[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84241-en":3,"doc-seo-84241-105":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},84241,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Fast Rates for Semi-Supervised Learning via Data-Augmentation Graph Regularization","Self-supervised learning can match supervised accuracy using only a fraction of labeled data, yet the labeled-sample efficiency has lacked a rigorous explanation. This work provides a theory for semi-supervised learning by building an augmentation-induced similarity graph on unlabeled samples and running graph-Laplacian-regularized learning on it. A fast transductive rate O(1/nL) is proved in labels, replacing the typical supervised scaling O(1/√nL). The bound makes augmentation quality explicit through a graph-cut mass term RDA(y).","arXiv :2607 .075 13v 1 [ cs .LG] 8 Jul 2026  \nFast Rates for Semi-Supervised Learning via Data-Augmentation Graph Regularization  \nAdam M. Oberman  \n[adam.oberman@mcgill.ca](adam.oberman@mcgill.ca)  \nDepartment of Mathematics and Statistics, McGill University  \nMila, Quebec AI Institute  \nLawZero  \nThursday 9th July, 2026  \nAbstract  \nSelf-supervised learning matches supervised accuracy from a fraction of the labels, but the labeled-sample efficiency behind this has lacked a theoretical explanation. We provide one. Data augmentation induces a similarity graph on the unlabeled data, so downstream learning on that graph is graph-Laplacian-regularized learning. We prove a fast transductive rate, O(1/nL ) in the number of labels, in place of the supervised O(1/ √nL  ), by carrying the leave-one-out stability apparatus of Johnson and Zhang (JMLR 2007) over to the augmentation graph, and without the unrealistic assumptions of limit-based analyses (exact kernel, generalizing features) . The bound makes augmentation quality explicit: the expected error is at most C/nL + RDA (y), where the data-augmentation alignment error RDA (y) is the graph-cut mass of augmentations that cross a label boundary, so good augmentations let few labels suffice. The analysis uses a streamlined loss that drops the projector, negative-sample, and orthogonality overhead of standard objectives yet still recovers the top-K ideal features in the infinite-data limit, the augmentation-kernel eigenspace studied by Zhai et al. The result explains the observed accuracy-versus-label-count curve rather than only bounding a generalization gap.  \nKeywords: semi-supervised learning, data augmentation, graph Laplacian regularization, algorithmic stability, fast rates  \n1 Introduction  \nSelf-supervised learning matches supervised accuracy from a small fraction of the labels. SimCLR [Chen et al., 2020a] matches a supervised ResNet-50 from a linear probe and is strong at 1% and 10% of ImageNet labels; SimCLRv2 [Chen et al., 2020b] reaches 80 .9% top-1 with only 10% of the labels, an order-of-magnitude label saving; and CPC v2 [H´enaff et al., 2020] named the “dataefficient” framing with two to five times fewer labels. This accuracy-versus-label-count curve isone of the most reported empirical facts about representation learning, yet it remains theoretically under-explained.  \nThe feature-learning side of the story is by now well understood. Different self-supervised losses (spectral contrastive, contrastive and non-contrastive, kernel-PCA) all recover the leading eigenfunctions of the augmentation similarity kernel k DAF . But that theory lives in the infinitedata limit: it characterizes the features a method would learn from unlimited unlabeled data, and says nothing about the finite labeled-sample behavior downstream, which is exactly what the curve  \nabove measures. Two assumptions used there are also unrealistic for the few-augmentation regime: that the expectation kernel is learned exactly, and that the factoring features generalize to unseen points.  \nThis paper supplies the missing labeled-sample guarantee. The data augmentations define a similarity graph on the unlabeled data, so downstream learning on that graph is graph-Laplacianregularized learning. For that setting a transductive fast rate is already available: Johnson and Zhang [2007] bound the transductive error of Laplacian-regularized multi-class graph learning by leave-one-out algorithmic stability and obtain an O(1/n) rate in the labeled count under balanced components, governed by a learning-theoretic graph cut. Our rate results (Theorems 6 and 9) are that apparatus, specialized and reinterpreted for the augmentation graph; the contribution isnot a new rate mechanism but its placement. Read on the augmentation graph, the cut becomesan augmentation-quality term RDA(y), the number of graph components becomes a property of the augmentation distribution, and the bound turns into a statement about how few label","cbCaif42FB1hYuvl","https://ap.wps.com/l/cbCaif42FB1hYuvl","pdf",442366,1,21,"English","en",105,"# Introduction\n## Contributions\n# Setup and notation","[{\"question\":\"What learning mechanism turns data augmentation into a semi-supervised method?\",\"answer\":\"Augmentations induce a similarity graph over unlabeled data, and downstream learning is performed with graph-Laplacian regularization on that augmentation graph.\"},{\"question\":\"What labeled-sample rate does the paper prove?\",\"answer\":\"The paper proves a fast transductive rate of order O(1/nL) in the number of labels, contrasting with the supervised scaling O(1/√nL).\"},{\"question\":\"How does augmentation quality affect the expected error bound?\",\"answer\":\"Expected error is bounded by C/nL plus an augmentation alignment term RDA(y), defined as the graph-cut mass of augmentations that cross a label boundary; better augmentations reduce this term.\"}]",1784194287,53,{"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},"fast-rates-for-semi-supervised-learning-via-data-augmentation-graph-regularization","",{"@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/fast-rates-for-semi-supervised-learning-via-data-augmentation-graph-regularization/84241/",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","2026-07-17","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 learning mechanism turns data augmentation into a semi-supervised method?","Question",{"text":75,"@type":76},"Augmentations induce a similarity graph over unlabeled data, and downstream learning is performed with graph-Laplacian regularization on that augmentation graph.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What labeled-sample rate does the paper prove?",{"text":80,"@type":76},"The paper proves a fast transductive rate of order O(1/nL) in the number of labels, contrasting with the supervised scaling O(1/√nL).",{"name":82,"@type":73,"acceptedAnswer":83},"How does augmentation quality affect the expected error bound?",{"text":84,"@type":76},"Expected error is bounded by C/nL plus an augmentation alignment term RDA(y), defined as the graph-cut mass of augmentations that cross a label boundary; better augmentations reduce this term.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]