  Subsampled Newton methods approximate Hessian matrices through subsampling
techniques, alleviating the cost of forming Hessian matrices but using
sufficient curvature information. However, previous results require $\Omega
(d)$ samples to approximate Hessians, where $d$ is the dimension of data
points, making it less practically feasible for high-dimensional data. The
situation is deteriorated when $d$ is comparably as large as the number of data
points $n$, which requires to take the whole dataset into account, making
subsampling useless. This paper theoretically justifies the effectiveness of
subsampled Newton methods on high dimensional data. Specifically, we prove only
$\widetilde{\Theta}(d^\gamma_{\rm eff})$ samples are needed in the
approximation of Hessian matrices, where $d^\gamma_{\rm eff}$ is the
$\gamma$-ridge leverage and can be much smaller than $d$ as long as $n\gamma
\gg 1$. Additionally, we extend this result so that subsampled Newton methods
can work for high-dimensional data on both distributed optimization problems
and non-smooth regularized problems.
