Prioritized Federated Learning: Leveraging Non-Priority Clients for Targeted Model Improvement
Authors: Aditya Narayan Ravi
Ilan Shomorony
Venue: Accepted by TMLR ()

Abstract: Federated Learning (FL) is a distributed machine learning approach to learn models on decentralized heterogeneous data, without the need for clients to share their data. Many existing FL approaches assume that all clients have equal importance and construct a global objective based on all clients. We consider a version of FL we call Prioritized FL, where the goal is to learn a weighted mean objective of a subset of clients, designated as priority clients. An important question arises: How do we choose well-aligned non-priority clients to participate in the federation, while discarding misaligned clients? We present FedALIGN (Federated Adaptive Learning with Inclusion of Global Needs) to address this challenge. The algorithm employs a matching strategy that chooses non-priority clients based on how similar the model’s loss is on their data compared to the global data, thereby ensuring the use of non-priority client gradients only when it is beneficial for priority clients. This approach ensures mutual benefits as non-priority clients are motivated to join when the model performs satisfactorily on their data, and priority clients can utilize their updates and computational resources when their goals align. We present a convergence analysis that quantifies the trade-off between client selection and speed of convergence. Our algorithm shows faster convergence and higher test accuracy than baselines for various synthetic and benchmark datasets.

Prioritized Federated Learning: Leveraging Non-Priority Clients for Targeted Model Improvement
Aditya Narayan Ravi
anravi2@rllinissedu
Electricol and Computer Engineering
University of Mlinois Urbena- Champaain
Ilan Shomorony
ilans edu
Electrical and Computer Engineering
University of Blinois Urbana- Chammpagnm
Reviewed on OpenReview: // openreveee......e.....me net/ forum? id= FRBdvo6g88
Abstract
Federated Learning (FL) is distributed machine learning approach to learn models decentralized heterogeneous data, without the need for clients to share their data. Many existing FL approaches assume that all clients have equal importance and construct a global objective based on all clients. We consider a version of FL, we call Prioritized FI., where the goal is to learn a weighted mean objective of subset of clients, designated as priority clients. An important question arises: How do we choose well-aligned non-priority clients to participate in the federation, while discarding misaligned clients? We present FedALIGN (Federated Adaptive Learning with Inclusion of Global Needs) to address this challenge. The algorithm employs matching strategy that chooses non-priority clients based on how similar the model's loss is on their data compared to the global data, thereby ensuring the use of non-priority client gradients only when it is beneficial for priority clients. This approach ensures mutual benefits as non-priority clients are motivated to join when the model performs satisfactorily on their data, and priority clients can utilize their updates and computational resources when their goals align. We present convergence analysis that quantifics the trade-off between client selection and speed of convergence. Our algorithm shows faster convergence and higher lest accuracy than baselines for various synthetic and benchmark datasets.
1 Introduction
Federated Learning (FL) (McMahan al., 2017 Kairouz et al. 2021 is distributed learning setting where set of clients, with highly heterogeneous data, limited computation, and communication capabilities, are connected to central server. For N clients, the goal of FL is for the central server to learn model that minimizes
F(w) D
(1)
where R' is the model parameter. The objective function F& (w) is the local objective function of the kth client, and pk is the fraction of data at the kth client. A key constraint of FL is that the server wishes to minimize F(w) without directly observing the data at each of the N clients.
A popular method lo solve 11 in the FL setting is Federated Averaging (FedAvg) (Brendan McMahan el al., 2017). FedAvg, E iterations of Stochastic Gradient Descent (SGD) are performed locally at the clients before cach communication round. After the fth local iteration, if corresponds to communication round (i.c., (mod E) 0), a participating client & sends the updated model parameter w to the server, which aggregates them w. Σ Epw,, updates the global model, and shares it with all clients.

