Accelerating incremental gradient optimization with curvature information

Abstract

Abstract This paper studies an acceleration technique for incremental aggregated gradient (IAG) method through the use of curvature information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using Hessian information. We propose and analyze two methods utilizing the new technique, the curvature-aided IAG (CIAG) method and the accelerated CIAG (A-CIAG) method, which are analogous to gradient method and Nesterov’s accelerated gradient method, respectively. Setting $$\kappa$$κ to be the condition number of the objective function, we prove the R linear convergence rates of $$1 - \frac{4c_0 \kappa }{(\kappa +1)^2}$$1-4c0κ(κ+1)2 for the CIAG method, and $$1 - \sqrt{\frac{c_1}{2\kappa }}$$1-c12κ for the A-CIAG method, where $$c_0,c_1 \le 1$$c0,c1≤1 are constants inversely proportional to the distance between the initial point and the optimal solution. When the initial iterate is close to the optimal solution, the R linear convergence rates match with the gradient and accelerated gradient method, albeit CIAG and A-CIAG operate in an incremental setting with strictly lower computation complexity. Numerical experiments confirm our findings. The source codes used for this paper can be found on http://github.com/hoitowai/ciag/.