Papers › Accelerating Incremental Gradient Optimization with Curvature Information
Accelerating Incremental Gradient Optimization with Curvature Information
Hoi-To Wai, Wei Shi, Cesar A. Uribe, Angelia Nedich, Anna Scaglione
This paper studies an acceleration technique for incremental aggregated gradient ({\sf IAG}) method through the use of \emph{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 ({\sf CIAG}) method and the accelerated CIAG ({\sf A-CIAG}) method, which are analogous to gradient method and Nesterov's accelerated gradient method, respectively. Setting κ to be the condition number of the objective function, we prove the R linear convergence rates of 1 - (4c₀ κ)/((κ+1)²) for the {\sf CIAG} method, and 1 - √(c₁/2κ) for the {\sf A-CIAG} method, where 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 {\sf CIAG} and {\sf 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 \url{http://github.com/hoitowai/ciag/}.
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