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The data across the clients are generally heterogencous, in which case the model updates from the different clients are not well aligned. In Li et al. (2020a), the authors use a heterogeneity measure I, given by
r F" F-= Ex- P
(2)
which captures the gap wetween the global objective's minimum F and a weighted sum of the minima Fi of each local objective Fg. A convergence analysis for FedAvg (in the strongly convex case (Li et al. 2020a) implies that the expected convergence error E[F(wT)] F at time step T scales as
E[F(wr)] - F IC/T)
(3)
where C' is a constant. Intuitively, more severe misalignment between the local and global objectives leads to larger I and to worse training performance. In particular, this suggests that FL systems that operate with a set of clients with reasonably aligned objectives may perform better.
Prioritizing Clients in the Model. The idea of selecting a subset of clients to include in each communication round in order to tackle heterogeneity is not new (Fu et al.
2022 Karimireddy et al., 2021 Li et al. 2018). In addition to reducing heterogeneity, client selection strategies have been proposed in the FL literature to accelerate convergence rates (Cho et al.,
2020 Chen et al. 2020). promote fairness (Sultana et al.
2022 and bolster robustness (Nguyen et al. aggregation process based on the task at hand, yet they cannot account for real-world applications with intrinsic prioritization.
2021 These methods emphasize specific clients during the
Specifically, in various real-world applications, some clients are inherently prioritized, compared to others. In subseription-based services, for instance, it is often the case that some clients have a paid subscription, while others use a free version of the service. One example is online streaming services, which may provide perks such as ad-free content only to paid subscríbers. In such cases, the contral server may want optimize a model (e-g., a recommendation system) that prioritizes the paid subscribers, while still taking advanlage of the data provided by all clients. Another example are internet service providers, which have set of paying clients, but may also provide free-access Wi-Fi hotspots. In such cases, the free service may be provided in exchange for some input from the clients, such as data or computational resources. These clients are likely to have highly heterogeneous data, and may not be willing to directly share their data, which fits naturally into a FL setting.
In these cases, the global objective function F(w) should only take into consideration the priority clients, and it is important to understand whether the non-priority clients can be leveraged to improve the prioritized global objective. Is it possible to incentivize "well-aligned" non-priority clients to participate and contribute to model improvement, while discarding the updates from "misaligned" clients? To formalize this question, we propose the Prioritized Federated Learning (PFL) setting. We discuss additional motivating scenarios for PFL in Section 2
Proposed Model. In PFL, there are N clients, and subset P C [1 N] is designated as priorily clients. Similar to ! the goal is to minimize F(w) = ΣKEP (w)) The remaining clients are designated as non-priority clients. Non-priority clients, if chosen wiscly and incorporated into the learning process, can potentially accelerate the model's convergence through additional gradient steps. To facilitate this, we propose to devise an algorithm capable of discerning well-aligned non-priority clients and aggregating their updates along with those from the priority clients. The critical challenge here lies in finding the right balance: including more non-priority clients might expedite the convergence but could also introduce bias into the global model, especially if these clients are not well-aligned with the primary objective. Conversely, implementing more stringent solection criteria ensures only the most well-aligned clients contribute, but at the expense of faster convergence from additional updates. Against this backdrop, our paper will address two questions that arise in this setting: (1) What criterion should we use to effectively select well-aligned non-priority clients without compromising the integrity of our learning objective? (2) Can we establish theoretical framework that quantifies the trade-off between accelerated convergence and potential bias introduced by including updates from non-priority clients?
FedALIGN and Our Contributions. To address the questions proposed for this PFL sctting, we propose FedALIGN (Federated Adaptive Learning with Inclusion of Global Needs). FodALIGN's strategy for non- priority client inclusion can be combined with a variety of existing algorithms. In every communication

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round, the server transmits not only the global model but also its associated loss to all participating clients. Priority clients invariably contribute to the global model's updates. Non-priority clients, on the other hand, participate only if their local loss is on par with or lower than the communicated global loss. Notice that this is a key step in incentivizing client participation, since well-aligned non-priority clients only participate if they gel a satislacyory model. The server then incorporates only those updates from clients that are close to the global loss.
The core advantage of FedALIGN lies in its adaptive capacity. It exploits well-aligned non-priority clients when their contribution is beneficial, also incentivizing their participation. At the same time, it prevents potential harm to the model from misaligned clients, thereby protecting the interests of priority clients. Theoretical analyses suggest that FedALIGN effectively navigates data heterogeneity amongst priority clients, accelerating convergence at the expense of a manageable bias term. This bias can be subsequently removed in later communication rounds allowing FedALIGN to converge to the optimal solution.
We present the FedALIGN algorithm as modification of FedAvg (although it can be built upon other algorithms too, discussed in the supplementary material). To analyze its performance, we adapt the convergence analysis from Li et al. (2020a). For FedALIGN, the convergence error is given by:
C C+ 0TT T
B[F(wT)] F F
+ PT.
(4)
The factor 0r E [0, 1] captures the reduction in hetcrogeneity achieved by FedALIGN. The more non-priority clients are included in all communication rounds till T, the lower the value of 0r. This is balanced with bias term Pr, which captures the misalignment in the selected non-priority clients.
The terms OT and pr exhibit an inverse relationship and depend on the desired proximity between global and local losses. Specifically when 0r = and Pr = O (we show in Section 3 that this choice achieved by ignoring all non-priority clients), we recover enables us to fine-tune these terms, favoring a rapid convergence initially, and gradually adjusting the bias as needed. Indeed experiments implementing FedALIGN shows that exploiting this trade-off leads faster convergence and better performance over baseline algorithms, both on synthetic data and many benchmark datasots (FMNIST, EMNIST and CIFAR10).
the convergence rate for FedAvg. The tradeoff between Or and pr
Prioritization and Fairness in AI. In our exploration of PFL, it is crucial to talk about fairness. At the heart of this discussion is the fact that PFL is developed to model real-world scenarios such as subscription- based services where a large fraction of the population (non-subscribers) are inherently excluded from the lederation, since the goal of the server is to optimize the model for the priority clients only. In this context, FedALIGN can be seen as way to attain higher fairness in the PFL setting. To explain this note if one were to utilize FedAvg in PFL, there would be two possible options: (i) using FedAvg only the priority clients and none of the non-priority clients (say MolAvy--Noe) or (ii) using FedAvg all priority and non-priority clients (say Ael.
A natural definition of fairness in this case is the fraction of the overall population of clients that has access to a model that is useful to them. This definition fairness used is inspired by many works Zafar et al. (2017); Mohri et al. (2019); this definition, (i) would lead to little fairness (but better model performance for priority clients), and (ii) would lead to higher fairness, but worse model performance, again for priority clients. In real-world settings (ii) will not be accepted by priority clients, since they do not stand to benefit by including non- non- priority clients. Therefore FedALIGN attains higher fairness in the PFL setting (although not as high as a full FedAvg-All which is not realistic). The priority clients stand to benefit from this (as they will ultimately receive a better model) and will agree to the model being shared with non-priority clients, and non-priority clients stand to benefit (at least some fraction of them) from gaining access to a well-trained model, even though they are not subscribers/ baasscriberspriortty-iiiii prietos Thus FodALIGN naturally leads to the most realistically fair algorithm for the subscription based setting we model.
; Li et al. (2020b), but is adapted to be more relevant to the PFL setting. Under

