Improving Federated Learning Face Recognition via Privacy-Agnostic Clusters

1 INTRODUCTION . Face recognition technique offers great benefits when used in right context , such as public safety , personal security and convenience . However , misuse of this technique is a concern as it involves unique and irrevocable biometric data . The rapid commercial applications based on face recognition and facial analysis techniques have stimulated a global conversation on AI ethics , and have resulted in various actors from different countries issuing governance initiatives and guidelines . EU ’ s General Data Protection Regulation ( GDPR ) ( Voigt & Von dem Bussche , 2017 ) , California Consumer Privacy Act ( CCP ) and Illinois Personal Information Protection Act ( IPI ) enforces data protection “ by design and by default ” in the development of any new framework . On the nationwide “ 315 show ” of year 2021 , the China Central Television ( CCTV ) called out several well-known brands for illegal face collection without explicit user consent . As researchers , it is also our duty to prevent the leakage of sensitive information contained in public datasets widely used by the research community . Therefore , faces in ImageNet ( Deng et al. , 2009 ) were recently all obfuscated ( Yang et al. , 2021 ) and a large face dataset called MS-Celeb-1M ( Guo et al. , 2016 ) was pulled of the Internet . In the wake of growing social consensus on data privacy , the field of face recognition calls for a fundamental redesign about model training while preserving privacy . A potential solution is the paradigm called Federated Learning ( FL ) ( McMahan et al. , 2017a ) . Given C clients with local datasets { D1 , D2 , · · · , DC } as shown in Fig . 1a , FL decentralizes the training process by combining local models fine-tuned on each client ’ s private data and thus hinders privacy breaches . Typical examples of these clients include personal devices containing photo collections of a few family members , or open-world scenarios such as tourist attractions visited by tens of thousands of people . In most circumstances , we can safely assume very few classes would co-exist in two or more clients . Despite the numerous FL-based applications in various domains ( Kairouz et al. , 2019 ) ranging from health to NLP , there are very little progress ( Aggarwal et al. , 2021 ; Bai et al. , 2021 ; Liu et al. , 2021 ) in training face recognition models with FL schemes . Unlike other tasks , parameters of the last classifier for a face recognition model are crucial for recognition performance but strongly associated with privacy . These parameters can be regarded as mean embeddings of identities ( Wang et al. , 2018 ; Meng et al. , 2021b ; Shen et al. , 2020 ) ( also called as class centers ) , from where individual privacy could be spied out as studied by plenty of works ( Kumar Jindal et al. , 2018 ; Boddeti , 2018 ; Mai et al. , 2020 ; Dusmanu et al. , 2021 ) . That prevents the FL approach from broadcasting the whole model among clients and the central server , and consequently leads to conflicts in the aggregation of local updates . As depicted in the global feature distribution of Fig . 