{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T02:17:04Z","timestamp":1777861024771,"version":"3.51.4"},"reference-count":35,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T00:00:00Z","timestamp":1773878400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T00:00:00Z","timestamp":1773878400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62172110"],"award-info":[{"award-number":["62172110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Expert Systems"],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Federated learning (FL) faces significant challenges under Non\u2010IID data, primarily due to misaligned local gradients that point in conflicting directions across clients. This inconsistency creates a fundamental trade\u2010off between global model accuracy and fairness, as clients with large\u2010magnitude gradients often dominate the aggregation process, marginalising minority participants. Existing methods typically fail to explicitly resolve these directional conflicts, leading to suboptimal convergence or exacerbated bias. To address this, we propose FedBEF, a theoretically grounded aggregation framework that jointly optimises convergence speed and fairness. Unlike heuristic weighting schemes, FedBEF derives optimal client weights by minimising an upper bound of the global loss, formulated as a constrained quadratic program. The resulting weights explicitly penalise gradient conflicts (e.g., by discouraging large angular deviations) while promoting a coherent descent direction that aligns with the true global gradient. Moreover, to improve robustness against noisy or outlier updates\u2014particularly pronounced under partial client participation and extreme heterogeneity, we introduce a Similar Neighbour Gradient (SNG) mechanism combined with adaptive momentum. By clustering clients based on gradient cosine similarity and smoothing within neighbourhoods, SNG effectively suppresses erratic updates without requiring additional communication. More importantly, we prove that FedBEF converges to a stationary point under non\u2010convex settings. Extensive experiments on CIFAR\u201010 and CIFAR\u2010100 demonstrate its superiority over state\u2010of\u2010the\u2010art baselines such as FedProx and FedALA. Notably, under severe heterogeneity (CIFAR\u201010, Dirichlet ), FedBEF achieves up to a 7.13% higher average accuracy. It also significantly enhances fairness, as evidenced by a substantial reduction in the coefficient of variation (standard deviation over mean) of per\u2010client accuracies.<\/jats:p>","DOI":"10.1111\/exsy.70247","type":"journal-article","created":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T10:46:59Z","timestamp":1773917219000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["<scp>FedBEF<\/scp>\n                    : Federated Learning With Balance of Performance and Fairness"],"prefix":"10.1111","volume":"43","author":[{"given":"Xiangkun","family":"Qiu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics Guangdong University of Technology  Guangzhou China"}]},{"given":"Heng\u2010Ying","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Guangdong University of Technology  Guangzhou China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9399-9612","authenticated-orcid":false,"given":"Fangqing","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Guangdong University of Technology  Guangzhou China"}]},{"given":"Hai\u2010Lin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Guangdong University of Technology  Guangzhou China"}]}],"member":"311","published-online":{"date-parts":[[2026,3,19]]},"reference":[{"key":"e_1_2_10_2_1","first-page":"2575","volume-title":"International Conference on Artificial Intelligence and Statistics","author":"Charles Z.","year":"2021"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2024.3514302"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/ISPA-BDCloud-SocialCom-SustainCom52081.2021.00042"},{"key":"e_1_2_10_5_1","unstructured":"Guo P. S.Zeng Y.Wang H.Fan F.Wang andL.Qu.2025.\u201cSelective Aggregation for Low\u2010Rank Adaptation in Federated Learning.\u201dThe Thirteenth International Conference on Learning Representations."},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIFS.2022.3207911"},{"key":"e_1_2_10_7_1","unstructured":"Hsu T. M. H. H.Qi andM.Brown.2019.\u201cMeasuring the Effects of Non\u2010Identical Data Distribution for Federated Visual Classification.\u201darXiv Preprint arXiv:1909.06335."},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TNSE.2022.3169117"},{"key":"e_1_2_10_9_1","unstructured":"Hu Z. andY.Yu.2025.Leveraging Variable Sparsity to Refine Pareto Stationarity in Multi\u2010Objective OptimizationThe Thirteenth International Conference on Learning Representations."},{"key":"e_1_2_10_10_1","doi-asserted-by":"publisher","DOI":"10.1561\/2200000083"},{"key":"e_1_2_10_11_1","first-page":"5132","volume-title":"International Conference on Machine Learning","author":"Karimireddy S. P.","year":"2020"},{"key":"e_1_2_10_12_1","doi-asserted-by":"crossref","unstructured":"Li Q. B.He andD.Song.2021.\u201cModel\u2010Contrastive Federated Learning.\u201dInProceedings of the IEEE\/CVF conference on computer vision and pattern recognition 10713\u201310722.","DOI":"10.1109\/CVPR46437.2021.01057"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2020.2975749"},{"key":"e_1_2_10_14_1","unstructured":"Li T. M.Sanjabi A.Beirami andV.Smith.2019.\u201cFair Resource Allocation in Federated Learning.\u201darXiv Preprint arXiv:1905.10497."},{"key":"e_1_2_10_15_1","unstructured":"Li X. K.Huang W.Yang S.Wang andZ.Zhang.2019.\u201cOn the Convergence of Fedavg on Non\u2010Iid Data.\u201darXiv Preprint arXiv:1907.02189."},{"key":"e_1_2_10_16_1","doi-asserted-by":"publisher","DOI":"10.1109\/TMC.2024.3365295"},{"key":"e_1_2_10_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11633-022-1341-4"},{"key":"e_1_2_10_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2024.128019"},{"key":"e_1_2_10_19_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2019.2940446"},{"key":"e_1_2_10_20_1","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2021.3076684"},{"key":"e_1_2_10_21_1","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v34i08.7021"},{"key":"e_1_2_10_22_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11280-022-01046-x"},{"key":"e_1_2_10_23_1","first-page":"1273","volume-title":"Artificial Intelligence and Statistics","author":"McMahan B.","year":"2017"},{"key":"e_1_2_10_24_1","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2020.3015958"},{"key":"e_1_2_10_25_1","first-page":"19683","volume-title":"International Conference on Machine Learning","author":"Sharma P.","year":"2022"},{"key":"e_1_2_10_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.future.2024.07.046"},{"key":"e_1_2_10_27_1","unstructured":"Wang J. andG.Joshi.2018.\u201cCooperative Sgd: A Unified Framework for the Design and Analysis of Communication\u2010Efficient Sgd Algorithms.\u201darXiv Preprint arXiv:1808.07576."},{"key":"e_1_2_10_28_1","first-page":"7611","article-title":"Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization","volume":"33","author":"Wang J.","year":"2020","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_2_10_29_1","unstructured":"Wang Y. 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