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2 Problem Setting and Related Works
In PFL, we assume that there are N clients, and client &o has the local dataset BR, with [B&| = Dr. The clients are split into two sets: priority clients P C [1 : N] and non-priority clients. All clients are connected to a central server, who wishes to minimize the objective
ΣKEP Ph
(5)
where Ph: := D Dj is the fraction of data at the kth client and Fs(w) :- EeEB, ((w:E) is the local objective function of the kth client. Note that since we normalize the data with the total data possessed only by the priority clients, ΣtEP Pk= = 1, but Σ p& # in general. We denote t to indicate the number of local updates. In each communication round, each client that participates in that round, independently runs E local epochs. Aggregation happens when & (mod E) =0. 0.
Other Settings with Client Prioritization. The need to prioritize a subset of clients may arise in contexts other than subscription services discussed in Section I For instance, existing FL settings that focus on optimizing objectives defined by their clients may face challenges when integrating additional clients. While these new clients may be valuable to the FL sctting, their inclusion often necessitates shift in the lcarning objective. FedALIGN provides a partial solution to this predicament, enabling the inclusion of well-aligned clients without necessitating alterations to the learning objective.
Another context that naturally lcads to prioritization of some clients, is when some clients are slow, unreliable, or resource-poor. This leads to the "straggler effect", ", which can significantly hamper the overall learning process (Wang et al. 2020a| Hard et al. 2019 Nishio and Yonetani 2019). The straggler effect occurs because the global model update must wait until slower clients complete their local computations, and the overall update time is when the overall computation time is dictated by the slowest participants. In many scenarios it is usually known which clients are likely to be slow. An example is households with multiple smart devices, where models are trained with few powerful devices like smart phones and laptops, with additional less powerful devices like IOT devices (with potentially different data) also available for computation. In this case it may be fruitful to exclude stragglers from the global objective and only include their updates if they are well-aligned to this objective. Moreover in our supplementary material we extend our convergence analysis to the case where non-priority clients are free to participate in aggregation as they want to, naturally modelling stragglers who may only be able to provide updates in a few rounds.
Selfish Federated Learning. To the best of our knowledge, only one work (Anonymous) 2022 has explored model similar to ours, which they call Selfish Federated Learning. In the Selfish paradigm, it is assumed that non-priority clients always contribute to the federation irrespective of what global model they receive, an assumption that may not hold true in practice, since clients need to be incentivized to participate. Secondly, the previous model assumes that the number of priority clients are much smaller than the total clients. This assumption can be overly restrictive, limiting the model's 's applicability in varied scenarios. Furthermore, their algorithm exhibits convergence only under highly stringent learning rates, thereby further confining its practical utility. Furthermore, regarding client participation, the model requires involvement from all non-priority clients in each communication round. This expectation is impractical and significantly impacts communication efficiency. Lastly, it presumes identical data distribution among priority clients, which essentially aligus it with the framework of Personalized FL, rendering the premise of their approach somewhat trivial. Our work, in contrast, imposes no such assumptions, but instead looks at a more pragmatic scenario as outlined in the introduction. While an other study (Chayti et al. 2021) has investigated scenarios where clients not involved in the global objective can still contribute to its improvement, these works necessitate full participation. These approaches are incompatible with federated learning contexts. Our model and algorithm naturally addresses these challenges.
Personalized FL. Various FL strategics have been explored to create personalized models. One common approach ves adapting a global FL model to each local client's specifie requirements (Wang et al. 2020b
Yang et al. 2021 Chai et al. 2020 Smith et al. 2018), while another leverages inter-client relationships to fine-tune personalized models, often achieved by introducing adjustable regularization terms into each client's 's local objective (Kairouz et al.
2019 Karimireddy et al.] 2021 Cheng et al., (021).. In these scenarios, a universally applicable model is usually maintained, which is subsequently refined to suit individual clients'