1a , both clients try to spread out their own classes in the same area ( pointed by the arrow ) of the normalized feature space . Thus , the training loss could oscillate to achieve consensus given the sub-optimal solutions from multiple clients . On the other hand , a large batch with sufficient negative classes is necessary to learn a discriminative embedding space for advanced face recognition algorithms . During the conventional FL updates , each class is only aware of local negative classes while those from other clients are untouchable . This further limits performances of FL approaches in face recognition . The privacy-utility paradox motivates us to introduce PrivacyFace , a framework improves federated learning face recognition by broadcasting sanitized information of local class globally . In the framework , a novel algorithm called Differentially Private Local Clustering ( DPLC ) first generates privacy-agnostic clusters of class centers while any specific individual in the cluster can not be learned , irrespective of attacker ’ s prior knowledge , information source and other holds . Recall that the privacy cost of a differential privacy scheme is propotional to the l2-sensitivity while inversely propotional to the query number . Our DPLC reaches a low l2-sensitivity by restricting the cluster size as well as covering sufficient class centers . In addition , the number of necessary centers to communicate in DPLC is irrelevant to the number of classes in the training data . These characteristics jointly equip DPLC with much smaller privacy cost than the naive alternative , which sanitizes each class center individually by Gaussian noise . In our experiments , DPLC ’ s privacy cost is only 1.7e-7 of that of the naive approach . That persuasively reveals the high security level of our approach . The second part of PrivacyFace is the consensus-aware face recognition loss . Following principles of Federated Averaging ( FedAvg ) ( McMahan et al. , 2017a ) , a server iteratively gathers feature extractors and privacy-agnostic clusters from clients , averages parameters of feature extractors and distributes them to clients . Accordingly , the consesus-aware loss notifies each client not to embed samples in the inappropriate zone ( differential private clusters marked by DP ) of the feature space during the local optimization , as shown in Fig . 1b . This process aids each client to train more discriminative features as well as align all consensuses . Compared to the conventional approach , our PrivacyFace boosts performances by +9.63 % and +10.26 % for TAR @ FAR=1e-4 on IJB-B and IJBC respectively with only single-digit privacy cost . Moreover , the additional computational cost as well as communication cost are negligible ( e.g. , the extra clusters to broadcast only occupy 16K storage while the backbone already takes 212M ) . In a word , PrivacyFace is an efficient algorithm which improves conventional federated learning face recognition by a large margin on performances , while requires little privacy cost as well as involves lightweight computational/communication overheads . 2 PRELIMINARIES . 2.1 DEEP FACE RECOGNITION . Most of early works in deep face recognition rely on metric-learning based loss , including contrastive loss ( Chopra et al. , 2005 ) , triplet loss ( Schroff et al. , 2015 ) and N-pair loss ( Sohn , 2016 ) . These methods are usually inefficient in training on large-scale datasets . One possible reason is that their embedding spaces at each iteration are constructed only by a positive sample and limited negative ones . Therefore , the main body of research ( Ranjan et al. , 2017 ; Liu et al. , 2017 ; Wang et al. , 2018 ; Deng et al. , 2019 ; Xu et al. , 2021 ; Meng et al. , 2021a ; c ) has focused on devising more effective classification-based loss and achieved leading performances on a number of benchmarks . Suppose that we are given a training dataset D with N face samples { xi , yi } Ni=1 of n identities , where each xi is a face image and yi ∈ { 1 , · · · , n } denotes its associated class label . Considering a feature extractor φ generating the embedding fi = φ ( xi ) and