# federated_recommendation_with_additive_personalization__26d741c7.pdf

Published as a conference paper at ICLR 2024

FEDERATED RECOMMENDATION WITH ADDITIVE PERSONALIZATION

Zhiwei Li1 Guodong Long1 Tianyi Zhou2

1 Australian AI Institute, Faculty of Engineering and IT, University of Technology Sydney 2 Department of Computer Science, University of Maryland, College Park zhw.li@outlook.com, guodong.long@uts.edu.au, zhou@umiacs.umd.edu

Building recommendation systems via federated learning (FL) is a new emerging challenge for next-generation Internet service. Existing FL models share item embedding across clients while keeping the user embedding private and local on the client side. However, identical item embedding cannot capture users individual differences in perceiving the same item and may lead to poor personalization. Moreover, dense item embedding in FL results in expensive communication costs and latency. To address these challenges, we propose Federated Recommendation with Additive Personalization (Fed RAP), which learns a global view of items via FL and a personalized view locally on each user. Fed RAP encourages a sparse global view to save FL s communication cost and enforces the two views to be complementary via two regularizers. We propose an effective curriculum to learn the local and global views progressively with increasing regularization weights. To produce recommendations for a user, Fed RAP adds the two views together to obtain a personalized item embedding. Fed RAP achieves the best performance in FL setting on multiple benchmarks. It outperforms recent federated recommendation methods and several ablation study baselines. Our code is available at https://github.com/mtics/Fed RAP.

1 INTRODUCTION

Recommendation systems have emerged as an important tool and product to allocate new items a user is likely to be interested in and they deeply change daily lives (Deng et al., 2019). These systems typically rely on servers to aggregate user data, digital activities, and preferences in order to train a model to make accurate recommendations (Das et al., 2017; Zhang et al., 2023b). However, uploading users personal data, which often contains sensitive privacy information, to central servers can expose them to significant privacy and security risks (Chai et al., 2020). Furthermore, recent government regulations on privacy protection (Voigt & Von dem Bussche, 2017) necessitate that user data need to be stored locally on their devices, rather than being uploaded to a global server. As a potential solution to the above problem, federated learning (FL) (Mc Mahan et al., 2017) enforces data localization and trains a globally shared model in a distributed manner, to be specific, by alternating between two fundamental operations, i.e., client-side local model training and serverside aggregation of local models. It achieves great success in some applications such as Google keyboard query suggestions (Hard et al., 2018; Chen et al., 2019). However, the heterogeneity across clients, e.g., non-identical data distributions, can significantly slow down the FL convergence, resulting in client drift or poor global model performance on individual clients. A variety of non IID FL methods (Zhao et al., 2018; Li et al., 2020; Karimireddy et al., 2020b; Ma et al., 2022; Yan & Long, 2023) is developed to address the challenge by finding a better trade-off between global consensus and local personalization in model training. However, most of them mainly focus on similar-type classification tasks, which naturally share many common features, and few approaches are designed for recommendation systems, which have severer heterogeneity among users, focus more on personalization, but suffer more from local data deficiency.

Federated Recommendation. In the quest to foster knowledge exchange among heterogeneous clients without breaching user privacy, scholars are delving into federated recommendation systems. This has given rise to the burgeoning field of Federated Recommendation Systems (FRSs) (Zhang

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et al., 2021). FRSs deal with client data originating from a single user, thereby shaping the user s profile (Imran et al., 2023; Yuan et al., 2023b). The user profile and rating data stay locally on the client side while the server is allowed to store candidate items information. To adhere to FL constraints while preserving user privacy, FRSs need to strike a fine balance between communication costs and model precision to yield optimal recommendations (Tan et al., 2022b). Several recent approaches (Lin et al., 2020; Li et al., 2020; Liang et al., 2021; Yuan et al., 2023a; Zhang et al., 2023c) have emerged to confront these hurdles. Most of them share a partial model design (Singhal et al., 2021; Pillutla et al., 2022) in which the item embedding is globally shared and trained by the existing FL pipeline while the user function/embedding is trained and stays local. However, they ignore the heterogeneity among users in perceiving the same item, i.e., users have different preference to each item and they may focus on different attributes of the item. While PFed Rec (Zhang et al., 2023a) considers personalized item embedding, it neglects the knowledge sharing and collaborative filtering across users via global item embedding. Moreover, FL of item embedding requires expensive communication of a dense matrix between clients and a server, especially for users interested in many items.

Main Contributions. To bridge the gap described above in FRSs, we propose Federated Recommendation with Additive Personalization (Fed RAP), which balances global knowledge sharing and local personalization by applying an additive model to item embedding and reduces communication cost/latency with sparsity. Fed RAP follows horizontal FL assumption (Alamgir et al., 2022) with distinct user embedding and unique user datasets but shared items. Specifically, the main contributions of Fed RAP are as follows:

Unlike previous methods, Fed RAP integrates a two-way personalization: personalized and private user embedding, and item additive personalization via summation of a user-specific item embedding matrix D(i) and a globally shared item embedding matrix C updated via server aggregations.

Two regularizers are applied: one encouraging the sparsity of C (to reduce the communication cost/overhead) and the other enforcing difference between C and D(i) (to be complementary).

In earlier training, additive personalization may hamper performance due to the time-variance and overlapping between C and D(i); to mitigate this, regularization weights are incrementally increased by a curriculum transitioning from full to additive personalization.

Therefore, Fed RAP is able to utilize locally stored partial ratings to predict user ratings for unrated items by considering both the global view and the user-specific view of the items. In experiments on six real-data benchmarks, Fed RAP significantly outperforms SOTA FRS approaches.