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local data. Furthermore, clustering techniques (Sattler et al. 2019) have been employed, wherein clients are categorized based on the similarity of their data, leading to the development of models tailored to each cluster's unique characteristics. The end goal remains the same: the customization of models for each participating client during the aggregation phase. For a comprehensive review of personalized FL, we refer to the survey in Tan et al. (2022). Our approach deviates from these strategies two critical ways: (1) Our global objective is constructed solely from pre-sclected subset of clients, who might exhibit substantial heterogeneity, and (2) Our training does not explicitly weigh the importance of non-priority clients. Instead, train a single model that inherently benefits non-priority clients that are well-aligned.
Tackling Heterogencity in FL. Many methods aim to tackle data heterogencity in FL. FedProx rox (Kiirouz et al. 2019) adds proximal terms to local objectives and SCAFFOLD (Kariniredyy et al. 2021 client drift. There are many other relevant works which aim to tackle heterogeneity (Reddi et al.,
tackles
2020
Haddadpour and Mahdavi 2019; Khaled et al. 2019 [Stich and Karimireddy 2021 Woodworth et al.
2020
Koloskova et al.) [2021] Zhang et al.) 2021 Pathak and Wainwright 2020). These methods aims to tune a global model to work well on all local clients. FedALIGN also largets heterogeneity, but specifically tries to reduce the heterogeneity for priority clients by leveraging non non-priority clients. The selection approach of
FedALIGN may be combined with other algorithms to improve their performance in this sctting.
Client: Selection in Federated Learning. Several existing methods employ sample re-weighting strategies based on individual client utility. Some approaches define client utility based on the norm of gradients (Zeng jet al., 2021), while others leverage local loss values as the utility measure (Chen et al. 2020). One particular study biases weights in favor of clients with higher losses for faster convergence towards the global objective, albeit at the expense of an enduring bias term (Cho et al.] 2020). Alternatively, some approaches increase weights on clients with lower losses, mitigating the risk of clients with noisy data adversely influencing the global model (Ghadikolaei et al., 2019). Our approach differs from these strategies. We employ a loss matching strategy, wherein non non-priority clients only share models when the received global model yields sufficienlly low loss on their local dataset.
While these client, selection algorithms may provide ideas for algorithm development in PFT., they are designed for the classical setting of FL and do not provide theoretical guarantees for PFL and are likely ill-suited for PFL.
Table 1: Comparison of Client Selection Strategies
Method
Server level decisions
Loss-based Utility
Gradient- based Utility
Zeng al.
22021
X
Chen et al.
(2020)
/
X
Cho et al.
[2220]
/
Ghadikolaci ct al.
2019
X
Wang et al. (2020a)
V
Fu et al. (0023)
X
FodALIGN (Our Method)
While client selection as a concept is similar to our problem setting, the methods used for client selection in classical settings often use updates from all the clients while making decisions, which is unrealistic in the PFL setting. In FL, there are two kinds of communication: downstream from server to clients and upstream from clients to server Sattler et al. (2020). In FedALIGN, the server broadcasts downstream while only aligned non-priority clients communicate upstream. This is different from current methods, which typically require full communication from all clients. This makes FedALIGN communication effcient in the PFL setting. For example, Wang et al. (2020a) presents a loss-based selection algorithm by picking clients with the highest. loss. This however requires you lo make server level decisions on which client has the highest, loss, implying that the non non-priority clients would all be participating in every communication round, which is unrcalistic in the PFL setting, and FodALIGN is specifically designed to not require all con-non-priorte clients to participate. Similarly, there are other works involving gradient- based selection criteria (Fu et al. 2023), which also require such server-level decisions.
all clients,

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Notably, gradient comparisons are more effective than loss comparisons since gradient alignment is better indicator of model aligmment at a given time (Zeng et al, 2021 1. Moreover, absolute loss based methods like those proposed by Cho et al. (2020) offer an advantage in terms of efficiency, as they eliminate the bottleneck caused by repeatedly determining the client with the lowest loss. Sorting clients based on absolute loss values, rather than comparison with relative loss values like FedALIGN, is compulationally more efficient. It. is thus important to acknowledge that other methods have their own advantages and may be more useful in different. contexts. For example, absolute loss-based methods Cho al. (2020); Chen et al. (2020); Ghadikolaei et al. (2019) can be particularly effective in scenarios where rapid convergence is critical, such as in sensitiee applications where communication is not the bottleneck, but computation is. Thus, while our approach offers significant benefits in terms of communication efficiency (since we do not require server level decisions) in the PFL setting, specific methods may prove more advantageous in other scenarios such computational efficiency.
3 FedALIGN and Convergence
In this section we describe FodALIGN and prove its convergence guarantec. Our results are presented for the full client participation setting, though the analysis can be easily extended to the case where subset of clients (both priority and non- prioriti are picked uniformly at random. Further we can extend the result for more general case, where non-priority clients follow an arbitrary participation rule or are given the choice to participate in any given round. This is relevant in the stragglers setting, for example. These cases are considered in the supplementary material.
3.1 FedALIGN
FedALIGN introduces a simple decision rule to select clients basod on the relative discrepancy between local and global loss. Non-priority clients are conditionally included in the aggregation process during a communication round corresponding to the Ith local round, if for the provided global model Wt, and chosen threshold e the absolute value of the difference between non-priority client's local loss and the global loss is below e;; i.e., [F(w:)- F& (w:)) S q.- This method ensures that updates contributing to the global model do not diverge significantly from the overarching learning objective, which is determined by the priority clients. By strategically aligning the updates from non-priorin clients with the global learning objective, FedALIGN can effectively harness the collective learning power of the entire network, while maintaining focus on the primary learning task. The exact steps taken are provided in Algorithm [
In practice, during communication round, along with the model paramcter Wt. the server sends the value of accuracy the global data. The non-priority clients send back updates, only if the accuracy (accuracy is used as proxy for loss) of the received model on it's local data is high enough. This is done to ensure that clients only communicate when they are well-aligned. The server then decides to aggregate it to the global model if it is well aligned, based on how close the accuracies are; Le., the full condition [F(w))) - F& (w.)) S E is satisfied.
3.2 Convergence Analysis
Before stating the main theoretical result the convergence of FedALIGN, we present key assumptions and notation. We make the following assumptions on local objective functions Fi...., Fn. Assumptions [ and are standard in optimization, while Assumptions 3 and 4 are standard in distributed learning and FL
literature Li et al. 2020al Cho et al. 2020 Zhang et al. 2013 Stich 2019
Assumption 1. (Smoothness) For each f E (Fi....,F f is L-smooth, that is, for all u and v, f(v) S (u) (v u)' + =1l2.
Assumption 2. (Converity) For each J JE (F1,..., is y-sttrongly convex, that is, for all and v, ((v) f(1) + (v- u)" f 2ll
Assumption 3. (Unbiased and Bounded Variance of Gradients) For a mini-batch &n randomly sampled from B& from user k. the SGD is unbiased; i.e., E[FF&(WH . gradients is bownded: F Ex) FF&((&))) S d² for k = I,...,N.
{x) Fww(w)). Also, the variance of random

