Fine-Tuning Personalization in Federated Learning to Mitigate Adversarial Clients

Authors: youssef  allouah
abdellah  el
rachid  guerraoui
nirupam  gupta
rafael  pinot
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 19:10:21

Abstract: Federated learning (FL) is an appealing paradigm that allows a group of machines
(a.k.a. clients) to learn collectively while keeping their data local. However, due
to the heterogeneity between the clients’ data distributions, the model obtained
through the use of FL algorithms may perform poorly on some client’s data.
Personalization addresses this issue by enabling each client to have a different
model tailored to their own data while simultaneously benefiting from the other
clients’ data. We consider an FL setting where some clients can be adversarial, and
we derive conditions under which full collaboration fails. Specifically, we analyze
the generalization performance of an interpolated personalized FL framework in the
presence of adversarial clients, and we precisely characterize situations when full
collaboration performs strictly worse than fine-tuned personalization. Our analysis
determines how much we should scale down the level of collaboration, according
to data heterogeneity and the tolerable fraction of adversarial clients. We support
our findings with empirical results on mean estimation and binary classification
problems, considering synthetic and benchmark image classification datasets

Main paper: 
 introduction: 
 Federated learning (FL) is the de facto standard for a group of machines (also referred to as clients) to learn a common model on their collective data (Kairouz et al., 2021) . The benefit of this method is twofold: the clients (1) retain control over their local data (as they do not communicate them to a central server) and (2) share the computational workload during the learning procedure. Although FL is a promising learning paradigm, it also comes with its own technical challenges, the first of which is data heterogeneity. Indeed, each client holds only a portion of the common dataset, which is not necessarily a faithful representation of the entire population. This could lead to disagreements amongst clients on model updates, rendering the task of learning an accurate model in this context cumbersome. Nevertheless, when all machines correctly follow a prescribed algorithm, this problem can be solved using either a distributed implementation of the stochastic gradient descent (SGD) algorithm or variants of the federated averaging scheme (McMahan et al., 2017; Khaled et al., 2020; Karimireddy et al., 2020) . Another negative impact of data heterogeneity is the non-uniformity of model accuracy. Specifically, while an FL algorithm outputs a model that is accurate on average, the accuracy of this model on different clients' local data could vary significantly.Personalization. Personalized FL attempts to address the aforementioned shortcoming on nonuniform model accuracy by having clients collaborate to design individual models that generalize well over their local data distributions (Vanhaesebrouck et al., 2017; Sattler et al., 2020; Fallah et al., 2020; Hanzely et al., 2020) . This approach is highly relevant to many modern machine learning applications, as it tailors the training of each model to the needs of the respective client. To get a more precise view of this approach, consider a general supervised learning task with X denoting the input space and Y the output space. Personalized FL involves n clients which can communicate, through a central trusted machine (a.k.a. the server). The clients hold local datasets S 1 , . . . , S n , comprising m data points, drawn i.i.d. from their respective local data distributions D 1 , . . . , D n with support in Z := X × Y. Given a parameter set Θ ⊆ R d , we consider hypothesis functions of the form h θ : X → Y ′ , parameterized by θ ∈ Θ, and we denote by H the corresponding hypothesis class. We allow Y ′ ̸ = Y to encompass cases in which the model outputs logits/probits instead of classes. The objective of each client i is to find a model θ * i ∈ Θ minimizing its local risk function R i (θ) := E (x,y)∼Di [ℓ(h θ (x), y)] , ∀θ ∈ Θ,where ℓ : Y ′ ×Y → R + is a non-negative point-wise loss function measuring how well a hypothesis h θ fits the output y on an input x. The mapping (θ, x, y) → ℓ(h θ (x), y) is assumed to be differentiable, with respect to θ. To minimize the local risk above, each client i only has access to its local dataset S i yielding a local empirical loss L i (θ) := 1 m (x,y)∈Si ℓ(h θ (x), y), and to the information sent by other clients on their respective datasets. This optimization problem can be solved in several ways, and we have at out disposal a rich literature on personalized FL schemes (Mansour et al., 2020; Fallah et al., 2020; Marfoq et al., 2021) . We focus on a specific personalization scheme wherein each client aims to solve for the λ-interpolated empirical loss minimization problem:EQUATIONwhere L(θ) := 1 n n i=1 L i (θ), ∀θ ∈ Θ, and λ ∈ [0, 1]. When λ = 1, the problem reduces to the standard FL objective and corresponds to a full collaboration amongst the clients. The case of λ = 0 represents the absence of collaboration, i.e., each client minimizes its local empirical loss. We use the terminology fine-tuned personalization to refer to the objective of solving for (2) when λ ∈ (0, 1) is determined by optimizing the learning performance in retrospect.Robustness. Standard personalized FL approaches usually assume that all clients correctly follow a prescribed protocol, and hence do not consider the possibility of adversarial clients. Robustness to such clients, a.k.a "Byzantine robustness", refers to the design of FL schemes that yield accurate models even when some clients arbitrarily deviate from the algorithm and can send potentially corrupted information to other clients: This could result from poisonous data or code, or attacks from malicious players. Although Byzantine robustness has received significant attention in the FL community (Guerraoui et al., 2024) , it remains largely understudied in the context of personalized FL. In fact, prior work has shown that the presence of adversarial clients induces a fundamental optimization error that grows with the heterogeneity across clients' local data (Karimireddy et al., 2022; Allouah et al., 2024) . This optimization error is at odds with the improvement in generalization, which is offered by collaborating with other correct clients, to a point where collaboration is rendered