Non-asymptotic convergence analysis of inexact gradient methods for machine learning without strong convexity

Abstract

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results by developing a framework for analysing the non-asymptotic convergence rates of IGMs when they are applied to a class of structured convex optimization problems that includes least squares regression and logistic regression. We then demonstrate the power of our framework by proving, in a unified manner, new linear convergence results for three recently proposed algorithms—the incremental gradient method with increasing sample size [R.H. Byrd, G.M. Chin, J. Nocedal, and Y. Wu, Sample size selection in optimization methods for machine learning, Math. Program. Ser. B 134 (2012), pp. 127–155; M.P. Friedlander and M. Schmidt, Hybrid deterministic–stochastic methods for data fitting, SIAM J. Sci. Comput. 34 (2012), pp. A1380–A1405], the stochastic variance-reduced gradient (SVRG) method [R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction, Advances in Neural Information Processing Systems 26: Proceedings of the 2013 Conference, 2013, pp. 315–323], and the incremental aggregated gradient (IAG) method [D. Blatt, A.O. Hero, and H. Gauchman, A convergent incremental gradient method with a constant step size, SIAM J. Optim. 18 (2007), pp. 29–51]. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods.