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In this paper, we improve the learning theory of kernel distribution regression. We address kernels based on Hilbertian embeddings that encompass most, if not all, of the existing approaches. We introduce the novel near-unbiased condition on the Hilbertian embeddings that enables us to provide new error bounds on the effect of the two-stage sampling, thanks to a new analysis. We show that this near-unbiased condition holds for three important classes of kernels, based on optimal transport and mean embedding. As a consequence, we strictly improve the existing convergence rates for these kernels. Our setting and results are illustrated by numerical experiments.<\/jats:p>","DOI":"10.1214\/25-aos2537","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T05:25:22Z","timestamp":1757654722000},"page":"1753-1782","source":"Crossref","is-referenced-by-count":2,"title":["Improved learning theory for kernel distribution regression with two-stage sampling"],"prefix":"10.1214","volume":"53","author":[{"given":"Fran\u00e7ois","family":"Bachoc","sequence":"first","affiliation":[{"name":"IMT, Universit\u00e9 Paul-Sabatier & Institut universitaire de France (IUF)"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Louis","family":"B\u00e9thune","sequence":"additional","affiliation":[{"name":"Apple"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alberto","family":"Gonz\u00e1lez-Sanz","sequence":"additional","affiliation":[{"name":"Department of Statistics, Columbia University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Michel","family":"Loubes","sequence":"additional","affiliation":[{"name":"Regalia Team, INRIA & IMT"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"67","doi-asserted-by":"publisher","unstructured":"<small>PORTNOY, S.<\/small> (2012). 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