2 RELATED WORK

Personalized Federated Learning. To mitigate the effects of heterogeneous and non-IID data, many works have been proposed (Yang et al., 2018; Zhao et al., 2018; Ji et al., 2019; Karimireddy et al., 2020a; Huang et al., 2020; Luo et al., 2021; Li et al., 2021; Wu & Wang, 2021; Hao et al., 2021; Wang et al., 2022; Long et al., 2023; Luo et al., 2023; Ma et al., 2023). In this paper, we focus on works of personalized federated learning (PFL). Unlike the works that aim to learn a global model, PFL seeks to train individualized models for distinct clients (Tan et al., 2022a), often necessitating server-based model aggregation (Arivazhagan et al., 2019; T Dinh et al., 2020; Collins et al., 2021). Several studies (Ammad-Ud-Din et al., 2019; Huang et al., 2020; T Dinh et al., 2020) accomplish personalized federated learning by introducing various regularization terms between local and global models. Meanwhile, some works (Flanagan et al., 2020; Li et al., 2021; Luo et al., 2022; Tan et al., 2023) focus on personalized model learning, with the former promoting the closeness of local models via variance metrics and the latter enhancing this by clustering users into groups and selecting representative users for training, instead of random selection.

Federated Recommendation Systems. FRSs are unique in that they typically have a single user per client (Sun et al., 2022), with privacy-protective recommendations garnering significant research attention. Several FRS models leverage stochastic gradient updates with implicit feedback (Ammad Ud-Din et al., 2019) or explicit feedback (Lin et al., 2020; Liang et al., 2021; Perifanis & Efraimidis, 2022; Luo et al., 2022). Fed MF (Chai et al., 2020) introduces the first federated matrix factorization algorithm based on non-negative matrix factorization (Lee & Seung, 2000), albeit with limitations

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in privacy and heterogeneity. Meanwhile, PFed Rec (Zhang et al., 2023a) offers a bipartite personalization mechanism for personalized recommendations, but neglects shared item information, potentially degrading performance. Although existing works have demonstrated promising results, they typically only consider user-side personalizations, overlooking diverse emphases on the same items by different users. In contrast, Fed RAP employs a bipartite personalized algorithm to manage data heterogeneity due to diverse user behaviors, considering both unique and shared item information.

3 FEDERATED RECOMMENDATION WITH ADDITIVE PERSONALIZATION

Taking the fact that users preferences for items are influenced by the users into consideration, in this research, we assume that the ratings on the clients are determined by the users preferences, which are affected by both the user information and the item information. Thus, the user information is varied by different users, and the item information on each client should be similar, but not the same. Given partial rating data, the goal of Fed RAP is to recommend the items users have not visited.

Notations. Let R = [r1, r2, . . . , rn]T {0, 1}n m denote that the input rating data with n users and m items, where ri {0, 1}m indicates the i-th user s rating data. Since each client only have one user information as stated above, ri also represents the i-th client s ratings. Moreover, we use D(i) Rm k to denote the local item embedding on the i-th client, and C Rm k to denote the global item embedding. In the recommendation setting, one user may only rate partial items. Thus, we introduce Ω= {(i, j) : the i-th user has rated the j-th item} to be the set of rated entries in R.

3.1 PROBLEM FORMULATION

To retain shared item information across users while enabling item personalization, we propose to use C to learn shared information, and employ D(i) to capture item information specific to the i-th user on the corresponding i-th client, where i = 1, 2, . . . , n. Fed RAP then uses the summation C and D(i) to achieve an additive personalization of items. Considering R {0, 1}n m, we utilize the following formulation for the i-th client to map the predicted rating ˆrij into [0, 1]:

min U,C,D(i) X

(i,j) Ω (rij log ˆrij + (1 rij) log (1 ˆrij)). (1)

Here, ˆrij = 1/(1 + e <ui,(D(i)+C)j>) represents the predicted rating of the i-th user for the j-th item. Eq. (1) minimizes the reconstruction error between the actual rating ri and predicted ratings ˆrij based on rated entries indexed by Ω. To ensure the item information learned by D(i) and C differs on the i-th client, we enforce the differentiation among them with the following equation:

i=1 ||D(i) C||2 F . (2)

Eqs. (1) and (2) describe the personalization model for each client, and aim to maximize the difference between D(i) and C. We note that in the early stages of learning, the item information learned by {D(i)}n i=1 and C may be inadequate for recommendation, and additive personalization may thus reduce performance. Thus, considering Eqs. (1), (2), and the aforementioned condition, we propose the following optimization problem L in Fed RAP:

min U,C,D(i)

(i,j) Ω (rij log ˆrij +(1 rij) log (1 ˆrij)) λ(a,v1)||D(i) C||2 F )+µ(a,v2)||C||1,

(3) where a is the number of training iterations. We employ the functions λ(a,v1) and µ(a,v2) to control the regularization strengths, limiting them to values between 0 and v1 or v2, respectively. Here, v1 and v2 are constants that determine the maximum values of the hyperparameters. By incrementally boosting the regularization weights, Fed RAP gradually transitions from full personalization to additive personalization concerning the item information. As there may be some redundant information in C, we employ the L1 norm on C (||C||1) to induce C sparsity, which eliminates the unnecessary information from C and helps reduce the communication cost between the server and the client. The whole framework of Fed RAP is depicted in Fig. 1.

Published as a conference paper at ICLR 2024

Client i ui

Additive Personalization

Server Aggregation

Upload Download

Enforce Difference

Reconstruction Error

Enforce Sparsity

ui User embedding

ri User ratings r i Predicted user ratings

D(i) Local item embedding

C Global item embedding

Figure 1: The framework of Fed RAP. For each user-i, Fed RAP utilizes its ratings ri as labels to locally train a user embedding ui and a local item embedding D(i). By adding D(i) to the global item embedding C from the server, i.e., additive personalization, Fed RAP creates a personalized item embedding based on both shared knowledge and personal perspective and thus produces better recommendations. Since clients and server only communicate C, Fed RAP enforce its sparsity to reduce the communication cost and encourage its difference to D(i) so they are complementary.

3.2 ALGORITHM

Algorithm Design. To address the objective described in Eq. (3), we employ an alternative optimization algorithm to train our model. As shown in Alg. 1, the overall workflow of the algorithm is summarized into several steps as follows. We start by initializing C at the server and ui and D(i) at their respective clients (i = 1, 2, . . . , n). Each client is assumed to represent a single user. For every round, the server randomly selects a subset of clients, represented by Sa, to participate in the training. Clients initially receive the sparse global item embedding C from the server. They then invoke the function Client Update to update their parameters accordingly with the learning rate η. Clients upload the updated {C(i)}ns i=1 to the server for aggregation. The server then employs the aggregated C in the succeeding training round. Upon completion of the training process, Fed RAP outputs the predicted ratings ˆR to guide recommendations. Fed RAP upholds user privacy by keeping user-specific latent embeddings on clients and merely uploading the updated sparse global item embedding to the server. This approach notably reduces both privacy risk and communication costs. Furthermore, Fed RAP accommodates user behavioral heterogeneity by locally learning user-specific personalized models at clients. As such, even though user data varies due to individual preferences, Fed RAP ensures that data within a user adheres to the i.i.d assumption.

Cost Analysis. Regarding the time cost, following Algorithm 1, the computational complexity on the i-th client for updating ui, D(i), and C requires a cost of O(km). Once the server receives C from n clients, it takes O(kmn) for the server to perform aggregations. Consequently, the total computational complexity of Algorithm 1 at each iteration is O(kmn). In practice, the algorithm typically converges within 100 iterations. As for the space cost, each client needs to store ui Rk, D(i) R(m k), and C R(m k), requiring O((2nm + n)k + nm) space. The server, on the other hand, needs to store the updated and aggregated global item embedding, requiring O((nm + m)k) space. Thus, Fed RAP demands a total of O((3nm + n + m)k + nm) space.

3.3 DIFFERENTIAL PRIVACY

Despite the fact that Fed RAP only conveys the global item embedding C between the server and clients, there still lurks the potential risk of information leakage.

Property 1 (Revealing local gradient from consecutive C) If a server can obtain consecutive C(i) returned by client i, it can extract the local gradient with respect to C from the latest update of the corresponding client.

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Algorithm 1 Federated Recommendation with Additive Personalization (Fed RAP)

Input: R, Ω, v1, v2, k, η, t1, t2 Initialize:C Global Procedure:

1: for each client index i = 1, 2, . . . , n in parallel do 2: initialize ui, D(i); 3: end for 4: for a = 1, 2, . . . , t1 do 5: Sa randomly select ns from n clients 6: for each client index i Sa in parallel do 7: download C from the server; 8: C(i), ˆri Client Update(ui, C, D(i)); 9: upload C(i) to the server; 10: end for 11: C 1 ns Pns i=1 C(i); Server Aggregation 12: end for 13: return: ˆR = [ˆr1, ˆr2, . . . , ˆrn]T

Client Update:

1: for b = 1, 2, . . . , t2 do 2: calculate the partial gradients ui L, CL and D(i)L by differentiating Eq. (3); 3: update ui ui η ui L; 4: update C(i) C η CL; 5: update D(i) D(i) η D(i)L; 6: end for 7: ˆri = σ( ui, D(i) + C(i) ); 8: return: C(i), ˆri

Discussion 1 If client i participates in two consecutive training rounds (denoted as a and a+1), the server obtains C(i) returned by the client in both rounds, denoted as C(i) (a) and C(i) (a+1), respectively.

According to Alg. 1, it can be deduced that C(i) (a+1) C(i) (a) = η C(i) (a+1), where η is the learning rate. Since there must exist a constant equal to η, the server can consequently obtain the local gradient of the objective function with respect to C for client i during the (a + 1)-th training round.

Property 1 shows that a server, if in possession of consecutive C(i) from a specific client, can derive the local gradient relative to C from the most recent client update, which becomes feasible to decipher sensitive user data from the client s local gradient (Chai et al., 2020). This further illustrates that a client should not participate in training consecutively twice. Despite our insistence on sparse C and the inclusion of learning rate η in the gradient data collected by the server, the prospect of user data extraction from C(i) (a+1) persists.

To fortify user privacy, we employ the widely used (ϵ, δ)-differential privacy protection technique (Minto et al., 2021) on C, where ϵ measures the potential information leakage, acting as a privacy budget, while δ permits a small probability of deviating slightly from the ϵ-differential privacy assurance. To secure (ϵ, δ)-Differential Privacy, it s imperative for Fed RAP to meet certain prerequisites. Initially, following each computation as outlined in Alg. 1, Fed RAP secures a new gradient w.r.t. C on the i-th client, denoted by C(i), which is limited by a pre-defined threshold τ:

C(i) C(i) min{1, τ || C(i)||F }. (4)

Post gradient clipping in Eq. (4), C(i) is utilized to update C on the client i. Upon receipt of all updated {C(i)}ns i=1, the server aggregates to derive a new C.

Property 2 (Sensitivity upper bound of C) The sensitivity of C is upper bounded by 2ητ

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Discussion 2 Given two global item embeddings C and C , learned from two datasets that differ only in the data of a single user (denoted as u), respectively, we have:

||C C ||F = || η

ns ( C(u) C (u))||F η

ns || C(u)||F + η

ns || C (u)||F 2ητ

Property 2 posits that the sensitivity of C cannot exceed 2ητ

ns . To ensure privacy, we introduce noise derived from the Gaussian distribution N(0, e2) to disrupt the gradient, where e is proportionate to the sensitivity of C. As stated in Mc Mahan et al. (2018), applying the Gaussian mechanism ( 2ητ

ns , z) during a training round ensures a privacy tuple of ( 2ητ

ns , z). Following Abadi et al. (2016), Fed RAP leverages the composability and tail bound properties of moment accountants to determine ϵ for a given budget δ, thus achieving (ϵ, δ)-differential privacy.

4 EXPERIMENTS

4.1 DATASETS

A thorough experimental study has been conducted to assess the performance of the introduced Fed RAP on six popular recommendation datasets: Movie Lens-100K (ML-100K), Movie Lens-1M (ML-1M), Amazon-Instant-Video (Video), Last FM-2K (Last FM) (Cantador et al., 2011), Ta Feng Grocery (Ta Feng), and QB-article (Yuan et al., 2022). The initial four datasets encompass explicit ratings that range between 1 and 5. As our task is to generate recommendation predictions on data with implicit feedback, any rating higher than 0 in these datasets is assigned 1. Ta Feng and QBarticle are two implicit feedback datasets, derived from user interaction logs. In each dataset, we include only those users who have rated at least 10 items.

4.2 BASELINES

The efficacy of Fed RAP is assessed against several cutting-edge methods in both centralized and federated settings for validation.

NCF (He et al., 2017). It merges matrix factorization with multi-layer perceptrons (MLP) to represent user-item interactions, necessitating the accumulation of all data on the server. We set up three layers of MLP according to the original paper.

Light GCN (He et al., 2017). It is a collaborative filtering model for recommendations that combines user-item interactions without additional complexity from side information or auxiliary tasks.

Fed MF (Chai et al., 2020). This approach employs matrix factorization in federated contexts to mitigate information leakage through the encryption of the gradients of user and item embeddings.

Fed NCF (Perifanis & Efraimidis, 2022). It is the federated version of NCF. It updates user embedding locally on each client and synchronizes the item embedding on the server for global updates.

PFed Rec (Zhang et al., 2023a). It aggregates item embeddings from clients on the server side and leverages them in the evaluation to facilitate bipartite personalized recommendations.

The implementations of NCF, Light GCN, Fed MF and PFed Rec are publicly available in corresponding papers. Although Fed NCF does not provide the code, we have implemented a version based on the code of NCF and the paper of Fed NCF. Moreover, we developed a centralized variant of Fed RAP, termed Cent RAP, to showcase the learning performance upper bound for Fed RAP within the realm of personalized FRSs.

4.3 EXPERIMENTAL SETTING

According to the previous works (He et al., 2017; Zhang et al., 2023a), for each positive sample, we randomly selected 4 negative samples. We performed hyperparameter tuning for all methods, the parameter v1 of our method Fed RAP in the range {10i|i = 6, . . . , 0}, and the parameter v2 of Fed RAP in the range {10i|i = 3, . . . , 3}. Given that the second and third terms in Eq. 3 gradually come into effect during training, we pragmatically set the function λ(a,v1) = tanh( a

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µ(a,v2) = tanh( a

10) v2, where a is the number of iterations. For PFed Rec and Fed RAP, the maximum number of server-client communications and client-side training iterations were set to 100 and 10 respectively. For NCF, Fed MF and Fed NCF, we set the number of training iterations to 200. To be fair, all methods used fixed latent embedding dimensions of 32 and a batch size of 2048. We did not employ any pre-training for any of the methods, nor did we encrypt the gradients of Fed MF. Following precedent (He et al., 2017; Zhang et al., 2023a), we used a leave-one-out strategy for dataset split evaluation.

Evaluation Metrics. In the experiments, the prediction performances are evaluated by two widely used metrics: Hit Rate (HR@K) and Normalized Discounted Cumulative Gain (NDCG@K). These criteria has been formally defined in He et al. (2015). In this work, we set K = 10, and repeat all experiments by five times. We report the average values and standard deviations of the results.

Ta Feng ML-100K ML-1M Video Last FM QB-article 0

Ta Feng ML-100K ML-1M Video Last FM QB-article 0

NCF Light GCN Fed MF Fed NCF PFed Rec Cent RAP Fed RAP

Figure 2: Evaluation metrics (%) on six real-world recommendation datasets. Cent RAP is a centralized version (upper bound) of Fed RAP. Fed RAP outperforms all the FL methods by a large margin.

4.4 MAIN RESULTS AND COMPARISONS

Fig. 2 illustrates the experimental results in percentages of all the comparing methods on the six real-world datasets, exhibiting that Fed RAP outperforms the others in most scenarios and achieves best among all federated methods. The performance superiority probably comes from the ability of Fed RAP on bipartite personalizations of both user information and item information. Moreover, the possible reason why Fed RAP performs better than PFed Rec is that Fed RAP is able to personalize item information while retaining common information of items, thus avoiding information loss. Cent RAP exhibits slightly better performance on all datasets compared to Fed RAP, demonstrating the performance upper bound of Fed RAP on the used datasets. In addition, to investigate the convergence of Fed RAP, we compare the evaluations of all methods except Cent RAP for each iteration during training on the ML-100K dataset. Because NCF, Fed MF and Fed NCF are trained of 200 iterations, we record evaluations for these methods every two iterations. Fig. 3 shows that on both HR@10 and NDCG@10, Fed RAP achieves the best evaluations in less than 10 iterations and shows a convergence trend within 100 iterations. The experimental results demonstrate the superiority of Fed RAP. But since Fed RAP is more complex than PFed Rec, it needs more iterations to converge.

