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Practical stochastic bilevel optimization problems become challenging in optimization or learning scenarios where the number of variables is high or there are constraints. In this paper, we introduce a bilevel stochastic gradient method for bilevel problems with nonlinear and possibly nonconvex lower-level constraints. We also present a comprehensive convergence theory that addresses both the lower-level unconstrained and constrained cases and covers all inexact calculations of the adjoint gradient (also called hypergradient), such as the inexact solution of the lower-level problem, inexact computation of the adjoint formula (due to the inexact solution of the adjoint equation or use of a truncated Neumann series), and noisy estimates of the gradients, Hessians, and Jacobians involved. To promote the use of bilevel optimization in large-scale learning, we have developed new low-rank practical bilevel stochastic gradient methods (BSG-N-FD and\u00a0BSG-1) that do not require second-order derivatives and, in the lower-level unconstrained case, dismiss any matrix\u2013vector products.\n<\/jats:p>","DOI":"10.1007\/s10898-025-01502-8","type":"journal-article","created":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T01:56:26Z","timestamp":1747965386000},"page":"569-614","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1436-5348","authenticated-orcid":false,"given":"T.","family":"Giovannelli","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0003-8636-0348","authenticated-orcid":false,"given":"G. 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