vacuous. The main motivation for our work is to quantify the impact of adversarial clients on the relevance of collaboration for any correct client.Given the set of n clients, we aim to tolerate the presence of at most f < n/2 adversarial clients. Such clients can respond arbitrarily to the queries made by the server, e.g., a gradient computation on their local dataset. We also call these clients Byzantine adversaries. The identity of Byzantine adversaries is a priori unknown to the correct (i.e., non-adversarial) clients, otherwise, the problem is rendered trivial. As we explain above, when no client is adversarial, the objective of each client i is to minimize its own risk, defined in (1), by solving for the interpolated empirical loss minimization problem (2). However, in the presence of adversarial clients, a correct client cannot simply seek a solution to (2), as they can never have truthful access to the information about Byzantine adversaries' datasets. A more reasonable goal is to minimize an interpolation between their local and the correct clients' average loss functions. We formally define the robust-interpolated objective for client i as follows:EQUATIONwhere C represents the set of correct clients, L C (θ) := 1 |C| i∈C L i (θ), ∀θ ∈ Θ, and λ ∈ [0, 1] is called the collaboration level. Similar to (2), λ = 0 and λ = 1, respectively, reduce to local learning which can be solved with local SGD (Stich, 2018) , and Byzantine-robust FL which can be tackled using a robust variant of distributed gradient descent (Allouah et al., 2023) . Recall that C is a priori unknown to the correct clients, hence they can neither have access to L C nor to its gradients. A correct client can only approximate any information on L C , where the approximation error grows with both f /n and data heterogeneity, shown in (Karimireddy et al., 2022; Allouah et al., 2024) .We establish optimization and generalization guarantees on the interpolated personalized objective (3) in the presence of adversarial clients, and show how the degree of collaboration λ, asymptotically navigates the trade-off between the fundamental optimization error due to adversarial clients and the improved generalization performance thanks to the collaboration with correct ones. This sheds light on specific situations in which full collaboration performs strictly worse than fine-tuned personalization, and even sometimes worse than local learning. Precisely, we extend an important result from domain adaptation (Ben-David et al., 2010) , showing that data from a distinct distribution cannot be beneficial for the local learning task when the heterogeneity is above a certain threshold, which depends on the local sample size and the hypothesis class complexity. In the context of Byzantine robust distributed learning, we show that even if data heterogeneity is low enough, the level of collaboration should be rescaled by a parameter that both depends on the fraction of adversaries and the gradient dissimilarity between correct clients. Essentially, while the effect of collaboration could be captured by substraction between the local task complexity and the heterogeneity of local distributions in a Byzantine-free context, we show that it is further captured by a multiplicative factor in the presence of adversaries. The higher the dissimilarity between correct gradients (and the number of adversarial clients in the system), the smaller λ should be taken.Our results show that for personalized FL, in most realistic contexts where heterogeneity between clients is not negligible, full collaboration is not optimal in the presence of adversarial clients. Moreover, the presence of these adversarial clients considerably limits the level of collaboration that could be useful for the correct clients, to the point where it is often more efficient to learn locally if the local task is sufficiently simple.The remainder of the paper is organized as follows. Section 2presents the mean estimation setting as a warm-up and special case of the framework presented in Section 1.1. The goal is to give an intuition on settings in which personalization can or cannot help reduce the estimation error in the presence of adversarial clients. Section 3 presents our analysis in the general binary classification setting, where we quantify the tension between the optimization error and the generalization gap in the presence of Byzantine adversaries. We also empirically validate our theory on the MNIST dataset. We defer full proofs to Appendix C and our full experimental setup to Appendix D. We also include further information on related work in Appendix A. 
warm-up: mean estimation with adversaries: 
 Before diving into the details of our results in a general supervised learning setting, let us first focus on the simple task of Byzantine-robust federated mean estimation in which the concepts of heterogeneity and task complexity are easier to grasp. We consider n -f correct clients denoted C. Each client i ∈ C has access to m data point y(1) i , ..., y (m) i sampled i.i.d. from their a local distribution D i and independently from the other correct clients. We assume that each local distribution D i has a support in R d , an unknown finite mean µ i . Furthermore, we assume that all the distributions have the same finite variance σ 2 := E y∼Di ∥y -µ i ∥ 2 . In this context, the objective of each client i is to find their true mean µ i . To do so, we assume that each client attempts solve a problem of the form minEQUATION)In fact, the optimal solution to this problem is y λ * i := (1 -λ) y i + λ y C where yi := 1 m m j=1 y (j)i and y C := 1 n-f i∈C y i . Recall that due to the presence of Byzantine adversaries, correct clients cannot compute y C directly, but should instead use a robust aggregation rule F to estimate y C . Specifically, we consider the natural robust estimator of y i * 3 , where y C is replaced by a robust approximationEQUATIONTo formalize how good a robust aggregation rule is, we use the notion of (f, κ)-robustness (Allouah et al., 2023) , which we state below. Definition 1 ((f, κ)-robustness). Let f < n /2 and κ ≥ 0. An aggregation rule F is said to be (f, κ)-robust if for any vectors v 1 , . . . , v n ∈ R d , and any set U ⊆ [n] of size n -f ,∥F (v 1 , . . . , v n ) -v U ∥ 2 ≤ κ n -f i∈U ∥v i -v U ∥ 2 , with v U := 1 n -f i∈U v i .We evaluate the performance of the estimator y i by the mean squared error defined asE ∥y λ i -µ i ∥ 2,where the expectation is taken over the random samples y(1) i , ..., y (m) i drawn i.i.d. from D i and