Global convergence rate of incremental aggregated gradient methods for nonsmooth problems

Abstract

Abstract. We focus on the problem of minimizing the sum of smooth component functions (where the sum is strongly convex) and a non-smooth convex function, which arises in regularized empirical risk minimization in machine learning and distributed constrained optimization in wireless sensor networks and smart grids. We consider solving this problem using the proximal incremental aggregated gradient (PIAG) method, which at each iteration moves along an aggregated gradient (formed by incrementally updating gradients of component functions according to a deterministic order) and taking a proximal step with respect to the non-smooth function. While the convergence properties of this method with randomized orders (in updating gradients of component functions) have been investigated, this paper, to the best of our knowledge, is the first study that establishes the convergence rate properties of the PIAG method for any deterministic order. In particular, we show that the PIAG algorithm is globally convergent with a linear rate provided that the step size is sufficiently small. We explicitly identify the rate of convergence and the corresponding step size to achieve this convergence rate. Our results improve upon the best known condition number dependence of the convergence rate of the incremental aggregated gradient methods used for minimizing a sum of smooth functions.

On Stochastic and Deterministic Quasi-Newton Methods for Nonstrongly Convex Optimization: Asymptotic Convergence and Rate Analysis

“…For example, averaging SA schemes achieve a rate of O M √ k , where M is an upper bound on the norm of the subgradient (see [21,22]). In past few years, fast incremental gradient methods with improved rates of convergence have been developed (see [5,9,26,29]). Of these, addressing the merely convex case, SAGA with averaging achieves a sublinear convergence rate O N k where N denotes the number of blocks, while in the presence of strong convexity, non-averaging variants of SAGA and IAG admit a linear convergence rate assuming that the function satisfies some smoothness conditions.…”