0 20 40 60 80 100 Iteration

0 20 40 60 80 100 Iteration

Fed MF Fed NCF PFed Rec Fed RAP

Figure 3: Convergence and efficiency comparison of four methods on the ML-100K dataset.

0 10 20 30 40 50 60 70 80 90 100 a

Fed RAP Fed RAP-sin

Fed RAP-square Fed RAP-fix

Fed RAP-frac

Figure 4: Different curriculum for λ(a, v1) in Eq. (3) with v1 = 0.1 and a to be the iteration.

4.5 ABLATION STUDY

To examine the effects of Fed RAP s components, we introduce several variants: Fed RAP-C, Fed RAP-D, Fed RAP-No, Fed RAP-L2, Fed RAP-fixed, Fed RAP-sin, Fed RAP-square, and Fed RAPfrac, defined as follows: (1) Fed RAP-C: Uses a common item embedding C for all users, ignoring personalization; (2) Fed RAP-D: Creates user-specific item embeddings D(i), without shared item information; (3) Fed RAP-No: Removes item sparsity constraint on C, i.e., ||C||1 is omitted; (4)

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Fed RAP-L2: Uses Frobenius Norm, ||C||2 F , instead of ||C||1 for constraint; (5) Fed RAP-fixed: Fixes λ(a,v1) and µ(a,v2) to be two constants v1 and v2, respectively; (6) Fed RAP-sin: Applies λ(a,v1) = sin( a

10) v1 and µ(a,v2) = sin( a

10) v2; (7) Fed RAP-square: Alternates the value of λ(a,v1) and µ(a,v2) between 0 and v1 or v2 every 10 iterations, respectively; (8) Fed RAP-frac: Utilizes λ(a,v1) = v1 a+1 and µ(a,v2) = v2 a+1. Here, a denotes the number of iterations and v is a constant. Fed RAP-C and Fed RAP-D validate the basic assumption of user preference influencing ratings. Fed RAP-No and Fed RAP-L2 assess the impact of sparsity in C on recommendations. The last four variants explore different weight curricula on Fed RAP s performance. We set v1 and v2 to 0.1 for all variants, and Fig. 4 depicts the hyperparameter trends λ(a,v1) of for the final four variants. The trend of µ(a,v2) is the same as the corresponding λ(a,v1) of each variant.

Ablation Study of Fed RAP. To verify the effectiveness of Fed RAP, we train all variants as well as Fed RAP on the ML-100K dataset. In addition, since PFed Rec is similar to Fed RAP-D, we also compare it together in this case. As shown in Fig. 5, we plot the trend of HR@10 and NDCG@10 for these six methods over 100 iterations. From Fig. 5, we can see that Fed RAP-C performs the worst, followed by Fed RAP-D and PFed Rec. These indicate the importance of not only personalizing the item information that is user-related, but also keeping the parts of the item information that are non-user-related. In addition, both Fed RAP-C and Fed RAP-D reach convergence faster than Fed RAP. Because Fed RAP-C trains a global item embedding C for all clients, and Fed RAP-D does not consider C, while Fed RAP is more complex compared to these two algorithms, which explains the slow convergence speed of Fed RAP because the model is more complex. The figure also shows that the performance of Fed RAP-D and PFed Rec is similar, but the curve of Fed RAP-D is smoother, probably because both have the same assumptions and similar theories, but we add regularization terms to Fed RAP. The results of Fed RAP-No, Fed RAP-L2 and Fed RAP are very close, where the performance of Fed RAP is slightly better than the other two as seen in Fig. 5(b). Thus, we choose to use L1 norm to induce C to become sparse. Also, a sparse C helps reduce the communication overhead between the server and clients.

0 20 40 60 80 100 Iteration

0 20 40 60 80 100 Iteration

Fed RAP-C Fed RAP-D

Fed RAP-No Fed RAP-L2

Fed RAP PFed Rec

Figure 5: Ablation study investigating the effectiveness of Fed RAP on the ML-100K dataset.

0 20 40 60 80 100 Iteration

0 20 40 60 80 100 Iteration

Fed RAP-fixed Fed RAP-sin

Fed RAP-square Fed RAP-frac

Figure 6: Comparison of different curricula for λ(a,v1) and µ(a,v2) in Fed RAP on the ML100K dataset.

Ablation Study on Curriculum. In this section, we investigate the suitability of additive personalization in the initial training phase by analyzing the performance of Fed RAP and its four weight variations (Fed RAP-fixed, Fed RAP-sin, Fed RAP-square and Fed RAP-frac) on HR@10 and NDCG@10 in the validation set of the ML-100K dataset. Fig. 6 shows that Fed RAP-fixed, with constant weights, underperforms Fed RAP, which adjusts weights from small to large. This confirms that early-stage additive personalization leads to performance degradation, as item embeddings have not yet captured enough useful information. Furthermore, Fed RAP-frac, which reduces weights from large to small, performs the worst, while Fed RAP performs the worst, reiterating that early-stage additive personalization can decrease performance. Therefore, we choose to use the tanh function.