independently for every i ∈ C. Then the following holds. Proposition 1. Consider the mean estimation setting described. For any i ∈ C, let y λ i be as defined in (5) with an aggregation rule F that satisfies (f, κ)-robustness. Then the following holds true:E y λ i -µ i 2 ≤ 3 1 - 1 n -f σ 2 Γ(λ, κ) m + 3λ 2 (∥µ i -µ C ∥ 2 + κ∆ 2 ). with µ C := 1 n-f i∈C µ i , ∆ 2 := 1 n-f j∈C ∥µ j -µ C ∥ 2 , and Γ(λ, κ) := λ 2 (κ + 1) -2λ + n-f n-f -1. Tightness of the bound. While the right-hand side of Proposition 1 may not be tight in general, we note that it is tight for λ = 0 and λ = 1, in the homogeneous setting (∆ = 0), provided that we use robust averaging with κ ∈ O(f /n). Indeed, when λ = 0 the squared error in estimating the mean of a distribution with variance σ 2 from m i.i.d. samples is in Ω(σ 2 /m), refer (Wu, 2017) . Furthermore, when λ = 1, the squared error in estimating the mean of a distribution with variance σ 2 is in Ω( f +1 n σ 2 m ), refer (Zhu et al., 2023) . Interpretation. Note that, in the above, the terms that represent the hardness of the local mean estimation task and the heterogeneity among correct clients are σ 2 m , and ∥µ i -µ C ∥ 2 +κ∆ 2 respectively. Indeed σ 2 is the variance of each distribution, hence the local mean estimation task is essentially as hard as σ 2 is large with respect to the number of points m each client has access to. Similarly, ∆ 2 essentially computes how far apart the true means of the correct clients are on average, and ∥µ i -µ C ∥ 2 penalizes especially the distance to the average of the correct clients for the one we observe (i.e., i ∈ C). By minimizing the right-hand side of Proposition 1, we getEQUATION)which approaches 0 as ∥µ i -µ C ∥ 2 + κ∆ 2 grows and 1/(1 + κ)when ∥µ i -µ C ∥ 2 + κ∆ 2 ≪ σ 2 /m.Note that κ can be as small as O ( f /n), hence the above means that, when heterogeneity is limited and with a small enough fraction of adversarial clients, high level of collaborations can be used by each correct client. On the other hand, correct clients need not gain much by collaborating with other clients in highly heterogeneous regimes.To validate our theoretical observations on Byzantine-robust federated mean estimation, we run a series of experiments using simulated datasets sampled from 1-dimensional Gaussian distributions. Specifically, for each of the n -f correct clients, we consider that their distribution is such thatD i = N (µ i , σ 2 ), where the unknown local means (µ i ) i∈C have been sampled i.i.d. from a Gaussian distribution N (10, σ 2 h )4 , where σ h determines the expected squared distance between the local means of each correct client. Each honest client i ∈ C samples m datapoints from D i and the error is computed with respect to µ i . To evaluate the theoretical expression of λ * , defined in (6), in this scenario, we replace the distances of the form (µ i -µ C ) 2 by the variance (1 -1 /n-f)σ 2 h . We present in Figure 1 the average error (squared distance to the true mean) of the estimator defined as per (5) on 20 runs, for different values of λ. The robust aggregation being used here is an NNM pre-aggregation rule (Allouah et al., 2023) followed by trimmed mean (Yin et al., 2018) and adversarial clients implement the sign-flipping attack. Below we analyze our result presented in the first row of Figure 1 .Heterogeneity. First, we study the impact of the heterogeneity level in Figure 1 . Doing so, we fix n = 600, f = 100, m = 20, σ = 15 and vary the level of heterogeneity σ h ∈ {0, 1, 2, 3, 4}. We observe that for low levels of heterogeneity, the optimal choice for λ is close to 1. In this case, personalization does not really help.Byzantine fraction. Second, we analyze the impact of the fraction of Byzantine adversaries. For Figure 1 , we fix n = 600, m = 20, σ = 15, σ h = 2 and vary f ∈ {0, 50, 100, 150}. As per our theoretical analysis, as the Byzantine fraction increases κ should increase; hence the correct clients need to rely less and less on the global aggregate. However, simply using a local estimator can be detrimental as showcased by the red curve, which shows the case f = 150. Then, by choosing the right collaboration, one reduces the error by 50% compared to the local estimator.Task complexity. Finally, we study the impact of the task complexity, characterized by the quantity σ 2 /m. We fix n = 600, f = 100, m = 20, σ h = 2 and vary σ ∈ {1, 7, 13, 19, 25, 31}. We observe, as expected, that the optimal level of collaboration level of collaboration moves away from 0 as the task gets harder. However, we also observe that, since the fraction of Byzantine clients is non-negligible, the optimal choice of collaboration λ, even for very large values of σ, never goes to 1.Correspondence between the upper-bound and empirical observations. The second line of Figure 1 shows a comparison between the optimal theoretical value predicted by our analysis and the empirical minimizer of the error on average. 5 We see that the value we predicted for λ * and the actual empirical best choice for λ has very similar trends in all the settings we consider.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: We discuss the limitations of our work all throughout the paper (e.g. line 235, footnoteGuidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
binary classification with adversaries: 
 In this section, we consider the more general problem of binary classification. We study the learningtheoretic setup given in Section 1 with binary output space Y := {0, 1} and Y ′ = [0, 1]. Throughout, we place ourselves in a hypothesis space H of finite pseudo-dimension (Vidyasagar, 2003; Mohri et al., 2018) , denoted Pdim(H), which we recall reduces to the VC dimension for the 0 -1 loss. We further make the following assumptions on the loss functions. These assumptions are standard in the Byzantine robustness literature see, e.g., (Karimireddy et al., 2022; Allouah et al., 2023) .Assumption 1 (L-smoothness, µ-strong convexity). Each loss function L i , i ∈ C, is L-smooth and µ-strongly convex. That is, for all θ, θ ′ ∈ Θ, we have µ 2 ∥θ -θ ′ ∥ 2 ≤ L i (θ) -L i (θ ′ ) -⟨∇L i (θ ′ ), θ -θ ′ ⟩ ≤ L 2 ∥θ -θ ′ ∥ 2 .Assumption 2 (Bounded heterogeneity). There exists a real value G such that for all θ ∈ Θ, we have1 |C| i∈C ∥∇L i (θ) -∇L C (θ)∥ 2 ≤ G 2 .We additionally make the following assumption on the parameter set Θ to ensure that the Euclidean projection on Θ (denoted Π Θ ) is well-defined, unique and that the minimizer of the interpolated loss is not pathological, i.e., is not in the border of Θ. Assumption 3. The parameter set