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Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"In a recent paper, Bubeck, Lee, and Singh introduced a new first order method for minimizing smooth strongly convex functions. Their geometric descent algorithm, largely inspired by the ellipsoid method, enjoys the optimal linear rate of convergence. We show that the same iterate sequence is generated by a scheme that in each iteration computes an optimal average of quadratic lower models of the function. Indeed, the minimum of the averaged quadratic approaches the true minimum at an optimal rate. This intuitive viewpoint reveals clear connections to the original fast-gradient methods and cutting plane ideas, and leads to limited-memory extensions with improved performance.<\/jats:p>","DOI":"10.1137\/16m1072528","type":"journal-article","created":{"date-parts":[[2018,2,1]],"date-time":"2018-02-01T12:29:50Z","timestamp":1517488190000},"page":"251-271","source":"Crossref","is-referenced-by-count":25,"title":["An Optimal First Order Method Based on Optimal Quadratic Averaging"],"prefix":"10.1137","volume":"28","author":[{"given":"Dmitriy","family":"Drusvyatskiy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maryam","family":"Fazel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Scott","family":"Roy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,1]]},"reference":[{"key":"atypb1","unstructured":"Z. Allen-Zhu and L. Orecchia,\n Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent\n , preprint,arXiv:1407.1537, 2016."},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"H. Attouch, J. Peypouquet, and P. Redont,\n Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity\n , Math. Program. (2016), pp. 1-53,https:\/\/doi.org\/10.1007\/s10107-016-0992-8.","DOI":"10.1007\/s10107-016-0992-8"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-006-9127-8"},{"key":"atypb4","unstructured":"S. Bubeck and Y. Lee,\n Black-Box Optimization with a Politician\n , preprint,arXiv:1602.04847, 2016."},{"key":"atypb5","unstructured":"S. Bubeck, Y. Lee, and M. Singh,\n A Geometric Alternative to Nesterov's Accelerated Gradient Descent\n , preprint,arXiv:1506.08187, 2015."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/1961189.1961199"},{"key":"atypb7","unstructured":"S. Chen and S. Ma,\n Geometric Descent Method for Convex Composite Minimization\n ,arXiv:1612.09034, 2017."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/0108053"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/15M1009597"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"Y. Nesterov,\n Introductory Lectures on Convex Optimization\n ,\n A Basic Course\n , Appl. Optim. 87, Kluwer Academic, Boston, 2004,https:\/\/doi.org\/10.1007\/978-1-4419-8853-9.","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb11","first-page":"543","volume":"269","author":"Nesterov Y. E.","year":"1983","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"atypb12","unstructured":"J. Nocedal and S. Wright,\n Numerical Optimization\n , 2nd ed., Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2006."},{"key":"atypb13","unstructured":"W. Su, S. Boyd, and E. Candes,\n A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights\n , in Advances in Neural Information Processing Systems 27, 28th Annual Conference, Montreal, 2014, Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Weinberger, eds., Curran Associates, Red Hook, NY, 2014, pp. 2510-2518,http:\/\/papers.nips.cc\/paper\/5322-a-differential-equation-for-modeling-nesterovs-accelerated-gradient-method-theory-and-insights.pdf."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1072528","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:07Z","timestamp":1787336107000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1072528"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1072528"],"URL":"https:\/\/doi.org\/10.1137\/16m1072528","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}