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We find that noisy tensor completion particularly benefits from online algorithms by avoiding the trimming procedure and ensuring sharp entrywise statistical error, which is often technically challenging for offline methods. The regret of oRGrad is analyzed, revealing a fascinating trilemma concerning the computational convergence rate, statistical error, and regret bound. By selecting an appropriate constant step size, oRGrad achieves an O(T1\/2) regret. We then introduce the adaptive-oRGrad algorithm, which can achieve the optimal O(logT) regret by adaptively selecting step sizes, regardless of whether the time horizon is known. The adaptive-oRGrad algorithm can attain a statistically optimal error rate without knowing the horizon. Comprehensive numerical simulations corroborate our theoretical findings. We show that oRGrad significantly outperforms its offline counterpart in predicting the solar F10.7 index with tensor predictors that monitor space weather impacts.<\/jats:p>","DOI":"10.1214\/25-aos2588","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:48:10Z","timestamp":1782395290000},"page":"1153-1177","source":"Crossref","is-referenced-by-count":1,"title":["Online tensor learning: Computational and statistical trade-offs, adaptivity and optimal regret"],"prefix":"10.1214","volume":"54","author":[{"given":"Jingyang","family":"Li","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, School of Management, Fudan University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian-Feng","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Michigan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dong","family":"Xia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"33","doi-asserted-by":"publisher","unstructured":"Kolda, T. 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