Θ is compact and convex. Moreover, for every i ∈ C, λ ∈ [0, 1], the minimizer of L λ i is in the interior of Θ. Additionally, for our generalization analysis, we make the following assumption on the loss function:Assumption 4. The loss function is bounded in [0, 1].To conduct an analysis of the effect of collaboration on the generalization performance of correct clients when trying to solve (1), we proceed in two steps. We first evaluate the optimization error that a standard gradient descent algorithm incurs in our setting. Then we bound the generalization gap induced when minimizing (1) and combine the two bounds into our main result in Theorem 1.We focus our analysis on a simple personalized variant of the robust distributed gradient descent algorithm, which is the standard algorithm in the Byzantine-robust FL literature and is shown to achieve tight optimization bounds (Allouah et al., 2023) . Our variant, presented in Algorithm 1, essentially corresponds to gradient descent on the function L λ i , but using a robust estimate R t i (iteration 7) of the gradient of L C , which cannot be computed exactly due to Byzantine adversaries. This robust estimate is computed using a robust aggregation F , e.g., trimmed mean or median.On the execution of the algorithm. Executing Algorithm 1 in practice implies that each client acts as a local server, and runs an independent federated learning procedure. We further assume that the clients have access to a communication protocol that allows them to broadcast their model to the other clients. This can either be done through the use of the central server or directly through a decentralized communication (i.e., without a server). This algorithm is not communication efficient and we believe that it could be improved in that direction. Nevertheless, it allows us to discuss the information-theoretic aspects of the problem in a simple manner. We first present the optimization error of Algorithm 1 in Lemma 1 below. Lemma 1. Let assumptions 1, 2, and 3 hold. Consider Algorithm 1 with learning rate η = 1 2L , λ ∈ [0, 1], and assume the aggregation function F to be (f, κ)-robust. For any T ≥ 1, we have:L λ i (θ T i ) -L λ i, * ≤ 5Lλ 2 κG 2 µ 2 + 1 - µ 2L T L µ L 0 ,whereL 0 := L λ i (θ 0 i ) -L λ i, * and L λ i, * := min θ∈Θ L λ i (θ).Algorithm 1 Interpolated Personalized Gradient Descent for client i ∈ C Require: Initialization θ 0 i , aggregation rule F , learning rate η, number of iterations T , and collaboration parameter λ.1: for t = 1 . . . T do 2: Broadcast θ t-1 i to all clients 3: for j = 1 . . . n, j ̸ = i do 4: Receive g t i,j = ∇L j (θ t-1 i) from client j ▷ adversarial clients send corrupted gradients 5:end for6: Compute local gradient g t i,i = ∇L i (θ t-1 i ) 7:Robustly aggregateR t i = F (g t i,1 , . . . , g t i,n ) 8:Update and project local parametersθ t i = Π Θ θ t-1 i -η (1 -λ)g t i,i + λR t i 9: end forOur optimization error bound recovers the classical GD linear rate when λ = 0 (local learning), and achieves an asymptotic error in O(κG 2 ) when λ = 1 (full collaboration), which is provably tight under Assumption 2 (Karimireddy et al., 2022) . Interestingly, for λ ∈ (0, 1), the asymptotic error smoothly interpolates in O(λ 2 κG 2 ), which is expected as lesser collaboration limits the influence of the adversary on the optimization error.Although the presence of adversarial clients impairs the optimization process, one expects a generalization benefit from collaboration if data distributions are similar enough. We formalize this intuition by bounding the generalization gap, of the personalized learning problem (2). We first bound the generalization gap on the interpolated loss L λ i using a result adapted from (Blitzer et al., 2007) . Lemma 2. Let Assumption 4 hold. For any δ > 0, θ ∈ Θ and λ ∈ [0, 1], we have with probability at least 1 -δ (over the choice of samples) that|L λ i (θ) -R λ i (θ)| ≤ 2β    1 -λ + λ n-f 2 m + λ 2 m(n -f )   , where R λ i (θ) := E[L λ i (θ)], ∀θ ∈ Θ, and β := Pdim(H) log em Pdim(H) + log(1/δ).When specializing the result of Lemma 2 to λ = 0 and λ = 1, we recover the standard generalization gaps O Pdim(H) /m and O Pdim(H) /m(n-f) for local learning and federated learning (with correct clients only), respectively. However, in addition to the generalization gap bounded in Lemma 2, we need to bound the gap between the interpolated risk (3) and the original local risk (1). In fact, these two objectives are statistically different when there is data heterogeneity among clients since the interpolated risk (3) involves the average of local loss functions. To quantify this difference, we leverage tools from domain adaptation theory (Ben-David et al., 2010) and consider a function Φ measuring the discrepancy between two statistical distributions. Formally, we require for every i ∈ C,|R i (θ) -R C (θ)| ≤ Φ(D i , D C ), ∀θ ∈ Θ.(7) For example, for the case of 0-1 loss in binary classification, for any two distributions D 1 and D 2 , (Blitzer et al., 2007) propose the discrepancy measure to be Φ(D 1 , D 2 ) = d H∆H (D 1 , D 2 ) := 2 sup h∈H |P D1 (I(h)) -P D2 (I(h))|, where I(h) := {x ∈ X : h(x) = 1}.For more general losses, we can use hypothesis space-dependent Integral Probability Metrics (Sriperumbudur et al., 2009; Bao et al., 2023) , which we include in Appendix B.2. We now combine our bounds on the optimization error and generalization gap into Theorem 1 below. Theorem 1. Let assumptions 1, 2, 3, and 4 hold, and let Φ be a function such that (7) holds. Consider Algorithm 1 with learning rate η = 1 2L , λ ∈ [0, 1], and assume the aggregation function F to be (f, κ)-robust. Then, for any δ > 0, T ≥ 1, we have with probability at least 1 -δ (over the choice of samples) thatEQUATIONwhereβ := Pdim(H) log em Pdim(H) + log(1/δ) and L λ i, * := min θ∈Θ L λ i (θ).Interpretation. The bound in Theorem 1 features a trade-off between data heterogeneity, sample size, and model complexity. Indeed, we minimize the bound in Theorem 1 and obtain the following closed-form approximation by ignoring constant and logarithmic terms, and assuming that n, T ≫ 1 (see Appendix C for details)EQUATIONwhere Π [0,1] denotes the projection over [0, 1]. This expression suggests that the optimal collaboration parameter decreases with data heterogeneity. Indeed, if Φ(D i , D C ) > Pdim(H) /m, then λ * = 0 and local learning, as expected, is optimal. Otherwise, the optimal collaboration parameter can be non-zero. This validates the fact that correct clients cannot improve their local generalization when their local distributions