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Inexact proximal stochastic second-order methods for nonconvex composite optimization

We propose a framework of Inexact Proximal Stochastic Second-order (IPSS) methods for solving nonconvex optimization problems, whose objective function consists of an average of finitely many (weakly) smooth functions and a convex but possibly nonsmooth function. At each iteration, IPSS inexactly solves a proximal subproblem constructed by using some positive definite matrix which could capture the second-order information of original problem. Proper tolerances are given for the subproblem solution in order to maintain global convergence and the desired overall complexity of the algorithm. Under mild conditions, we analyze the computational complexity related to the evaluations on the component gradient of the smooth function. We also investigate the number of evaluations of subgradient when using an iterative subgradient method to solve the subproblem. In addition, based on IPSS, we propose a linearly convergent algorithm under the Polyak-Łojasiewicz condition. Finally, we extend the analysis to problems with weakly smooth function and obtain the computational complexity accordingly.