a classifier layer with weights W = [ w1 , · · · , wn ] , a series of face recognition losses can be summarized as Lcls ( φ , W ) = − N∑ i=1 log eu ( wyi , fi ) eu ( wyi , fi ) + ∑n j=1 , j 6=yi e v ( wj , fi ) . ( 1 ) The specific choices of the similarity functions u ( w , f ) , v ( w , f ) yield different variants , e.g . : CosFace ( Wang et al. , 2018 ) : u ( w , f ) = s · ( cos θ −m ) , v ( w , f ) = s · cos θ , ( 2 ) ArcFace ( Deng et al. , 2019 ) : u ( w , f ) = s · cos ( θ −m ) , v ( w , f ) = s · cos θ , ( 3 ) where s , m are hyper-parameters and θ = arccos ( wTf/‖wTf‖ ) is the angle between w and f . 2.2 DIFFERENTIAL PRIVACY . Differential Privacy ( DP ) ( Dwork et al. , 2006 ) is a well-established framework under which very little about any specific individual can be learned in the process irrespective of the attacker ’ s prior knowledge , information source and other holds ( Feldman et al. , 2017 ; Nissim et al. , 2016 ; Stemmer & Kaplan , 2018 ) . We state the relevant definitions and theories ( Dwork & Roth , 2014 ) below . Definition 1 ( Differential Privacy ) . A randomized algorithm M : X → Y is ( , δ ) -DP if for every pair of neighboring datasets X , X ′ ∈ X ( i.e. , X and X ′ differ in one row ) , and every possible output T ∈ Y the following equality holds : Pr [ M ( X ) ∈ T ] ≤ e Pr [ M ( X ′ ) ∈ T ] + δ . Here , δ ≥ 0 are privacy loss parameters , which we consider a privacy guarantee meaningful if δ = o ( 1n ) , where n is the size of dataset . Definition 2 ( l2-sensitivity ) . The l2-sensitivity of a function f : X → Rd is ∆2 ( f ) = max X , X′∈X X , X′are neighbors ‖f ( X ) − f ( X ′ ) ‖2 . ( 4 ) Definition 3 ( Gaussian Mechanism ) . Suppose a function f : X → Rd and Id is the ddimensional identity matrix . The mechanism M ( X ) = f ( X ) + N ( 0 , σ2Id ) is ( , δ ) -DP if σ ≥ ∆2 ( f ) √ 2 ln ( 1.25δ ) . Definition 4 ( Composition of Differentially Private Algorithms ( Dwork & Roth , 2014 ; Dwork & Lei , 2009 ) ) . Suppose M = ( M1 , M2 , · · · , Mk ) is a sequence of algorithms , where Mi is ( i , δi ) differentially private , and the Mi ’ s are potentially chosen sequentially and adaptively . Then M is ( ∑k i=1 , ∑k i=1 δk ) -differentially private . 3 PRIVACYFACE . This section details the PrivacyFace framework as illustrated in Fig . 2 . At its core is a novel clustering algorithm that extracts non-sensitive yet informative knowledge about the local class distribution ( Fig . 2a ) . After the central server broadcasting local DP-guaranteed outputs ( Fig . 2b ) , each client optimizes over a consensus-aware objective that takes into account both the local data and the privacy-agnostic clusters to learn a discriminative embedding space ( Fig . 2c ) for face recognition . p local class centers p local class centers p local class enters 3.1 DIFFERENTIALLY PRIVATE LOCAL CLUSTERING . Our target is to improve performances of federated learning face recognition by communicating auxiliary and privacy-agnostic information among clients . To distill useful information from each client and resolve the privacy-utility paradox , we propose a specialized algorithm called Differential Private Local Clustering ( DPLC ) with rigorous theoretical guarantees . Problem Design . To facilitate the discussion , let us first introduce the margin parameter ρ ∈ [ 0 , π ] , which defines the boundary of a cluster centered at p by saying that any w in the cluster satisfies1 arccos ( wTp ) ≤ ρ . We design the local clustering problem with two principles . First , the clustering results are expected to carry information only about population rather than individual from local dataset . Supported by Theorem 2 proved later , this