A Superlineary-Convergent Proximal Newton-Type Method for the Optimization of Finite Sums

We consider a new method for minimizing the sum of a large number of smooth functions which we call the cyclic Newton method. We show that the cyclic Newton method for minimizing the sum of n functions has an n-step quadratic local convergence rate under the assumptions analogous to those used in the classical convergence analysis of the standard Newton method. The proposed method is the first example of an incremental method with superlinear convergence. In the second part, we consider a generalization of the cyclic Newton method for composite minimization problems involving the sum of a large number of functions. In this case, one needs to solve an auxiliary quadratic composite minimization problem at each iteration of the method. In general, this problem does not have an exact analytical solution. We investigate the question of the accuracy with which one has to solve the auxiliary problem in order to retain the superlinear convergence of the method.