  The cubic regularized Newton method of Nesterov and Polyak has become
increasingly popular for non-convex optimization because of its capability of
finding an approximate local solution with second-order guarantee. Several
recent works extended this method to the setting of minimizing the average of N
smooth functions by replacing the exact gradients and Hessians with subsampled
approximations. It has been shown that the total Hessian sample complexity can
be reduced to be sublinear in N per iteration by leveraging stochastic variance
reduction techniques. We present an adaptive variance reduction scheme for
subsampled Newton method with cubic regularization, and show that the expected
Hessian sample complexity is O(N + N^{2/3}\epsilon^{-3/2}) for finding an
(\epsilon,\epsilon^{1/2})-approximate local solution (in terms of first and
second-order guarantees respectively). Moreover, we show that the same Hessian
sample complexity retains with fixed sample sizes if exact gradients are used.
The techniques of our analysis are different from previous works in that we do
not rely on high probability bounds based on matrix concentration inequalities.
Instead, we derive and utilize bounds on the 3rd and 4th order moments of the
average of random matrices, which are of independent interest on their own.
