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 introduces a federated analytics technique for computing the Pearson correlation for two random variables. The method is based on repeated uses of secure aggregation, as well as stable random projections. The protocol is argued to be secure in a semi-honest security model.