are too dissimilar compared to the complexity of the hypothesis class and the number of data points each of them has. Furthermore, even if the heterogeneity Φ(D i , D C ) sufficiently small compared to Pdim(H) /m, the level of collaboration must be re-evaluated according to f n G 2 ; hence depending on both the fraction of adversaries and the dissimilarity of the gradient between correct clients. Setup. We empirically investigate the impact of Byzantine adversaries on the generalization performance, in our personalization framework, using the Phishing dataset (Chiew et al., 2019) and the MNIST dataset (LeCun and Cortes, 2010) . Other experiments are included in Appendix D. For the different experiments, we use the state-of-the-art defense under data heterogeneity: NNM preaggregation (Allouah et al., 2023) rule followed by trimmed mean as robust aggregation (Yin et al., 2018) . We simulate the Byzantine attack with the Sign Flipping attack (Li et al., 2020) . Throughout the experiments, we fix n = 20, and only vary the local dataset size m, the fraction of Byzantine adversaries f , and the heterogeneity level α which we will explain in the next paragraph. The clients execute Algorithm 1 for T iterations which depends on the dataset being used. Figure 2 , as well as the figures in Appendix D, show the average results and error bars for 5 random runs.To simulate heterogeneity, we use a Dirichlet distribution (Hsu et al., 2019) to generate datasets with unbalanced class fractions for each client. The parameter α determines the heterogeneity degree (the smaller the bigger the heterogeneity). We subsequently sample the test datasets using the same class distribution as the train datasets for each client, and we evaluate the trained models on these local datasets. We defer the implementation details to Appendix D. Full collaboration can be suboptimal. This can be gleaned from settings where the adversarial fraction is substantial (over 6 adversarial clients, e.g. green and red curves in Figure 2 ), when the heterogeneity is large (small values of α, e.g. blue and orange curves in Figure 6 in Appendix D) and when the local dataset is large enough (e.g. Figure 2b and Figure 2c ). In situations where the adversarial fraction is critical (f = 9, corresponding to red curves in Figure 2 ), robust federated learning completely fails, achieving accuracy scores under 50% on the local test datasets. Figure 5 in Appendix D further illustrates this point, showing that even local learning can be better than state-of-the-art robust Federated Learning in these circumstances.Fine-tuning the collaboration helps get the best of both worlds. Figure 2 (Bottom) shows that in data scarcity scenarios, for moderate values of the number of adversarial clients (f = 3 or f = 6), using a collaboration degree strictly between 0 and 1 yields better accuracy on the local test dataset.The same effect also appears in Figure 2 (Top) for f = 3 and Figure 4 in Appendix D. Additionally, even for extreme adversarial fractions (f = 9 in our experiments), the gain from decreasing the collaboration degree can be substantial compared to simply using Robust Federated Learning methods. This suggests that fine-tuning the collaboration degree can help dampen the detrimental effects of adversarial clients.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. Justification: We include a section about compute resources and runtime in Appendix D.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
conclusion & future work: 
 In this paper, we have taken a first step towards understanding how fine-tuning personalization in FL can mitigate the impact of adversarial clients. The results we obtain suggest that the use of personalization can improve the performance of FL algorithms in the presence of adversarial clients, but also that the level of collaboration needs to be chosen carefully. Our theoretical analysis accounts for the necessity to strike a balance between the optimization error, which is impacted by adversarial clients, and the generalization gap, which is likely to benefit from a larger pool of data. We identify the main factors that impact the local performance, namely data heterogeneity, the fraction of adversarial clients, and data scarcity. This work could be extended in several directions. Firstly, as explained in Section 3, the study of Algorithm 1 primarily serves our understanding of the robust interpolated optimization problem in (3) from an information-theoretic perspective. However, it may not be efficient enough to cope with high-dimensional learning tasks involving a number of clients. An interesting future direction could be to investigate alternative algorithms with lower communication and computational costs, which would be able to handle these high-dimensional problems. Second, our study focuses on a personalization strategy that interpolates between client loss functions. Although this strategy is simple to interpret and allows us to derive optimization and generalization bounds, it is only one of many possible personalized FL schemes. Another interesting avenue would be to investigate whether the insights we drew from our interpolated problem could be derived in a similar fashion from other personalization frameworks such as regularization, layer specialization, or clustering.  
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 Appendix: Personalized federated learning. As federated learning methods are typically challenged by data heterogeneity (Kairouz et al., 2021) , many personalization methods have recently been proposed to learn different models tailored for each client. A first category of personalized learning algorithms is based upon local fine-tuning (Fallah et al., 2020; Dinh et al., 2020) , where the goal is to collaboratively find a good global model from which local fine-tuning can result in an improved local performance. A second category of personalized algorithms is based upon regularization between local models (Li et al., 2021; Kundu et al., 2022) , i.e., adding a regularization term penalizing a distance between local client models. Another line of research in personalized federated learning considers partial personalization where the parameters are split into globally shared and individual local parameters (Bietti et al., 2022; Mishchenko et al., 2023) , this, however, creates the additional challenge of choosing this split. Personalization can also be implemented through clustering methods, where participants selectively collaborate with a smaller subset of clients to