Ablation Study on Sparsity of C. Since Fed RAP only exchanges C between server and clients, retaining user-related information on the client side, we further explore the impact of sparsity constraints on C in Fed RAP by comparing the data distribution of C and {D(i)}n i=1 learned in the final iteration when training both Fed RAP and Fed RAP-L2 on the ML-100K dataset. We select three local item embeddings for comparison: D(43), D(586), and D(786), along with the global item embedding C. For visualization, we set latent embedding dimensions to 16 and choose 16 items. Fig. 7 reveals that Fed RAP s global item embedding C is sparser than that of Fed RAP-L2. We also notice that 4.19% of Fed RAP s C parameters exceed 1e-1 and 67.36% exceed 1e-2, while for Fed RAP-L2,

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Figure 7: Sparse global matrix C and row-dense personalized matrix D(i) for item embedding.

these values are 8.07% and 88.94%, respectively. This suggests that Fed RAP s sparsity constraint reduces communication overhead. Additionally, both methods learn similar D(i) (i = 46, 586, 785), with notable differences between distinct D(i). This supports our hypothesis that user preferences influence item ratings, leading to data heterogeneity in federated settings and validating Fed RAP s ability to learn personalized information for each user.

Fed RAP with Differential Privacy. To protect user privacy, we introduce noise from the Gaussian distribution to the gradient w.r.t. C on clients and compare the performance of Fed RAP with and without noise. In this work, we set τ = 0.1, and fix the noise variance z = 1 for Gaussian Mechanism following the previous work (Mc Mahan et al., 2018). We use the Opacus libaray (Yousefpour et al., 2021) to implement the differential privacy. Table 1 presents results for these approaches on six datasets, showing reduced Fed RAP performance after noise addition. However, as Fig 2 indicates, it still outperforms benchmark methods. Moreover, since Fed RAP only requires the transmission of C between the server and clients, adding noise solely to C is sufficient for ensuring privacy protection. This results in a mitigated performance decline due to differential privacy compared to adding noise to all item embeddings.

Table 1: Fed RAP vs. Fed RAP-noise (Fed RAP with injected noise for differential privacy)

Dataset Ta Feng ML-100K ML-1M Video Last FM QB-article Metrics HR@10 NDCG@10 HR@10 NDCG@10 HR@10 NDCG@10 HR@10 NDCG@10 HR@10 NDCG@10 HR@10 NDCG@10 Fed RAP 0.9943 0.9911 0.9709 0.8781 0.9324 0.7187 0.4653 0.2817 0.2329 0.1099 0.5398 0.2598 Fed RAP-noise 0.9536 0.9365 0.9364 0.8015 0.9180 0.7035 0.4191 0.2645 0.2268 0.1089 0.5020 0.2475 Degradation 4.09% 5.51% 3.55% 8.72% 1.54% 2.11% 9.93% 6.11% 2.62% 0.91% 7.00% 4.73%

5 CONCLUSIONS

In this paper, based on the assumption that user s ratings of items are influenced by user preferences, we propose a method named Fed RAP to make bipartite personalized federated recommendations, i.e., the personalization of user information and the additive personalization of item information. We achieves a curriculum from full personalization to additive personalization by gradually increasing the regularization weights to mitigate the performance degradation caused by using the additive personalization at the early stages. In addition, a sparsity constraint on the global item embedding to remove useless information for recommendation, which also helps reduce the communication cost. Since the client only uploads the updated global item embedding to the server in each iteration, thus Fed RAP avoids the leakage of user information. We demonstrate the effectiveness of our model through comparative experiments on six widely-used real-world recommendation datasets, and numerous ablation studies. Also, due to the simplicity of Fed RAP, it would be interesting to explore its applications in other federated scenarios.

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A THEORETICAL ANALYSIS

In this section we give further details on the theoretical results presented in Section 3.4. We start by giving the exact conditions needed for Theorem 1 to hold.

A.1 FULL DISCUSSION OF THEOREM 1

To delineate Theorem 1, we first re-define the formula for updating C(i) at the i-th client based on Alg. 1: C(i) (a+1) = C(a) η C(i) (a+1), (6)

where η is the learning rate, and C(i) stands for the gradients of Eq. (3) w.r.t. C on the client i in the a + 1-th iteration. When the server receipts all the updated {C(i) (a+1)}na i=1, it then perform the aggregation to get C(a+1). Thus, if the client i participate the training in two consecutive rounds,

we can get the difference between C(i) (a+1) and C(i) (a) by the following fomula:

C(i) (a+1) C(i) (a) = η C(i) (a+1). (7)

Since there exists a constant equal to the pre-defined learning rate η, an attacker can easily obtain the gradient C(i) (a+1) of client i in the a-th round from the server. According to Chai et al. (2020), if

the client i participates in a sufficient number of training rounds consecutively, the C(i) obtained by the server would leak information from that client s local data. This emphasizes that a client should avoid participating in training consecutively.

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A.2 OPTIMIZATION ANALYSIS

While the optimization of the objective function is not jointly convex in all variables, it exhibits convexity with respect to each individual variable when the others are held fixed. Consequently, we can pursue an alternate optimization strategy for each of the variables within the objective function in Eq. (3). Denoting x1 = (D(i) + C)j, x2 = 1 + exp( u T i x1) and x3 = 1 + exp( x T 1 ui), we can develop an alternating algorithm to update the parameters U, C, {Di}n i=1.

A.2.1 UPDATE U

Since the updating happens on each client, when C and Di are fixed, the problem w.r.t. ui becomes:

(i,j) Ω (rij log 1/(1+e <ui,(D(i)+C)j>)+(1 rij) log(1 1/(1+e <ui,(D(i)+C)j>))). (8)

Assume that x1 = (D(i) + C)j and x2 = 1 + exp( u T i x1), we have the partial derivative of L(ui) as following:

(i,j) Ω ((e( u T i x1) (1 rij))/(x2 2 (1 1/x2)) x1 + (rij e( u T i x1))/x2 x1). (9)

We then update ui on each client by gradient descent:

ui ui η ui L. (10)

A.2.2 UPDATE C

On each client, when fixing ui and D(i), the problem becomes:

(i,j) Ω (rij log 1/(1 + e <ui,(D(i)+C)j>) + (1 rij) log(1 1/(1 + e <ui,(D(i)+C)j>)))

(i,j) Ω λ(a,v1)||D(i) C||2 F + µ(a,v2)||C||1.