principle ensures PrivacyFace to gain insight about the underlying distribution , while preserving the privacy of individual record . Second , the spaces confined by the clusters should be tight because they represent the inappropriate zones to escape from in other clients ’ perspectives . A cluster with large margin ρ would occupy a huge proportion of the sphere , leaving limited room for other embeddings ( e.g. , half of sphere would be occupied if ρ = π/2 ) . More specifically , we quantify the occupancy ratio by the following theorem : Theorem 1 . Assume there is a unit and d-dimensional sphere Sd and an embedding point p ∈ Sd . Embeddings with angles less than ρ to p ( i.e. , { f : arccos ( fTp ) ≤ ρ } ) occupy 12Isin2 ( ρ ) ( d−1 2 , 1 2 ) of the surface area of Sd . Here Ix ( a , b ) is the regularized incomplete beta function . Proved in Sec . A.1.3 of the appendix , Theorem 1 indicates that the occupancy ratio increases monotonically with respect to the margin ρ given a feature dimension d. Fig . 3 plots a series of functions between cluster margin ρ and occupancy ratio under different dimensions . Take the rightmost curve corresponding to a typical setting d = 512 in face recognition as an example . The occupancy ratios are 0.055 , 5 · 10−5 , 4 · 10−10 when ρ = 1.5 , 1.4 , 1.3 respectively . In another word , the whole feature space would be fully occupied if we sample 20 , 000 clusters over the original face classes when ρ = 1.4 . It is obvious that the more space left for optimization , the higher likelihood to achieve better performance . Therefore , we seek for a margin-constrained clustering of local data with fixed ρ ≤ 1.4 when d = 512 for better privacy-utility trade-offs . A counter example to the latter principle is the classical k-means algorithm , which have been extended to differentially private versions in a number of works ( Nissim et al. , 2016 ; Feldman et al. , 2017 ; Stemmer & Kaplan , 2018 ) . However , k-means naturally has no control about the scopes of the generated clusters , not to mention the difficult hyper-parameter k to pick . That motivates us to propose a specified clustering algorithm for our task . 1We adopt cosine similarity by assuming all vectors lie on a unit sphere in face recognition . An Approximated Solution . Given the margin ρ , the problem of finding clusters to cover all class centers is a generalization of the well-studied 2D disk partial covering problem which is unfortunately NP-complete ( Xiao et al. , 2004 ) . We take a greedy approach by iteratively finding a cluster centered at p∗ with margin ρ to cover the most class centers wi and weeding out those covered before the next round . We cast the problem of identifying such a cluster at each iteration as : Definition 5 ( Spherical Cap Majority Covering Problem ) . Denote 1 ( · ) as an indicator function . Assuming a unit d-dimensional sphere Sd and class centers { w1 , w2 , · · · , wn } ∈ Sd , the target is to find a cluster center p∗ ∈ Sd where p∗ = arg maxp∈Sd ∑n i=1 1 ( arccos ( w T i p ) ≤ ρ ) . Finding the optimal p∗ is also NP-complete . To address this , we first sort out the densest area S in terms of the neighbor count for eachw . An efficient approximation of p∗ is then the average of class centers in the area , p = 1|S| ∑ i∈S wi . Lines 3-5 of Algorithm 1 summarize the detailed procedure , by which the approximate cluster centers pi can be quickly enumerated as depicted in Fig . 