  For optimization on large-scale data, exactly calculating its solution may be
computationally difficulty because of the large size of the data. In this paper
we consider subsampled optimization for fast approximating the exact solution.
In this approach, one gets a surrogate dataset by sampling from the full data,
and then obtains an approximate solution by solving the subsampled optimization
based on the surrogate. One main theoretical contributions are to provide the
asymptotic properties of the approximate solution with respect to the exact
solution as statistical guarantees, and to rigorously derive an accurate
approximation of the mean squared error (MSE) and an approximately unbiased MSE
estimator. These results help us better diagnose the subsampled optimization in
the context that a confidence region on the exact solution is provided using
the approximate solution. The other consequence of our results is to propose an
optimal sampling method, Hessian-based sampling, whose probabilities are
proportional to the norms of Newton directions. Numerical experiments with
least-squares and logistic regression show promising performance, in line with
our results.