limit their exposure to heterogeneity (Ghosh et al., 2020; Werner et al., 2023; Bao et al., 2023) . (Mansour et al., 2020) proposed and analyzed three different interpolation methods to formalize personalization, including the one we consider in this paper, which they refer to as data interpolation. Although they discuss the generalization properties of this model, they do not address Byzantine robustness.Personalization for Byzantine robustness. A few works in the personalized learning literature suggest that personalization can improve the robustness to Byzantine participants (Li et al., 2021; Kundu et al., 2022; Mishchenko et al., 2023) . These works however only present a limited theoretical analysis in the general Byzantine context. For instance, contrary to our work, the theoretical results in (Li et al., 2021) only cover linear regression and random Gaussian noise attacks. Moreover, the results of (Mishchenko et al., 2023) concerning Byzantine robustness make very strong assumptions on the heterogeneity of loss functions, essentially assuming that they share a common minimum, which largely mitigates the effect of Byzantine adversaries on the optimization error (Allouah et al., 2024) . Both (Kundu et al., 2022; Mishchenko et al., 2023) only analyze the optimization error and do not study the generalization error, which we showcase as an important part of the trade-off.A close, but weaker version of strong convexity is the Polyak-Lojasiewicz condition defined below: Definition 2 (Polyak-Lojasiewicz (PL)). A function L : R d → R is said to satisfy Polyak-Lojasiewicz with parameter µ, noted µ-PL if ∀x ∈ R d :L(x) -inf R d L ≤ 1 2µ ∥∇L(x)∥ 2 (10)If a function L is µ-strongly convex, it satisfies µ-PL condition. Indeed a µ-strongly convex function L satisfies:EQUATIONThus, µ-PL can be obtained from the above by minimizing each side with respect to x.For the 0-1 loss, let d H be defined as follows :EQUATIONwhereI(h) = {x : h(x) = 1}.Define the symmetric difference hypothesis space : H∆H := {h(x) ⊕ h ′ (x)} : h, h ′ ∈ H. Then following (Blitzer et al., 2007) , d H∆H satisfies (7).For general losses, we define the following discrepancy:ϕ(D 1 , D 2 ) = max h∈H |E D1 L(h(x), y) -E D2 L(h(x), y)| C Proofs C.1 Proof of Mean Point EstimationProposition 1. Consider the mean estimation setting described. For any i ∈ C, let y λ i be as defined in (5) with an aggregation rule F that satisfies (f, κ)-robustness. Then the following holds true:E y λ i -µ i 2 ≤ 3 1 - 1 n -f σ 2 Γ(λ, κ) m + 3λ 2 (∥µ i -µ C ∥ 2 + κ∆ 2 ).withµ C := 1 n-f i∈C µ i , ∆ 2 := 1 n-f j∈C ∥µ j -µ C ∥ 2 , and Γ(λ, κ) := λ 2 (κ + 1) -2λ + n-f n-f -1 .Proof. We can bound the error as follows:EQUATION)where, for the second line we use the triangle inequality and the fact that (a+b+c) 2 ≤ 3(a 2 +b 2 +c 2 ), for all a, b, c ≥ 0.Upper-bounding the first term. The first term can be simplified as follows:E ∥(1 -λ)( y i -µ i ) + λ( y C -µ C )∥ 2 = E    1 -λ + λ |C| ( y i -µ i ) + λ |C| j̸ =i ( y j -µ j ) 2   Because the data points are sampled independently for each client, we have thatE ∥(1 -λ)( y i -µ i ) + λ( y C -µ C )∥ 2 = 1 -λ + λ |C| 2 E ∥ y i -µ i ∥ 2 + λ 2 |C| 2 E    j̸ =i ( y j -µ j ) 2    .Because the data points are sampled i.i.d. each client local distributing and independently for each client, we also haveE ∥(1 -λ)( y i -µ i ) + λ( y C -µ C )∥ 2 = 1 -λ + λ |C| 2 E ∥ y i -µ i ∥ 2 + λ 2 |C| 2 j̸ =i E ∥( y j -µ j )∥ 2 = 1 -λ + λ |C| 2 1 m E y∼D i|Y ∥y -µ i ∥ 2 + λ 2 |C| 2 j̸ =i 1 m E y∼D j|Y ∥y -µ j ∥ 2 = 1 -λ + λ |C| 2 + λ 2 (|C| -1) |C| 2 σ 2 m .Where the last line comes from the fact that E y∼D i|Y ∥y -µ i ∥ 2 = σ i = σ 2 for all i ∈ [n].Upper-bounding the second term. The second term can be controlled by the (f, κ)-robustness property of F as followsλ 2 E ∥R -y C ∥ 2 = λ 2 E ∥F ( y 1 , . . . , y n ) -y C ∥ 2 ≤ λ 2 κ |C| i∈C E ∥ y i -y C ∥ 2Then using similar arguments as for the first term we haveE ∥ y i -y C ∥ 2 = E ∥ y i -µ i + (µ i -µ C ) + µ C -y C ∥ 2 = E ∥ y i -µ i + µ C -y C ∥ 2 + ∥µ i -µ C ∥ 2 = E    1 - 1 |C| ( y i -µ i ) + 1 |C| j̸ =i (µ j -y j ) 2    + ∥µ i -µ C ∥ 2 = E 1 - 1 |C| ( y i -µ i ) 2 + E    1 |C| j̸ =i (µ j -y j ) 2    + ∥µ i -µ C ∥ 2 = 1 - 1 |C| 2 σ 2 m + (|C| -1) |C| 2 σ 2 m + ∥µ i -µ C ∥ 2 = 1 - 1 |C| 2 + (|C| -1) |C| 2 σ 2 m + ∥µ i -µ C ∥ 2Substituting this in the above we getλ 2 E ∥R -y C ∥ 2 ≤ κλ 2 1 - 1 |C| 2 + (|C| -1) |C| 2 σ 2 m + κλ 2 |C| j∈C ∥µ j -µ C ∥ 2 ≤ |C| -1 |C| κλ 2 σ 2 m + κλ 2 |C| j∈C ∥µ j -µ C ∥ 2 Conclusion. By substituting the above in (13), we get the following.E[ y λ i -µ i 2 ] ≤ 3 |C| -1 |C| κλ 2 + 1 -λ + λ |C| 2 + λ 2 (|C| -1) |C| 2 σ 2 m + 3λ 2 ∥µ i -µ C ∥ 2 + 3 κλ 2 |C| j∈C ∥µ j -µ C ∥ 2 (14) Using the notation ∆ 2 = 1 |C| j∈C ∥µ j -µ C ∥ 2 , we get E[ y λ i -µ i 2 ] ≤ 3 |C| -1 |C| λ 2 (κ + 1) + 1 -2 |C| -1 |C| λ σ 2 m + 3λ 2 (∥µ i -µ C ∥ 2 + κ∆ 2 ).Lemma 1. Let assumptions 1, 2, and 3 hold. Consider Algorithm 1 with learning rate η = 1 2L , λ ∈ [0, 1], and assume the aggregation function F to be (f, κ)-robust. For any T ≥ 1, we have:L λ i (θ T i ) -L λ i, * ≤ 5Lλ 2 κG 2 µ 2 + 1 - µ 2L T L µ L 0 ,whereL 0 := L λ i (θ 0 i ) -L λ i, * and L λ i, * := min θ∈Θ L λ i (θ).Proof. Let assumptions 1, 2, and 3 hold. First, we note that L λ i is L-smooth (and µ-strongly convex) as a convex combination of L i and L C , which are both L-smooth (and µ-strongly convex) by Assumption 1. In the remainder of this proof, we denote by θ λ i the minimizer of L λ i on Θ, i.e., L λ i ( θ λ i ) = L λ i, * . Recall also that, by Assumption 3, θ λ i is also a critical point of L λ i . Let t ∈ {0, . . . , T -1} and denote ξ t i := R t i -∇L C (θ t i ). We rewrite the update step as follows:θ t+1 i = Π Θ θ t i -η (1 -λ)∇L i (θ t i ) + λR t i = Π Θ θ t i -η ∇L λ i (θ t i ) + λ R t i -∇L C (θ t i ) .Since the Euclidean projection operator Π Θ is non-expansive, and θ λ i ∈ Θ by definition, we have∥θ t+1 i -θ λ i ∥ 2 ≤ ∥θ t i -θ λ i -η ∇L λ i (θ t i ) + λξ t i ∥ 2 .