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Algorithm 1: FedALIGN: Adaptive Client Selection in Prioritized Federated Learning
Input: Data Fraction: P, Communication Rounds: T, Initial model: wo: Threshold: c E [1 :T],
Local Epochs: E
Output: WT
W. <- Wo
for each communication round f = 0, E.... do
// Server executes:
1-+1// 1 1/ data fraction
Broadcast We, ((w)( to all clients
W++E +-
for each client E do
// Client k executes:
W -- ClientUpdato((k w.) // Server executes:
W4E t- W + Pk P&
end
for each client & $ P do
// Client executes: Compute local loss Fw(We)
if (w,,,) F(w)) + cz then wH+ + ClientUpdate(k,, w.) return (l(),)
// Server executes:
if (we))) 2 F(w.) E then WI+E &- Wi+E + p&
p p Pk
end
else
Remain silent for this round
end
end
// Server executes:
WIE t Wt+E/P
end
return
Function ClientUpdate w, 17, E, b)
B& < local dataset of client h
B - (split B, into batches of size b)
for I....,E do for each batch do (w)
end
end
return
Note that in general assume that each client has distinct variance bound ok- Assumption 4. (Non-Divergent Gradients) The stochastic gradient's expected squared norm is uniformly bounded, i.e., BillVF (IIIVF((w&:E))1) [x) G2 for l =l,...,, N.
As 12, we define r = F' Ekep Pk Fr be the heterogeneity parameter, but here only defined for the priority clients P. larger I implies higher degree of heterogeneity. When r 0, the local objectives and global objectives consistent. Recall that the index counts the number of local updates, and each

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client runs E local updates before sending the updates when (mod E) 0. Moreover also define T() := max l' (mod E) 0, I t) as the time at which the last communication round occurred. := Fx(w")- - Fi, which is the objective gap between the optimal global model and the
Further define I's := optimal local model.
Here we present the convergence result for FedALIGN the full participation scenario (partial participation is considered in the appendix) and compare it to existing results. Suppose FedALIGN runs for T iterations, for T such that T (mod E) O and outputs Wr the global model.
Theorem 1. (Convergence) Suppose that Assumplions SL rate T p(t+t)); where max {*r ( EE} and any E 2 0. The expected error following T total local iterations (equivalent to T/E communication rounds) for FedALIGN under full device participation satisfies
2
3 and 4 hold, and we set a decuying learning
1 [[F(WT)] F" T+y (C1 + C20,T)+ PT:
(6)
where Or E [0. and pr can be adjusted by selecting an appropriale in each time slep (pT and Or are defined in paragraphs after the theorem statement). Here, C1 and C2 are constants given by
G = 2T.
+ 4T,2 |[o-- -w"
(o²
C2 = 1212
122
This theorem is proved in the supplementary material. We now define and explain some of the important constants we introduce in the result.
Alignment of non-priority clients Or. The expression for Or is
0T 0r:B
T + 2 1m1 1+ Enep Ph P kr(())
where TR, : 1l (w?) F Fw < ) is an indicator random variable for whether a non-priority client is included in a given communication round. The alignment of non-priority clients, 0r, effectively measures the average inclusion of non-priority clients over T time steps. Lower values of fr correspond to a greater number of clients being included aggregation. Specifically the convergence rate improves to O(0r each time step. This kind of improvement due to client selection strategies has been considered before Cho
let al. 2020
Tunable bias pr. The expression for pr is
2L
Exep Palk Palkott kggp Pke (.)(
PT (T 27
E
(8)
This term is an average aggregation of the bias introduced in the first T - time steps.
The parameters Or and Pr effect the convergence of the learning model under consideration, opposite ways. The choice of 4 at cach time step plays crucial role in controlling this balance. If 4 is set too low, it risks under-utilizing the potential of well-aligned clients, thereby causing slower rale of convergence. On the other hand, if e, is set too high, it introduces high bias in the aggregated model updates, which negatively impact the model's performance.
Optimality of FeIALINN. The above dichotomy highlights the fine-tune CCheng el al. 2021 aspect of the FedALIGN algorithm. We can thus set e, such that is gradually reduced in the later rounds. This strategy ensures that the algorithm initially benefits from the contributions of larger number of clients (alboit potentially introducing some bias), but as the rounds progress and the model closer to convergence, the bias can gradually be eliminated by reducing et- This enables the algorithm to leverage the power of well-aligned

