  In this paper, we consider an unconstrained optimization model where the
objective is a sum of a large number of possibly nonconvex functions, though
overall the objective is assumed to be smooth and convex. Our bid to solving
such model uses the framework of cubic regularization of Newton's method. As
well known, the crux in cubic regularization is its utilization of the Hessian
information, 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 of
the objective. Based upon such sampling strategy, we develop accelerated
adaptive cubic regularization approaches and provide theoretical guarantees on
global 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 still
achieves an $O\left(\log(\epsilon^{-1})\epsilon^{-5/6}\right)$ iteration
complexity bound. The performances of the proposed methods on the regularized
logistic regression problems show a clear effect of acceleration in terms of
the epoch counts on several real data sets.