We first calculate the first two terms in Eq. equation 11:

(i,j) Ω ((x3 1) (1 rij)

(x2 3 (1 1 x3 )) ui + rij (x3 1)) ui) + 2λ(a,v1)(C D(i)). (12)

As for the constraint ||C||1, we need to use the soft-thresholding (Donoho, 1995) to update C:

C Sθ(C η CL), (13)

where θ is a pre-defined thresholding value, Sθ is the shrinkage operator, and it is defined as Sθ(v) = max(v θ, 0) max( v θ, 0).

A.2.3 UPDATE D(i)

When the others fixed, the problem w.r.t. D(i) becomes:

(i,j) Ω (rij log 1/(1 + e <ui,(D(i)+C)j>) + (1 rij) log(1 1/(1 + e <ui,(D(i)+C)j>)))

(i,j) Ω λ(a,v1)||D(i) C||2 F .

(14) Each client then obtains the partial derivative of L(D(i)) as following:

(i,j) Ω ((x3 1) (1 rij)

(x2 3 (1 1 x3 )) ui + rij (x3 1)) ui) + 2λ(a,v1)(D(i) C). (15)

We can update D(i) on the client i by the following equation:

D(i) D(i) η D(i)L. (16)

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B MORE EXPERIMENT DETAILS

In this section, we present some additional details and results of our experiments. To be specific, all models were implemented using Py Torch (Paszke et al., 2019), and experiments were conducted on a machine equipped with a 2.5GHz 14-Core Intel Core i9-12900H processor, a RTX 3070 Ti Laptop GPU, and 64GB of memory.

Table 2: The statistic information of the datasets used in the research. #Ratings is the number of the observed ratings; #Users is the number of users; #Items is the number of items; Sparsity is percentage of #Ratings in (#Users #Items).

Datasets #Ratings #Users #Items Sparsity Ta Feng 100,000 120 32,266 78.88% ML-100k 100,000 943 1,682 93.70% ML-1M 1,000,209 6,040 3,706 95.53% Video 23,181 1,372 7,957 99.79% Last FM 92,780 1,874 17,612 99.72% QB-article 266,356 24,516 7,455 99.81%

B.1 DATASETS

All the six popular recommendation datasets: Movie Lens-100K (ML-100K)1, Movie Lens-1M (ML1M)1, Amazon-Instant-Video (Video)2, Last FM-2K (Last FM)3, Ta Feng Grocery (Ta Feng)4, and QB-article5 are public. Table 2 presents the statistical information of the six datasets used in this study. All datasets show a high level of sparsity, with the percentage of observed ratings to the total possible ratings (users times items) all above 90%.

B.2 COMPRISON ANALYSIS

Tables 3 and 4 present experimental results comparing various centralized and federated methods across six real-world datasets, with the measurements focused on HR@10 and NDCG@10, respectively. In Table 3, it s evident that Fed RAP consistently outperforms other federated methods (Fed MF, Fed NCF, and PFed Rec) in terms of HR@10 across all datasets. Remarkably, Fed RAP also surpasses the centralized method NCF and approaches the performance of Cent RAP, a centralized variant of Fed RAP that inherently leverages more data than Fed RAP. Table 4 presents a similar trend in terms of NDCG@10. Again, Fed RAP demonstrates superior performance among federated methods, and closely competes with the centralized methods, even surpassing Cent RAP in some datasets. These results substantiate the effectiveness of Fed RAP in balancing personalized item recommendation with privacy preservation in a federated setting. It also highlights Fed RAP s capability to handle heterogeneous data across clients while still achieving competitive performance with centralized methods that have full access to all user data.

Table 3: Experimental results on HR@10 shown in percentages on six real-world datasets. Cent and Fed represent centralized and federated methods, respectively. The best results are highlighted in boldface.

Dataset Ta Feng ML-100K ML-1M Video Last FM QB-article

Cent NCF 98.33 1.42 65.94 0.38 62.52 0.70 39.11 0.95 20.81 0.69 40.94 0.16 Light GCN 99.33 0.51 99.54 0.99 99.43 0.31 51.60 0.54 28.56 0.97 61.09 0.63 Cent RAP 99.87 0.97 99.30 0.64 99.17 0.43 48.54 0.32 24.76 0.12 54.28 0.51

Fed MF 97.50 1.11 65.05 0.84 65.95 0.37 39.00 0.76 20.04 0.20 40.54 0.35 Fed NCF 87.54 1.01 60.89 0.24 61.31 0.32 39.25 1.01 20.14 0.49 40.52 0.57 PFed Rec 98.33 1.05 70.03 0.27 68.59 0.21 40.87 0.82 21.61 0.45 46.00 0.22 Fed RAP 99.43 1.44 97.09 0.56 93.24 0.26 46.53 0.48 23.29 1.19 53.98 0.57

1https://grouplens.org/datasets/movielens/ 2http://jmcauley.ucsd.edu/data/amazon/ 3https://grouplens.org/datasets/hetrec-2011/ 4https://github.com/Reco Hut-Datasets/tafeng 5https://github.com/yuangh-x/2022-NIPS-Tenrec

Published as a conference paper at ICLR 2024

Table 4: Experimental results on NDCG@10 shown in percentages on six real-world datasets. Cent and Fed represent centralized and federated methods, respectively. The best results are highlighted in boldface.