(15) By developing the right-hand side (15) and using Jensen's inequality, we get∥θ t+1 i -θ λ i ∥ 2 ≤ ∥θ t i -θ λ i ∥ 2 -2η θ t i -θ λ i , ∇L λ i (θ t i ) -2η θ t i -θ λ i , λξ t i + η 2 ∥∇L λ i (θ t i ) + λξ t i ∥ 2 ≤ ∥θ t i -θ λ i ∥ 2 -2η θ t i -θ λ i , ∇L λ i (θ t i ) -2η θ t i -θ λ i , λξ t i + 2η 2 ∥∇L λ i (θ t i )∥ 2 + 2η 2 λ 2 ∥ξ t i ∥ 2 .Furthermore, by Assumption 3, we know that θ λ i is a critical point of L λ i (Assumption 1), hence using the L-smooth of L λ i we have∥∇L λ i (θ t i )∥ 2 ≤ 2L(L λ i (θ t i ) -L λ i, * ). Moreover, since L λ i is µ-strongly convex (Assumption 1), we have θ t i -θ λ i , ∇L λ i (θ t i ) ≥ L λ i (θ t i ) -L λ i, * + µ 2 ∥θ t i -θ λ i ∥ 2. Substituting these in the above yields∥θ t+1 i -θ λ i ∥ 2 ≤ ∥θ t i -θ λ i ∥ 2 -2η(L λ i (θ t i ) -L λ i, * ) -ηµ∥θ t i -θ λ i ∥ 2 -2η θ t i -θ λ i , λξ t i + 4η 2 L(L λ i (θ t i ) -L λ i, * ) + 2η 2 λ 2 ∥ξ t i ∥ 2 . Given that η = 1 2L, we can simplify this inequality as follows,∥θ t+1 i -θ λ i ∥ 2 ≤ ∥θ t i -θ λ i ∥ 2 -ηµ∥θ t i -θ λ i ∥ 2 -2η θ t i -θ λ i , λξ t i + 2η 2 λ 2 ∥ξ t i ∥ 2 .We now use Young's inequality on scalar product to get -θ t i -θ λ i , λξ t i ≤ µ 4 ∥θ t i -θ λ i ∥ 2 + 4 µ λ 2 ∥ξ t t ∥ 2 .Substituting these in the above yields∥θ t+1 i -θ λ i ∥ 2 ≤ ∥θ t i -θ λ i ∥ 2 -ηµ∥θ t i -θ λ i ∥ 2 + η µ 2 ∥θ t i -θ λ i ∥ 2 + η 8 µ λ 2 ξ t t 2 + 2η 2 λ 2 ∥ξ t i ∥ 2 ≤ 1 - ηµ 2 ∥θ t i -θ λ i ∥ 2 + λ 2 η 8 µ + 2η 2 ∥ξ t i ∥ 2 .Substituting η = 1 2L in the above, and using the fact that L ≥ µ, we obtain∥θ t+1 i -θ λ i ∥ 2 ≤ 1 - µ 2L ∥θ t i -θ λ i ∥ 2 + λ 2 4 µL + 1 2L 2 ∥ξ t i ∥ 2 ≤ 1 - µ 2L ∥θ t i -θ λ i ∥ 2 + λ 2 5 µL ∥ξ t i ∥ 2 .By (f, κ)-robustness of F , and Assumption 2, we have∥ξ t i ∥ 2 ≤ κ |C| i∈C ∥∇L i (θ t i ) -∇L C (θ t i )∥ 2 ≤ κG 2 . Hence, we get ∥θ t+1 i -θ λ i ∥ 2 ≤ 1 - µ 2L ∥θ t i -θ λ i ∥ 2 + λ 2 5 µL κG 2 .Recursively using the above, we getEQUATION)Combining Assumption 1 and Assumption 3, we easily get that µ 2 ∥θ -θ λ i ∥ 2 ≤ L λ i (θ) -L λ i, * ≤ L 2 ∥θ -θ λ i ∥ 2 for all θ ∈ Θ.Using this, and specializing ( 16) for t = T -1, we haveL λ i (θ T i ) -L λ i, * ≤ 5Lλ 2 κG 2 µ 2 + 1 - µ 2L T L µ (L λ i (θ 0 i ) -L λ i, * ).We start by reminding Hoeffding inequality: Lemma 3 (Hoeffding inequality). If X 1 , . . . , X N are real-valued independent random variables, each almost surely belonging to interval [a i , b i ], then the sumEQUATIONBack to our problem: Lemma 2. Let Assumption 4 hold. For any δ > 0, θ ∈ Θ and λ ∈ [0, 1], we have with probability at least 1 -δ (over the choice of samples) that|L λ i (θ) -R λ i (θ)| ≤ 2β    1 -λ + λ n-f 2 m + λ 2 m(n -f )   , where R λ i (θ) := E[L λ i (θ)], ∀θ ∈ Θ, and β := Pdim(H) log em Pdim(H) + log(1/δ).Proof.EQUATIONLet us now consider the set of m |C| real-valued independent6 random variables defined as|C| 1 -λ + λ |C| ℓ(h θ (x), y) | (x, y) ∈ S i ∪ {λℓ(h θ (x), y) | (x, y) ∈ S j , ∀j ̸ = i}. As per As- sumption 4, we know that ℓ takes its values in [0, 1], hence this set is composed of m random variables with values in 0, |C| 1 -λ + λ |C| and (|C| -1)m others with values in [0, λ]. Finally, note that by definition E L λ i (θ) = R λ i (θ).Where the expectation is taken over the independent sampling of the local data sets (S i ) i∈C from the local data distribution of the correct clients. Thus using Hoeffding inequality, we getEQUATIONBy using the change of variablesε ′ = |C| 2 (1-λ+ λ |C| ) 2 |C| 2 +(|C|-1)λ 2 ε,we can use the Pseudo-dimension union bound (e.g. Theorem 11.8 from (Mohri et al., 2018) to get that with probability at least 1 -δ, ∀θ|L λ i (θ) -R λ i (θ)| ≤ 2 1 -λ + λ |C| 2 m + λ 2 (|C| -1) m|C| 2 Pdim(H) log em Pdim(H) + log(1/δ) ≤ 2 1 -λ + λ n-f 2 m + λ 2 m(n -f ) Pdim(H) log em Pdim(H) + log(1/δ) .C.4 Proof of Theorem 1Theorem 1. Let assumptions 1, 2, 3, and 4 hold, and let Φ be a function such that (7) holds. Consider Algorithm 1 with learning rate η = 1 2L , λ ∈ [0, 1], and assume the aggregation function F to be (f, κ)-robust. Then, for any δ > 0, T ≥ 1, we have with probability at least 1 -δ (over the choice of samples) thatEQUATIONwhereβ := Pdim(H) log em Pdim(H) + log(1/δ) and L λ i, * := min θ∈Θ L λ i (θ).Proof. In order to control the quantity R i (θ t i ) -R i (θ * i ), we can write:EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONEQUATIONwhere θ λ i is the minimizer in Θ of the empirical objective function L λ i , which we assume is strongly convex.For lines 1 and 6, we use the property of our loss function described in (7), to writeEQUATIONFor lines 2 and 5, since Assumption (4) holds true, we can use the results from Lemma 2. Let β := Pdim(H) log(2m |C|) + log(4/δ). By applying Lemma 2 with δ/2 for both (2) and (5), we get that with probability at least 1 -δ:(EQUATION)For line 3, recall that assumptions 1 and 2, that η = 1 2L , and that F is assumed to be (f, κ)-robust. Hence, Lemma 1 holds true. we use ( 16) in the proof of Lemma 1, to getL λ i θ t i -L λ i ( θ λ i ) ≤ 5Lλ 2 κG 2 µ 2 + 1 - µ 2L t L µ L λ i (θ 0 i ) -L λ i ( θ λ i )For line 4, we can simply disregard it as (4) contributes a negative amount to the inequality.Finally, combining all the above inequalities we getEQUATIONWe conclude by substituting |C| = n -f .C.5 Proof of ( 9)Equation (9) λ * ≈ Π [0,1]   Pdim(H) m -Φ(D i , D C ) f n G 2   , Proof.Here, our goal is to reasonably approximate the value of lambda λ * for which the right-hand side of ( 24) is minimized. To do so, we first ignore the asymptotically negligible terms. Essentially, we consider that T is large enough so that the term 1-µ 2L T L µ L λ i (θ 0 i ) -L λ i ( θ λ i) is close enough to zero. In the reminder, we denote by θ ∞ i the output of Algorithm 1 for T → ∞. Even after ignoring this term, seeking the optimal value λ * is the solution to a 4th-degree equation, which may not admit a close-form solution in general. To make the computation of an approximate minimizer more tractable, we proceed with the following simplification using the fact thatEQUATIONwith α := 6LκG 2 µ 2 . Now the right-hand side is a quadratic function of λ which can be easily minimized. Specifically, the solution of ( 25) isEQUATION)Assuming |C| is large enough, we simplify the above asEQUATIONIn this section, we give the implementation details of our experiments for each setting and we provide some additional figures. For all the experiments, we run Algorithm 1 with full gradients for a number T of iterations which we change depending on the dataset.We run our experiments on a server with the following specifications:• HPe DL380 Gen10• 2 x Intel(R) Xeon(R) Platinum 8358P CPU @ 2.60GHz• 128 GB of RAM• 740GB ssd disk • 2 Nvidia A10 GPU cardsFor most of the experiments, we only use one of the two GPUs.Using these compute resources, each experiment on the MNIST dataset (meaning each subfigure in Figure 2 (bottom)) took less than 72 hours to run. Each experiment on Phishing dataset took less than 8 hours to run.For the experiments on MNIST, we use the following Convolutional Neural Network:• Convolutional Layer (1, 32, 5, 1) + ReLU + Maxpooling We use T = 100 and the learning rate η = 0.05.We run some experiments on a binarized version of the MNSIT dataset, where the task is to output whether or not the MNIST class of the image, which corresponds to a digit between 0 and 9, is strictly smaller than 5.We use T = 150 and the learning rate η = 0.002. Justification: Our claims in the abstract and introduction are precise and consistent with our findings.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.This estimator need not be optimal a priori. But it represents the state-of-the-art solution for robust FL.We use a mean at 10 to break the symmetry of the mean simulation around 0. This is simply to present a case in which the sign-flipping attack is indeed disrupting the federated mean estimation procedureκ is replaced by f /n-2f, following(Allouah et al., 2023)The data sets have been sampled independently of each other and are composed of points sampled i.i.d from the local data distributions. 


