[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82787-en":3,"doc-seo-82787-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82787,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","FedSPM Routing-Enabled Federated Learning under Dual Heterogeneity via Semiparametric Mixture","Routing-prediction federated learning turns inter-client heterogeneity into system intelligence by routing each external query to the best-matched client for prediction. Prior work typically assumes intra-client homogeneity, which ignores latent subpopulations within local data and can harm both routing and prediction. FedSPM addresses dual heterogeneity using a routing-enabled semiparametric mixture, where each client is modeled by latent components and component feature densities are estimated through empirical likelihood density ratios. A federated EM algorithm optimizes a surrogate with proven convergence, and experiments show consistent gains on benchmarks and real medical data.","arXiv :2607 .04085v 1 [ cs .LG] 5 Jul 2026  \nFedSPM: Routing-Enabled Federated Learning under Dual Heterogeneity via Semiparametric Mixture  \nZijian Wang 1 Pengfei Li2 Guangyu Yang 1 Qiong Zhang 1 ∗  \n1Institute of Statistics and Big Data, Renmin University of China  \n2Department of Statistics and Actuarial Science, University of Waterloo  \nAbstract  \nRouting-prediction federated learning has emerged as a new paradigm that reframes inter-client heterogeneity as a resource for system-level intelligence: at inference time, the server routes each external query to the best-matched client for prediction.  \nExisting approaches, however, typically treat each client as internally homogeneous, overlooking latent subpopulations within local data. For example, patients with the same diagnosis at one hospital may exhibit morphologically distinct disease subtypes. The coexistence of inter-client and intra-client heterogeneity, which we call dual heterogeneity, can impair both routing and prediction. To address this challenge, we propose FedSPM, a routing-enabled semiparametric mixture framework that represents each client using client-specific latent components. Each component combines a predictive distribution for classification with a feature distribution for routing. To flexibly model feature distributions while effectively sharing information across clients, FedSPM models their density ratios relative to a common nonparametric measure estimated via empirical likelihood. We develop a federated expectation-maximization algorithm that optimizes a tractable surrogate and prove convergence of the exact profiled objective at the standard O(1/ √T)  rate when the surrogate errors are properly controlled. Experiments on controlled benchmarks and real-world medical data demonstrate consistent improvements in routing and prediction under dual heterogeneity. Code is available here.  \n1 Introduction  \nFederated learning (FL) [27] enables multiple clients to collaboratively train a model without centralizing their raw data. In a typical FL system, a server distributes a shared model to participating clients, each client updates the model using its local data, and the server aggregates the resulting locally updated models into an improved global model. This distributed training paradigm allows knowledge to be shared across clients such as hospitals [43] and mobile devices [20], making FL particularly promising when data are sensitive, geographically dispersed, or impractical to centralize [21, 14, 44] . Traditionally, inter-client heterogeneity, where data distributions vary across clients, is viewed as an obstacle in FL. Since clients optimize different local objectives, their gradients may drift in conflicting directions [15], thereby slowing convergence [24] and degrading global model performance [50] . In contrast, the recent routing-prediction FL paradigm [40] reframes inter-client heterogeneity as a useful signal of client specialization. Like personalized FL [23, 10, 3], it learns specialized models for individual client domains. Beyond personalization, it further estimates how well an external query matches each client’s data distribution. At inference time, the server uses these distributional match scores to route the query to the most suitable client, whose specialized model then makes the final prediction, thereby turning client-specific expertise into system-level intelligence.  \n∗ Correspondence to: Qiong Zhang ([qiong.zhang@ruc.edu.cn](qiong.zhang@ruc.edu.cn)) and Guangyu Yang ([yguangyu@ruc.edu.cn](yguangyu@ruc.edu.cn))  \nPreprint.  \nHowever, such a routing-prediction FL paradigm typically assumes that the data within each client are drawn from a homogeneous distribution, thereby overlooking intra-client heterogeneity. This assumption is often violated in practice, as local data may arise from a mixture of latent components [26, 41] . For example, within one hospital, dermoscopic images may involve different lesion types, anatomical ","cbCaiq59EtLdsEqo","https://ap.wps.com/l/cbCaiq59EtLdsEqo","pdf",2346008,1,24,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What problem does FedSPM address in routing-prediction federated learning?\",\"answer\":\"FedSPM targets dual heterogeneity, where both inter-client and intra-client heterogeneity exist. Intra-client heterogeneity causes local data to come from latent subpopulations, which can degrade both routing quality and prediction accuracy.\"},{\"question\":\"How does FedSPM model each client under dual heterogeneity?\",\"answer\":\"FedSPM represents every client using client-specific latent components. Each component provides a predictive distribution for classification and a feature distribution that supports routing for external queries.\"},{\"question\":\"What method does FedSPM use to learn the proposed semiparametric mixture model?\",\"answer\":\"FedSPM develops a federated expectation-maximization algorithm that optimizes a tractable surrogate objective. The method includes convergence guarantees for the exact profiled objective when surrogate errors are controlled.\"}]",1784182938,60,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"fedspm-routing-enabled-federated-learning-under-dual-heterogeneity-via-semiparametric-mixture","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/fedspm-routing-enabled-federated-learning-under-dual-heterogeneity-via-semiparametric-mixture/82787/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does FedSPM address in routing-prediction federated learning?","Question",{"text":75,"@type":76},"FedSPM targets dual heterogeneity, where both inter-client and intra-client heterogeneity exist. Intra-client heterogeneity causes local data to come from latent subpopulations, which can degrade both routing quality and prediction accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does FedSPM model each client under dual heterogeneity?",{"text":80,"@type":76},"FedSPM represents every client using client-specific latent components. Each component provides a predictive distribution for classification and a feature distribution that supports routing for external queries.",{"name":82,"@type":73,"acceptedAnswer":83},"What method does FedSPM use to learn the proposed semiparametric mixture model?",{"text":84,"@type":76},"FedSPM develops a federated expectation-maximization algorithm that optimizes a tractable surrogate objective. 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