Gradient Method With Inexact Oracle for Composite Non-Convex Optimization

In this paper, we develop new first-order method for composite non-convex minimization problems with simple constraints and inexact oracle. The objective function is given as a sum of "`hard"', possibly non-convex part, and "`simple"' convex part. Informally speaking, oracle inexactness means that, for the "`hard"' part, at any point we can approximately calculate the value of the function and construct a quadratic function, which approximately bounds this function from above. We give several examples of such inexactness: smooth non-convex functions with inexact H"older-continuous gradient, functions given by auxiliary uniformly concave maximization problem, which can be solved only approximately. For the introduced class of problems, we propose a gradient-type method, which allows to use different proximal setup to adapt to geometry of the feasible set, adaptively chooses controlled oracle error, allows for inexact proximal mapping. We provide convergence rate for our method in terms of the norm of generalized gradient mapping and show that, in the case of inexact H"older-continuous gradient, our method is universal with respect to H"older parameters of the problem. Finally, in a particular case, we show that small value of the norm of generalized gradient mapping at a point means that a necessary condition of local minimum approximately holds at that point.