Decision: This paper provides a nice analysis and discussion on the interplay between collaboration, personalization, and adversarial clients in federated learning. The focus is on the theory, with an intuitive special case of mean estimation presented in Section 2 and the full analysis of the generalization bound presented in Section 3. The theoretical findings have been complemented by experiments which show useful insights. The conclusions and avenues of future work have been well-discussed too. As a theory-focused paper, it would be nice if the technical challenges and novelty in the methodology can be explained more.

Reviews: review: paper_summary: paper_summary: This paper presents theoretical analysis and experimental validation results of the allowed level of collaboration in personalized FL with the presence of a fraction of Byzantine adversaries.
strength_weakness: strength_weakness: - The experimental validations can still be further improved from multiple aspects. For example, it may not be very convincing by using simulated datasets being generated by simple 1D sampling. The data heterogeneity settings should be more complicated and practical accordingly. More complicated models should also be used.
- The analysis results only apply to the simple problem of binary classification. What's its generalizability to more practical multi-class classification?
questions: questions: See weakness above.
limitations: limitations: See weakness above.
paper_summary: paper_summary: This paper considers an FL setting where some clients can be adversarial, and we derive conditions under which full collaboration fails. Specifically, they analyze the generalization performance of an interpolated personalized FL framework in the presence of adversarial clients. The authors claim that they precisely characterize situations when full collaboration performs strictly worse than fine-tuned personalization.
strength_weakness: strength_weakness: Section 2 doesn't make sense to me. Proposition 2 characterizes the difference between two variables, a local variable $\mu_i$, and a variable depending on collaboration (y^{\lambda}). 
This manuscript only considers deterministic cases. The assumption is strong, such as assumptions 2 and 4. 
The bound shown in the analysis is loose and the conclusion does not convince me.
questions: questions: In equation 9, when there is less data heterogeneity, \Psi() ->0 and G ->0, $lambda$  -> 1. Why do we need to collaborate when the data is homogeneous? I think we can train locally to avoid the adversary. 

Is assumption 3 necessary? Is it repetitive with Assumption 1?

Is the model only effective for binary classification problems?
limitations: limitations: Experiments are a bit simple.  Assumptions are strong.
paper_summary: paper_summary: This paper studies fine-tuning personalization in federated learning (FL) to mitigate the impact of adversarial clients. The authors leverage interpolation techniques for personalization, and they derive the closed-form approximation of the interpolation parameter $\lambda$. The study comprehensively considers both data heterogeneity and the presence of adversarial clients in the context of tailoring personalized FL.
strength_weakness: strength_weakness: 1. The proposed fine-tuning personalization strategy requires each client should broadcast its model. Besides, each client should send the gradients to other clients. This is not efficient and may incur other privacy issues.
2. The prediction tasks in this paper are simple. 
Other issues:
1. Then then --> Then the in 142.
questions: questions: 1. For more convincing, the authors should consider other more complicated datasets. 
2. The suggested fine-tuning strategy for personalization necessitates that each client share its gradients with others, which could potentially raise privacy concerns. Moreover, clients might be able to identify adversarial clients through the gradients accumulated during communication.
limitations: limitations: N/A.
paper_summary: paper_summary: This paper proposes a fundamental validation to understand the relationship between local models and the global model to mitigate the impact of adversarial clients. The level of collaboration needs to be chosen carefully because of the existence of adversarial clients. The theoretical analysis is provided to validate the statement, considering data heterogeneity, the fraction of adversarial clients, and data scarcity. Several simulated and open-source datasets are used to further demonstrate the effectiveness of the method.
strength_weakness: strength_weakness: 1. Some typos, such as the missing space behind "Section 2" in Line 112.
2. Although the theoretical proof is thorough, more experiments on different datasets, Byzantine attack methods, and defense methods should be evaluated.
3. For the simulated datasets in Section 2.2, cross-device (n=600/f=100) is employed, but the experimental validation in Section 3.3 uses cross-silo (n=20/f=0,3,6,9).
4. The models used for each dataset are not mentioned.
In summary, although theoretical proof is provided, the practical applicability of the method has not been sufficiently demonstrated.
questions: questions: Refer to Weaknesses.
limitations: limitations: yes

Comments: title: Rating
comment: We hope our response has addressed your doubts and concerns. In which case, we kindly urge you to reconsider the rating of our paper accordingly. We remain at your disposal for clarifying any additional concerns.

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Thanks for the rebuttal. I will keep my score.
Thanks for your reply. Based on this response, I checked the manuscript again. I have raised the score from 4 to 5.
Given the authors’ response, I will maintain my rating.