniques . We give formal security proof and utility analysis of FED-χ2 . We evaluate FED-χ2 in real-world use cases , and the findings suggest that FED-χ2 can substitute centralized χ2-test with comparable accuracy , and FED-χ2 can tolerate up to 20 % of clients dropout with minor accuracy drop . 2 RELATED WORK . Bonawitz et al . ( 2017 ) proposed the well-celebrated secure aggregation protocol as a low-cost way to calculate linear functions in a federated setting . It has seen many variants and improvements since then . For instance , Truex et al . ( 2019 ) and Xu et al . ( 2019 ) employed advanced crypto tools for secure aggregation , such as threshold homomorphic encryption and functional encryption . So et al . ( 2021 ) proposed TURBOAGG , which combines secure sharing with erasure codes for better dropout tolerance . To improve communication efficiency , Bell et al . ( 2020 ) and Choi et al . ( 2020 ) replaced the complete graph in secure aggregation with either a sparse random graph or a low-degree graph . Secure aggregation is deployed in a variety of applications . Agarwal et al . ( 2018 ) added binomial noise to local gradients , resulting in both differential privacy and communication efficiency . Wang et al . ( 2020 ) replaced the binomial noise with discrete Gaussian noise , which is shown to exhibit better composability . Kairouz et al . ( 2021 ) proved that the sum of discrete Gaussian is close to discrete Gaussian , thus discarding the common random seed assumption from Wang et al . ( 2020 ) . The above three works all incorporate secure aggregation in their protocols to lower the noise scale required for differential privacy . Chen et al . ( 2020 ) added an extra public parameter to each client to force them to train in the same way , allowing for the detection of malicious clients during aggregation . Nevertheless , designing secure federated correlation tests , despite its importance in real-world scenarios , is not explored by existing research in this field . On the other end of the spectrum , Wang et al . ( 2021 ) proved that stable projection is differentially private if the projection matrix is secret . In our protocol , the projection matrix is public information ; hence FED-χ2 does not consider the differential privacy guarantee . 3 FEDERATED CORRELATION TEST WITH MINIMAL LEAKAGE . In this section , we elaborate on the design of FED-χ2 , a secure federated protocol for χ2-test . Sec . 3.1 first formalizes the problem , establishes the notation system , and introduces the threat model . In Sec . 3.2 , we recast χ2-test as a second frequency moment estimation problem in the federated setting , and consequently , we are able to leverage stable projection to encode each client ’ s local information ( Sec . 3.3 ) , and then aggregate them using secure aggregation ( Sec . 3.4 ) . Sec . 3.5 , 3.6 , and 3.7 present security proof , utility analysis , communication analysis , and computation analysis of FED-χ2 . 3.1 PROBLEM SETUP . We now formulate the problem of the federated correlation test and establish the notation system . We use [ n ] to denote { 1 , · · · , n } . We denote vectors with bold lower-case letters ( e.g. , a , b , c ) and matrices with bold upper-case letters ( e.g. , A , B , C ) . We consider a population of n clients C = { ci } i∈ [ n ] . Each client has one share of local data composed of the triplets Di = { ( x , y , v ( i ) xy ) } , x ∈ X , y ∈ Y , v ( i ) xy ∈ { −M , · · · , M } , where x and y are categories of the contingency table , v ( i ) xy is the observed counting of the categories x and y in the local contingency table of the ith client , |X | = mx and |Y| = my are finite domains , and M is the maximum value |v ( i ) xy | can be . The global dataset is defined as D = { ( x , y , vxy ) : vxy =∑ i∈ [ n ] v ( i ) xy } . We focus on federated χ2-test and the data in contingency table is discrete . For the ease of presentation , we define the marginal statistics vx = ∑ y∈ [ |Y| ] vxy , vy = ∑ x∈ [ |X | ] vxy , and v = ∑ x∈ [ |X | ] , y∈ [ |Y| ] vxy . Besides , we define v̄xy = vx×vy v , denoting the expectation of vxy if x and y are uncorrelated . We define m = mxmy and use an indexing function I : [ mx ] × [ my ] → [ m ] to obtain a uniform indexing given the indexing of each variable . A centralized server S calculates the statistics for χ2-test sχ2 ( D ) = ∑ x∈ [ |X | ] , y∈ [ |Y| ] ( vxy−v̄xy ) 2 v̄xy on the global dataset to decide whether X and Y are correlated without collecting the raw data from clients . Overall , using MPC to conduct secure correlation tests in a federated scenario is highly expensive and impractical ( Boyle et al. , 2015 ; Damgård et al. , 2012 ) . Hence , in the present work , we trade off accuracy for efficiency , as long as the estimation error is small with a high probability . Formally , if FED-χ2 outputs ŝχ2 , whose corresponding standard centralized χ2-test output is sχ2 , the following accuracy requirement should be satisfied with small and δ. P [ ( 1− ) sχ2 ≤ ŝχ2 ≤ ( 1 + ) sχ2 ] ≥ 1− δ Threat Model . We assume that the centralized server S is honest but curious . It honestly follows the protocol due to regulatory or reputational pressure but is curious to discover extra private information from clients ’ legitimate updates for profit or surveillance purposes . As a result , client updates should contain as little sensitive information as feasible . We want to emphasize that , while the server may explore the privacy of clients , the server will honestly follow the protocol due to regulation or reputational pressure . The server won ’ t provide adversarial vectors to the clients . On the other hand , we assume honest clients . Specifically , we do not consider client-side adversarial attacks ( e.g. , data poisoning attacks ( Bagdasaryan et al. , 2020 ; Bhagoji et al. , 2019 ) ) . However , we allow a small portion of clients to drop out during the execution . Also , as we mentioned in Sec . 2 , we do not consider the differential privacy guarantee ; see further clarification in Appendix A . 3.2 FROM CORRELATION TEST TO FREQUENCY MOMENTS ESTIMATION . We first recast correlation test to a second frequency moments estimation problem as defined below . Given a set of key-value pairs S = { ki , vi } i∈ [ n ] , we re-organize it into a histogramH = { kj , vj =∑ ki=kj , i∈ [ n ] vi } , and estimate the αth frequency moments as Fα = ∑ j v α j . χ 2-test can thus be recast to a 2nd frequency moments estimation problem as follows : sχ2 ( D ) = ∑ x , y ( vxy − v̄xy ) 2 v̄xy = ∑ x , y ( vxy − v̄xy√ v̄xy ) 2 In federated setting , each client ci holds a local dataset Di = { ( x , y , v ( i ) xy ) } and computes a vector ui , where ui [ I ( x , y ) ] = v ( i ) xy−v̄xy/n√ v̄xy and ui has m elements . Thus , the challenge in federated χ2-test becomes calculating the following equation : sχ2 ( D ) = ∑ x , y ( vxy − v̄xy√ v̄xy ) 2 = || ∑ i∈ [ n ] ui||22

Summary: This paper discusses the problem of conducting correlation tests with sensitive data separately collected from $n$ clients, where centralizing data collection is risky but the available secure multiparty computation is costly in computation. The proposed test adjusts the classical chi-square test by first rewriting it as a second frequency moments estimation using $n$ vectors, each from a client. The proposed test prevent data recovering when performing the test by letting each client project its vector to a lower dimension. The higher the dimension is, the more accurate the test is, yet the higher the computation costs.