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clients to expedite convergence initially, and then gradually reduces the bias to ensure accurate model learning. Specifically we can ensure that. FedALIGN will converge to global optimum ér approaches O when T tends to infinity.
Corollary 2. (Optimal Convergence) Assuming that the conditions in Theorem hold and that + 0 as T + 00, we have
lim E[ T-x
Corollary 2 is straightforward consequence of Theorem and is proved in the supplementary material. Further, in the supplementary material, we conduct experiments that set eT to be distinct monotonically decreasing functions of T. We use our benchmark datasets to evaluate the effect of different decay rates of er on convergence.
Consistency of our result with FedAvg on priority clients. If we set 6, = 0 for all time steps in FedALIGN, fr (the aligmment of non-priority clients) becomes implying that all priority clients are included in the aggregation. At the same time, pr (the tunable bias) becomes 0, indicating that no bias is introduced in the model updates. Under these conditions, FedALIGN is identical to FedAvg on just the priority clients, and the convergence bound is precisely the result in Li et al. (2020a). It shows how FedALIGN builds upon FedAvg by introducing additional flexibility and control through the & parameter, which allows us to manage the tradc-off between bias and speed of convergence. We also highlight that we can modify other algorithms like FedPROX (Kairouz et al., 2019) to obtain a similar control. We show experimental results in our supplementary material to validate this.
4
Experiments
We conduct experiments various benchmark datasets: FMNIST, balanced EMNIST and CIFAR-10 (Xiao ct al.
2017). The datasets were preprocessed following the methods developed in Brendan McMahan et al. (2017) For FMNIST we use a logistic regression model, for EMNIST we use a two-layer neural net work and for the CIFAR-10 dataset we use a CNN. The dataset is generated by distributing uni-class shards to each client, following the steps in Brendan McMahan et al. (2017). More details about the architectures are deferred the supplementary material. In addition, in order to assess the effect of noisy and irrelevant data in the non-priority clients, we modify the SYNTH(1,1) 1) dataset considered in Li et al. (2018). We use logistic
regression model for this experiment.
We evaluate FedALIGN (built on FedAvg) against two baselines: (1) FedAvg on only the priority clients and (2) FedAvg on all clients. Further in our supplementary material we consider FedALIGN built on FedPROX LLi et al. 2018). We consider the full participation case, where all clients are sent a global model by the server and consider the case of partial participation (where only a randomly sampled subset of clients are chosen in each round) in the supplementary material.
Experiment results on benchmark data. For the learning rate (17), we adopted value of 0.1 for both FMNIST and EMNIST datasets, whereas for CIFAR-10, utilized a lower learning rate of 0.01. The selection of these learning rates was made after conducting a grid search of different learning rates on small sets of the data. The selected learning rates resulted in stable convergence and exhibited the best performance. All experiments were conducted with 5 different seeds for randomness, setting the local epoch E = 5. We expand on the impact of varying epoch choices in the supplementary material. The initial 10% of communication rounds were dedicated as warm-up rounds during which only priority clients contributed to the aggregation process. This design was to ensure the development of reasonably working model prior to the inclusion of non non-priority clients. We found by optimizing over grid of different e's that the selection e = 0.2 and eventually reducing it to 0, performs the best. The results depicted in Figure [ show the superiority of FedALIGN over the baseline methods. This is evident in the post-warm-up phase, where there is a noebbeele acceleration in convergence rates. Interestingly, it was observed that the inclusion of all clients often led to a degradation in overall performance, further affirming the importance of selecting well-aligned clients. We further explore alternative configurations of priority client selections with varying data fractions in the supplementary material.

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Figure Test accuracy for benchmark datasets, under full participation, with 2 heterogeneous priority clients and N 60 clients, with = 5. After the first warm-up rounds, a clear increase convergence rate observed for FedALIGN (green), while deterioration observed for All (orange) (FedAvg with all clients). FedALIGN also achieves higher final accuracy.
Experimental results on synthetic data. We simulate scenarios to analyze the performance of non-priority clients with varying degrees of alignment. For this purpose, utilize the SYNTH(1, dataset referred to in Li et al. (2018). We create a global dataset and distibute this data to non-priority clients and add two forms of noise: (1) Discrepancies in label assignments for data points, and (2) the addition of irrelevant data points, which are generated by independent distributions. We configure the noise to ensure that the non-priority clients experience varying degrees of noise, thereby influencing their alignment with the objective. The detailed methodology for the generation of this noise is provided in the supplementary matcrials, We apply logistic regression model to this data, maintaining learning rate of 0.1, with local models used to find the best and stable learning rates, as in the prior experiments. We consider 3 different average levels across non-priority clients: medium, low and high. We found that 0.2 is the best choice for the selection parameter for low and medium noise, while e = 0.4 is the best choice for the high noise experiment. From Figure 2 it becomes evident that FedALIGN consistently outperforms baselines, under this more complicated form of misaligument in the non-priotity clients. This underscores the robustness of FedALIGN even in challenging conditions.
5 Concluding Remarks
We proposed Prioritized Federated Learning, a FL setting that modols the real-world situation where some clients are given priority from the point of view model training, but other clients may choose to participate in the federation as well. We introduced simple algorithm (FedALIGN), that smartly selects non- con-priority clients to help achieve the goals of priority clients. This creates win-win situation where non-priority clients that align well with the system obtain strong global model that can enhance their local tasks. The convergence analysis of FedALIGN highlights an important trade-off: using more priority clients can speed up convergence but introduces a bias. We show that this bias can be eliminated in later rounds. Our extensive experiments support these findings.
Generalized PFL. One can conceptualize traditional FL and PFL. as two ends of a spectrum. On one end, traditional FL involves weighing the objectives generated by both priority and non-priority clients according to their respective data fractions. This ensures that each client, regardless of priority status, contributes proportionally to the overall model training. On the other end of the spectrum, PFL focuses exclusively the objectives generated by priority clients, effectively assigning a weight of zero to the non-priority clients. This approach prioritizes the specific needs and data characteristics of the priority clients, ensuring that the model tailored to their unique requirements. An intermediate approach would involve assigning varying weights to the objectives generated by priority and non-priority clients, with a greater emphasis on the priority clients. By adjusting these weights, we can control the influence of each client group on the model training process. This allows for flexible and nuanced approach, balancing the benefits of penalization for

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Figure Training loss and test accuracy for the SYNTH(1,1) dataset, under full participation, with heterogeneous priority clients and N 20 clients, with F. The first. graph considers the case where there is medium noise, the second considers the low noise case and the third considers the high noise case. After the first 20 warm-up rounds, clear improvement is observed for FedALIGN (green) for all these cases.