2a . In line 8 , the class centers would be further normalized in accordance with the DP mechanism . Algorithm 1 : Differentially Private Local Clustering ( DPLC ) Data : Class center embeddings W = { w1 , w2 , · · · , wn } encoded in the classifier layer . Parameters : Margin ρ ; Minimum cluster size T ; Maximum # queries Q ; Privacy budget , δ . Result : Privacy-agnostic cluster centers p̂ ’ s . Let I = { 1 , 2 , · · · , n } and calculate θi , j = arccos ( wTi wj ) for i , j ∈ { 1 , 2 , · · · , n } ; 1 for q = 1 , 2 , · · ·Q do2 For each { wi , i ∈ I } , find the indexes of its neighbors by Si = { j : θi , j ≤ ρ , j ∈ I } ; 3 Find a set with the most elements , i.e. , S = arg max |Si| ; 4 p← 1|S| ∑ i∈S wi ; // approximate solution for problem 55 if |S| ≥ T then6 Sample a v from distribution N ( 0 , σ2Id ) where σ = 2|S| √ ( 1− cos ( 2ρ ) ) ln ( 1.25δ ) ; 7 p̂q ← p+v‖p+v‖ , I ← I/ { arccos ( w T i p ‖p‖2 ) ≤ ρ , i ∈ I } ; 8 else9 Break out of the loop ; 10 Differential Privacy Endorsement . Although the centers p reveal the population-level property of the local training set , they are still outcomes of a deterministic algorithm , thereby vulnerable to adversary attack during the FL updating . We prove below that p can be perturbed to achieve DP : Theorem 2 . Define a function p , f ( S ) = 1|S| ∑ i∈S wi . Then the Gaussian Mechanism p̂ , M ( S ) = f ( S ) +N ( 0 , σ2Id ) is ( , δ ) -DP if σ ≥ 2|S|· √ ( 1− cos ( 2ρ ) ) ln ( 1.25δ ) . Proof . Let S be the set of indexes of class centers which have cosine similarities larger than cos ρ with respect to a centerwo . Assuming that S , S ′ are neighbors differed atw andw′ , where vectors are normalized , i.e. , ‖w‖ = ‖w′‖ = ‖wo‖ = 1 , then we have wTo w ≥ cos ρ and wTo w′ ≥ cos ρ. Lemma 2 in appendix states that w Tw′ ‖w‖2‖w′‖2 ≥ cos ( arccos wTo w ‖wo‖2‖w‖2 + arccos wTo w ′ ‖wo‖2‖w′‖2 ) for all { w , w′ , wo } . Therefore , the lower bound of wTw′ is wTw′ = wTw′ ‖w‖2‖w′‖2 ≥ cos ( arccos w T o w ‖wo‖2‖w‖2 + arccos wTo w ′ ‖wo‖2‖w′‖2 ) ≥ cos ( ρ+ ρ ) = cos ( 2ρ ) Following this inequality , the l2-sensitivity of f ( S ) is 1|S| √ 2− 2 cos ( 2ρ ) as ‖f ( S ) − f ( S ′ ) ‖2 = 1 |S|‖w−w ′‖2 = 1|S| √ 2− 2wTw′ ≤ 1|S| √ 2− 2 cos ( 2ρ ) . By Definition 3 , we easily conclude that M ( S ) = 1|S| ∑ i∈S wi +N ( 0 , σ2Id ) is ( , δ ) -DP if σ ≥ 2 |S|· √ ( 1− cos ( 2ρ ) ) ln ( 1.25δ ) . Additional proofs can be found in Sec . A.1.2 of appendix . Theorem 2 outlines the setting of noise variance σ given the privacy budget , δ and the cluster margin ρ . Basically , the lower bound of σ is proportional to 1|S| . That implies when the size of cluster |S| is large enough , adding small noise is sufficient to achieve promising privacy level . We found setting minimum cluster size to T , min |S| = 512 yields empirically stable performance . It is worth emphasizing that the number of original classes n does not directly influence the introduced noise σ . This advantage ensures that DPLC maintains a high utility when dealing with large-scale face recognition problems . Full algorithm . Algorithm 1 presents the full DPLC and we highlight several key properties below . Properties 1 , 2 state that DPLC is efficient and differentially private in theory . Besides the efficency and privacy , another major concern of the DP algorithm is the utility . Specifically , as p̂ is a combination of the source cluster center p and Gaussian noise v , we should prevent the noise v to have overwhelming effects to the output , which is further verified by our Property 3 and Property 4 . Property 1 . The complexity of the DPLC algorithm is O ( Q · n2 ) . Proof . The computational complexity of calculating θi , j is O ( n2 ) . In each loop , we also compare the value θi , j , ∀i , j ∈ S with ρ , whose worst time complexity is O ( n2 ) . Because remaining steps are of linear complexity , the total complexity is O ( n2 +Q · n2 ) = O ( Q · n2 ) . Property 2 . DPLC is a ( Q · , Q · δ ) -differentially private algorithm . Proof . The mechanism f ( S ) + N ( 0 , σ2Id ) is ( , δ ) -differentially private according to Lemma 2 . As DPLC queries at most Q results , it ’ s easy to conclude that our DPLC algorithm is ( Q · , Q · δ ) differentially private based on Definition 4 . Property 3 . Denote Φ as the cumulative distribution function of a standard Gaussian distribution . For vector magnitudes of p , v , we have ‖p‖2 ∈ ( cos ρ , 1 ] and P ( ‖v‖2 ≤ r ) ' Φ ( r 2 σ2 √ 2 ( d−1 ) − d−12 ) . Proof . For p , the upper bound is 1 as ‖p‖2 = ‖ 1|S| ∑ i∈S wi‖2 ≤ 1 |S| ∑ i∈S ‖wi‖2 = 1 . As S denotes the set of indexes of class centers which has cosine similarities larger than cos ρ with a center wo , the lower bound of ‖p‖2 can be calculated by ‖p‖2 = ‖p‖2‖wo‖2 ≥ pTwo = ( 1 |S| ∑ i∈S wi ) Two = 1 |S| ∑ i∈S wTi wo ≥ 1 |S| ( 1 + ( |S| − 1 ) cos ρ ) > cos ρ . Therefore , we have ‖p‖2 ∈ ( cos ρ , 1 ] . For v ∼ N ( 0 , σ2Id ) , let v = [ v1 , v2 , · · · , vd ] , we have viσ following the standard normal distribution for all i . Thus , the sum of square of viσ follows a χ 2-distribution with d− 1 degrees of freedom , i.e. , 1 σ2 ‖v‖ 2 2 = ∑d i=1 ( vi σ ) 2 ∼ χ2d−1 . The mean and variance of ‖v‖22 are E ( ‖v‖22 ) = σ2 ( d − 1 ) and V ar ( ‖v‖22 ) = 2σ4 ( d− 1 ) , based on properties of the χ2-distribution . By the central limit theorem , ‖v‖22 converges toN ( σ2 ( d− 1 ) , 2σ4 ( d− 1 ) ) when d is large enough . In practice , for d ≥ 50 , ‖v‖22 is sufficiently close to N ( σ2 ( d− 1 ) , 2σ4 ( d− 1 ) ) ( Box et al. , 1978 ) . Therefore , we have P ( ‖v‖2 ≤ r ) = P ( ‖v‖22 − σ2 ( d− 1 ) √ 2σ4 ( d− 1 ) ≤ r 2 − σ2 ( d− 1 ) √ 2σ4 ( d− 1 ) ) ' Φ ( r 2 σ2 √ 2 ( d− 1 ) − √ d− 1 2 ) . Note that in our work , σ can be a small value with little privacy cost . Thus , with high probability , the norm of v is close to zero . Property 4 . If ‖p‖2 ≥ ‖v‖2 , the cosine similarity between p and p̂ is always greater than√ 1− ‖v‖22/|p‖22 . Proof . Denote magnitudes of p , v as lp , lv and cosine similarity between p and v as x ( i.e. , x = p·v ‖p‖‖v‖ ∈ [ −1 , 1 ] ) . Let a = lr lp < 1 and s be the cosine similarity between p + v and p , then s = ( p+v ) Tp ‖p+v‖2‖p‖2 = l2p+lplvx lp √ l2p+l 2 v+2lplvx = 1+ax√ 1+a2+2ax . The first order derivative of s with respect to x is dsdx = a2 ( a+x ) ( 1+a2+2ax ) 3 2 . It ’ s easy to conclude that s takes the smallest value when x = −a , which can be reached as a is always smaller than 1 . In the end , we have s ≥ √ 1− l2v/l2p . In DPLC , we reasonably require large local clusters S at line 6 while preserving privacy in terms of , δ . That constraint ensures the synthesized noise v to have a small magnitude ( property 3 ) , which thereby results in slight offsets of the sanitized vector p̂ to the source p ( property 4 ) . With these theoretical guarantees , DPLC can obtain good end-to-end utility with low privacy cost .

Summary: The main contribution of the paper is a slight modification of the standard federated learning (FL) framework for the purpose of improving face recognition performance. Since face recognition involves learning a deep neural network for feature embedding and the weight vectors for mapping the feature embedding to an identity, the standard FL framework (e.g. FedAvg) that updates only the feature embedding network does not suffice. Hence, the paper proposes to cluster the weight vectors of each participant and transmit the cluster centers (albeit with Gaussian noise to ensure differential privacy) to the other participants. Once the cluster centers of other participants are known, the local weight vectors can be learned to avoid overlap with those received clusters.