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We introduce the concept of inexact first-order oracle of degree q for apossibly nonconvex and nonsmooth function, which naturally appears in thecontext of approximate gradient, weak level of smoothness and other situations.Our definition is less conservative than those found in the existingliterature, and it can be viewed as an interpolation between fully exact andthe existing inexact first-order oracle definitions. We analyze the convergencebehavior of a (fast) inexact proximal gradient method using such an oracle forsolving (non)convex composite minimization problems. We derive complexityestimates and study the dependence between the accuracy of the oracle and thedesired accuracy of the gradient or of the objective function. Our results showthat better rates can be obtained both theoretically and in numericalsimulations when q is large.

Proximal gradient methods with inexact oracle of degree q for composite optimization