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Our definition is less conservative than those found in the existing literature, and it can be viewed as an interpolation between fully exact and the existing inexact first-order oracle definitions. We analyze the convergence behavior of a (fast) inexact proximal gradient method using such an oracle for solving (non)convex composite minimization problems. We derive complexity estimates and study the dependence between the accuracy of the oracle and the desired accuracy of the gradient or of the objective function. Our results show that better rates can be obtained both theoretically and in numerical simulations when <jats:italic>q<\/jats:italic> is large.<\/jats:p>","DOI":"10.1007\/s11590-024-02118-9","type":"journal-article","created":{"date-parts":[[2024,5,1]],"date-time":"2024-05-01T12:51:57Z","timestamp":1714567917000},"page":"285-306","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Proximal gradient methods with inexact oracle of degree q for composite optimization"],"prefix":"10.1007","volume":"19","author":[{"given":"Yassine","family":"Nabou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fran\u00e7ois","family":"Glineur","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ion","family":"Necoara","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,4,30]]},"reference":[{"key":"2118_CR1","doi-asserted-by":"publisher","DOI":"10.1080\/10556788.2023.2261604","author":"A Agafonov","year":"2017","unstructured":"Agafonov, A., Kamzolov, D., Dvurechensky, P., Gasnikov, A., Takac, M.: Inexact tensor methods and their application to stochastic convex optimization. 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