{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T15:48:29Z","timestamp":1778860109155,"version":"3.51.4"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T00:00:00Z","timestamp":1652400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Federated learning is a framework for multiple devices or institutions, called local clients, to collaboratively train a global model without sharing their data. For federated learning with a central server, an aggregation algorithm integrates model information sent from local clients to update the parameters for a global model. Sample mean is the simplest and most commonly used aggregation method. However, it is not robust for data with outliers or under the Byzantine problem, where Byzantine clients send malicious messages to interfere with the learning process. Some robust aggregation methods were introduced in literature including marginal median, geometric median and trimmed-mean. In this article, we propose an alternative robust aggregation method, named \u03b3-mean, which is the minimum divergence estimation based on a robust density power divergence. This \u03b3-mean aggregation mitigates the influence of Byzantine clients by assigning fewer weights. This weighting scheme is data-driven and controlled by the \u03b3 value. Robustness from the viewpoint of the influence function is discussed and some numerical results are presented.<\/jats:p>","DOI":"10.3390\/e24050686","type":"journal-article","created":{"date-parts":[[2022,5,13]],"date-time":"2022-05-13T08:37:02Z","timestamp":1652431022000},"page":"686","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Robust Aggregation for Federated Learning by Minimum \u03b3-Divergence Estimation"],"prefix":"10.3390","volume":"24","author":[{"given":"Cen-Jhih","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Statistical Science, Academia Sinica, Taipei City 11529, Taiwan"}]},{"given":"Pin-Han","family":"Huang","sequence":"additional","affiliation":[{"name":"Data Science Degree Program, National Taiwan University, Taipei City 10617, Taiwan"}]},{"given":"Yi-Ting","family":"Ma","sequence":"additional","affiliation":[{"name":"Institute of Statistical Science, Academia Sinica, Taipei City 11529, Taiwan"}]},{"given":"Hung","family":"Hung","sequence":"additional","affiliation":[{"name":"Institute of Epidemiology and Preventive Medicine, National Taiwan University, Taipei City 10055, Taiwan"}]},{"given":"Su-Yun","family":"Huang","sequence":"additional","affiliation":[{"name":"Institute of Statistical Science, Academia Sinica, Taipei City 11529, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,13]]},"reference":[{"key":"ref_1","unstructured":"Kone\u010dn\u00fd, J., McMahan, H.B., Yu, F.X., Richtarik, P., Suresh, A.T., and Bacon, D. 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