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In this paper, we consider an unconstrained optimization model where theobjective is a sum of a large number of possibly nonconvex functions, thoughoverall the objective is assumed to be smooth and convex. Our bid to solvingsuch model uses the framework of cubic regularization of Newton's method. Aswell known, the crux in cubic regularization is its utilization of the Hessianinformation, which may be computationally expensive for large-scale problems.To tackle this, we resort to approximating the Hessian matrix via sub-sampling.In particular, we propose to compute an approximated Hessian matrix by either\textit{uniformly}\/ or \textit{non-uniformly}\/ sub-sampling the components ofthe objective. Based upon such sampling strategy, we develop acceleratedadaptive cubic regularization approaches and provide theoretical guarantees onglobal iteration complexity of $O(\epsilon^{-1/3})$ with high probability,which matches that of the original accelerated cubic regularization methods\cite{Jiang-2017-Unified} using the \textit{full}\/ Hessian information.Interestingly, we show that in the worst case scenario our algorithm stillachieves an $O\left(\log(\epsilon^{-1})\epsilon^{-5/6}\right)$ iterationcomplexity bound. The performances of the proposed methods on the regularizedlogistic regression problems show a clear effect of acceleration in terms ofthe epoch counts on several real data sets.

Accelerating Adaptive Cubic Regularization of Newton's Method via Random Sampling