Dataset Ta Feng ML-100K ML-1M Video Last FM-2K QB-article

Cent NCF 97.55 3.20 38.76 0.61 35.60 0.56 21.94 1.33 11.36 0.34 19.61 0.37 Light GCN 98.7 1.47 91.52 0.86 90.43 0.72 32.64 1.13 14.32 1.01 31.85 0.96 Cent RAP 99.54 1.35 86.65 0.51 86.89 0.46 28.83 2.10 11.30 0.26 26.09 0.54

Fed MF 96.72 2.13 38.40 0.97 38.77 0.29 25.03 0.57 9.87 0.25 17.76 0.18 Fed NCF 83.42 1.71 32.68 0.47 34.94 0.42 21.72 1.02 9.99 0.64 19.31 0.67 PFed Rec 98.03 0.62 41.38 0.33 40.19 0.21 24.04 0.36 10.88 0.10 22.28 0.28 Fed RAP 99.11 2.31 87.81 2.00 71.87 0.28 28.17 0.22 10.99 0.51 24.75 0.36

0.0 0.2 0.4 0.6 0.8 1.0

(a) C for the 43-th user

0.0 0.2 0.4 0.6 0.8 1.0

(b) C for the 586-th user

0.0 0.2 0.4 0.6 0.8 1.0

(c) C for the 785-th user

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

(g) C + D43

0.0 0.2 0.4 0.6 0.8 1.0

(h) C + D586

0.0 0.2 0.4 0.6 0.8 1.0

(i) C + D785

Figure 8: t-SNE visualization of item embeddings learnd by Fed RAP.

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B.3 ABLATION STUDY ON ITEM PERSONALIZATION

In this section, we visually demonstrate the benefits of Fed RAP for personalizing item information. Fed RAP not only personalizes user information, but also implements the additive personalization to item information, thus it achieves the bipartite personalization to perform user-specific recommendations. In this section, we investigate the roles of the global item embedding C and the local item embeddings {D(i)}n i=1 in Fed RAP. To visualize them, we select three local item embeddings D(43), D(586), D(785), and the global item embedding C from the models learned on the Movie Lens-100K dataset according to Sec. 4.3. Since in this paper, we mainly focus on the implicit feedback recommendation, i.e., each item is either rated or unrated by the users, we map these item embeddings into a 2-D space through t-SNE (Van der Maaten & Hinton, 2008), and the normalized results are shown in Fig. 8, where purple represents items that have not been rated by the user, and red represents items that have been rated by the user. From Figs. 8 (a)-(c), we can see that the items rated and unrated by users are mixed together in C, which depicts that C only keeps information about items shared among users. And Figs. 8 (d)-(f) show that the items learned by D(i) can by obviously divided into two clusters for the i-th user by Fed RAP. This demonstrates that D(i) can only learn information about items related to the i-th user, such as the user s preferences for items. Morover, we add C and D(i) to show the additive personalization of items for the i-th user, which is shown in Figs. 8 (g)-(i). These three figures again demonstrate the ability of Fed RAP to effectively personalize item information. This case supports the additive personalization of item information to help make user-specific recommendations.

Table 5: Comparison of Performance Degradation with and without Differential Privacy on HR@10.

w/ DP w/o DP Degradation Fed MF 0.6505 0.6257 3.82% Fed NCF 0.6089 0.4348 28.60% PFed Rec 0.7003 0.6352 9.30% Fed RAP 0.9709 0.9364 3.55%

Table 6: Comparison of Performance Degradation with and without Differential Privacy on NDCG@10.

w/ DP w/o DP Degradation Fed MF 0.3840 0.3451 10.13% Fed NCF 0.3268 0.2386 27.00% PFed Rec 0.4138 0.3549 14.23% Fed RAP 0.8781 0.8015 8.72%

B.4 ABLATION STUDY ON DIFFERENTIAL PRIVACY

We further evaluated DP versions of the used federated methods and compared them with the original methods in Table 5 and Table 6, which shows Fed RAP with DP can still preserve the performance advantage of Fed RAP and meanwhile achieve (ϵ, δ)-DP.

B.5 ABLATION STUDY ON FACTOR EFFECTS

Table 7: Sparsity of various subsets in ML-1M and Fed RAP s fine-grained performance (HR@10 and NDCG@10).

Subset Sparsity HR@10 NDCG@10

w/ 500 users 81.33% 1.0000 0.9446 95.49% 1.0000 0.8227 98.62% 0.9020 0.5686

w/ 1000 users 85.87% 1.0000 0.9064 97.52% 0.9990 0.7118 98.97% 0.8260 0.5207

w/ 1000 items 87.62% 0.9851 0.8931 94.86% 0.9398 0.7759 96.85% 0.9186 0.7003

To examine factors affecting Fed RAP s performance, we designed ML-1M dataset variations with different sparsity levels by selecting specific numbers of users or items. Table 7 displays Fed RAP s performance on these datasets, and shows that as sparsity increases, performance declines more significantly. Additionally, with similar sparsity, increasing user count leads to slightly reduced performance. Datasets with a fixed number of items consistently underperform compared to those with equal user counts. This table indicates that Fed RAP s effectiveness is influenced by dataset sparsity and the counts of users and items.

Published as a conference paper at ICLR 2024

C DISCUSSION ON LIMIATION

While Fed RAP has shown excellent performance in federated recommendation systems through techniques like additive personalization and variable weights, one limitation is that the current Fed RAP algorithm requires storing the complete item embedding matrix on each user s device. This could demand significant storage space, especially if the total number of items is large. A simple approach to alleviate this limitation is to only store the embeddings of items that each user has interacted with. However, this modification may require extra attention to the cold-start problem of new items for each user. Our future work will aim to address this limitation.