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priority clients with the broader generalization capabilities of traditional FL. Exploring this spectrum and adjusting the weights accordingly presents an exciting direction for future research.
On the willingness of non-priority clients to participate. We argue that the receipt of a well-performing global model is one of many incentives for non-priority clients to contribute to the aggregation. our supplementary material, we experimentally show numerous instances where the globally trained model outperforms locally trained counterparts. During instances where the received model does not surpass the performance of locally trained model, primarily due to the latter's fine-tuned adaptation to specific client characteristics, we posit that globally trained models help refine local models. This concept of leveraging external data for local fine-tuning has been the subject of extensive rescarch in Personalized FL (Tan et al. 2022). Furthermore, we assert that access to a global model that demonstrates satisfactory performance on local client data is invariably beneficial. contributes to the suite of tools available to the local client to potentially improve overall performance. compelling future direction is thus formalize these incentives as utility functions.

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Reviews:
[{"review_id": "VxPPBxfT1l", "reviewer": "Reviewer_hCpB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- The framework is interesting and could be leveraged to explore further research. For example, the authors mention several times the idea of ``incentivizing'' the clients which may be cast in a bandit framework. \n- The algorithm is simple which ensures a simple integration in usual frameworks.\n- The theoretical analysis provides meaningful practical insights on the implementation of the algorithm.\n- The experiments demonstrate the benefits of the proposed approach on problems that satisfy the assumptions.\n\n**Weaknesses**\n- The paper will benefit from finding common ground with previous approaches to better understand its benefits. The authors insist that direct comparisons are not possible. I believe that the paper would actually benefit from finding common grounds to understand how well the algorithm performs according to some metrics compared to others. To give an example of such a presentation, in the context of heterogeneity, numerous distributionally robust methods have been designed. Such methods present tradeoffs between accuracy on the whole population and accuracy on the tail population. So in such papers both metrics (on the whole population and on the tail population) are presented. All methods can run but present different advantages.\n- To give a full perspective on the approach, sensibilities to diverse hyperparameters should be evaluated and summarized. For example, sensibility to the thresholds $\\epsilon_t$ need to be discussed. A plot of the final accuracy (or the area under the curve of the accuracy along time to capture convergence speed) versus the choice of epsilon would be helpful. Similarly, as pointed out in the Appendix, the benefits of the approach can vary dramatically with the proportion of priority clients. A plot summarizing such sensibility would be more helpful than a collection of optimization curves for different proportions. Same for the sensibility to noise: the experiments are interesting but could be summarized in a more meaningful way.\n\nOverall the relevance of the approach may be questionable but it's ok for a publication at TMLR. Nevertheless a complete analysis of the approach with all its perks and pitfalls is necessary.\n\n**Additional remarks and questions**\n- Rather than coming up with their own definition of fairness with a lengthy argumentation (though an interesting one for philosophical discussion), the authors should consider appropriate references on fairness to strengthen their point.\n- Eq. (5) typo $F_i$ should be $F_k$\n- \"While client selection as a concept is similar to our problem setting, the methods used for client\nselection in classical settings often use updates from all the clients while making decisions, which is unrealistic\nin the PFL setting. Therefore these algorithms do not serve as natural/direct baselines for our setting\" -> I really don't understand why it would be unrealistic. The current algorithm also uses communications with all clients. The \"unrealistic part\" needs to be illustrated with a concrete experiment.\n- In Algo 1:\n  - Give names to the input parameters to facilitate its reading\n  - I think there are mistakes about the definition of $p$. It's supposed to be a data fraction (so between 0 and 1). But it is inialized at 1 and then updated as $p \\leftarrow p + p_k$ which makes it greater than 1.\n- \"We argue that the receipt of a well-performing global model provides a compelling incentive for non-priority clients to contribute to the aggregation\" What is a compelling incentive? The right framework may be to consider incentives as utility functions (see literature from economy). One may then analyze formally what a ``compelling incentive'' is. Right now the associated paragraph lacks scientific ground.\n- (Detail) \"We posit\", \"We argue\" (often in the text): A perk of working in applied mathematics is that arguments may be formalized and tested rigorously. Maybe consider putting comments without evidence in the conclusion. The points raised by the authors can be particularly relevant to build on the proposed approach\n- Why do we need a warmup phase? What happens without the warmup phase? What is the impact of varying the fraction of warmup steps? (Also maybe add a vertical line in the plots to delineate the warmup phase)\n- It seems to me that the proposed framework is close to a risk-seeking learning perspective where more weights are given to conforming examples. (The opposite, risk-sensitive learning, would give more weight to outliers so that the solution is good even on rarely seen examples). Here the authors hav a given prior on what conform means (that is the set of preferred clients). Could the authors comment on such a link and more generic formulation? It is generally odd to think that the proposed framework cannot be cast in any previous generic framework. The approach is surely new but it would help to consider its differences with other approaches in a common framework detailed mathematically.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a new framework for federated learning that targets a predetermined set of priority clients among a larger pool of clients. In this new framework, the authors propose a simple algorithm that can leverage the information of non-preferred clients to speed-up the convergence process. The idea is to incorporate non-preferred clients if they are well aligned with the set of priority clients. A theoretical analysis uncovers the trade-offs between increased convergence speed and additional bias introduced by non-preferred clients. Experiments demonstrate the benefits of the proposed approach on tailored problems.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See the main weaknesses part. The main requested changes are \n- Comparison with prior work on client selection by finding a common ground to understand the benefits of each approach from scientific evidence.\n- Analysis of sensibilities to the thresholds $\\epsilon_t$ and to the proportion of preferred clients. Also maybe summarize analysis of sensibility to alignment of non-preferred clients (i.e. Fig. 2)\n- Analysis to the sensitivity to the fraction of warmup steps. In particular an experiment without warm-up steps would be interesting.\n\nI would also appreciate answers to the additional remarks and questions to the best of the authors abilities.", "broader_impact_concerns": "The introduction of preferred clients in federated learning may have broader economical impacts if adopted.\nThe authors argue that such a framework give more fairness. The viewpoint from a researcher in economy may help discuss such an impact.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gpkIk1joCk", "reviewer": "Reviewer_TRig", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. Assumes the data distribution across clients is such that non-priority clients can be aligned with the global objective. This assumption is overly simplistic and fails to account for highly heterogeneous data distributions common in federated learning environments.\n2. The bias term theta_T is not rigorously defined or justified. The introduction of this term can lead to a model that is heavily biased towards certain client data, reducing overall model performance. It makes difficult for balancing the trade-off between accelerated convergence and the bias introduced by including non-priority clients.\n3. The convergence analysis is limited to full client participation scenarios, which is unrealistic. Partial participation, where only a subset of clients participates in each round, is more common and not adequately addressed.\n4. The algorithm does not scale well with the number of clients. As the number of clients increases, the computational and communication overhead of maintaining and updating the alignment criteria grows significantly.\n5. The epsilon_T was mentioned to be decayed to 0 but in the experiments it was set to be 0.2. \n6. The model adopted were two layer CNN and LR, which was not sufficient to justify the effectiveness of the methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The submission proposes Prioritized Federated Learning (PFL), introducing a novel algorithm FedALIGN (Federated Adaptive Learning with Inclusion of Global Needs). The algorithm addresses the critical issue of client selection in federated learning, specifically focusing on leveraging non-priority clients to enhance model performance for priority clients.\n\n1.Definition of PFL. The objective is to minimize the weighted mean objective of a subset of clients (priority clients).\n2. Introduction of FedALIGN. which includes non-priority clients based on their alignment with the global objective, measured by the similarity of the model’s loss on their data compared to the global data.\n3. FedALIGN employs a matching strategy, ensuring non-priority client gradients are utilized only when beneficial for priority clients.\n4. Adaptive capacity to exploit well-aligned non-priority clients while preventing misaligned clients from negatively impacting the model.\n5. Convergence analysis quantifying the trade-off between client selection and convergence speed.\n6. Introduction of parameters \\(\\theta_T\\) and \\(\\delta_T\\) to balance the inclusion of non-priority clients and the resulting bias.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Requested Changes\n\n1.  Provide a more comprehensive analysis of the proposed method's performance under highly heterogeneous data distributions. Include experiments and theoretical justifications for how FedALIGN handles scenarios with significant data heterogeneity among clients.\n\n2: Extend the convergence analysis to account for partial participation scenarios. Include theoretical proofs and empirical results demonstrating the performance and convergence behavior of FedALIGN under varying degrees of client participation.\n\n3: Propose and implement optimizations to reduce the computational and communication overhead of the alignment criteria. Provide scalability experiments with a large number of clients to demonstrate the improved efficiency and effectiveness of the proposed optimizations.\n\n4: Clarify the intended behavior of the parameter epsilon_T. If epsilon_T is meant to decay to 0, update the experiments to reflect this decay process. Provide a detailed explanation and justification for the choice of all hyperparas including epsilon_T values used in the experiments.\n\n6. Include experiments with more complex and diverse model architectures, such as deep neural networks (DNNs) and advanced convolutional neural networks (CNNs). Demonstrate the effectiveness of FedALIGN across a wider range of model complexities.", "broader_impact_concerns": "NAN", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1RKpWPq5vp", "reviewer": "Reviewer_ufLW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The problem setting addressed in this paper is compelling and relevant to real-world applications. The authors provide practical scenarios where the proposed algorithm could be beneficial. The main sections of the paper are well articulated and accessible, making the concepts and methodologies clear to the reader.\n\nA significant concern is the choice of the threshold value $\\epsilon_t$. The authors mention that a decreasing $\\epsilon_t$ from 0.2 to 0 performed best in their experiments. However, this selection strategy may not generalize well across different datasets or conditions.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces Prioritized Federated Learning (PFL), a variant of federated learning (FL) that optimizes the learning process by prioritizing certain clients. The proposed FedALIGN algorithm selects non-priority clients for inclusion based on the similarity of their data loss to the global model’s loss. This selection aims to enhance convergence rates and model accuracy by incorporating updates only from well-aligned non-priority clients. A theoretical analysis of FedALIGN is presented, demonstrating its potential for faster convergence with controlled bias, and these claims are validated through empirical results on synthetic and benchmark datasets. The findings suggest that FedALIGN effectively manages data heterogeneity.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It is requested that the authors elaborate on the method used to determine the optimal choice of $\\epsilon_t$ in their experiments. Additionally, please provide the computational complexity associated with this search process, which will help in assessing the practical applicability of FedALIGN in diverse settings.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Initially, some of the concerns were comparison with related methods and sensitivity analysis of hyperparameters.

After reviewing the revised version, the reviewers are generally satisfied with the paper.   Two reviewers recommended "leaning to accept" and one recommended "accept".   Since the reviewers did not point out additional revisions, I recommend "accept as is".
Decision: Accept (poster)