  We present two new remarkably simple stochastic second-order methods for
minimizing the average of a very large number of sufficiently smooth and
strongly convex functions. The first is a stochastic variant of Newton's method
(SN), and the second is a stochastic variant of cubically regularized Newton's
method (SCN). We establish local linear-quadratic convergence results. Unlike
existing stochastic variants of second order methods, which require the
evaluation of a large number of gradients and/or Hessians in each iteration to
guarantee convergence, our methods do not have this shortcoming. For instance,
the simplest variants of our methods in each iteration need to compute the
gradient and Hessian of a {\em single} randomly selected function only. In
contrast to most existing stochastic Newton and quasi-Newton methods, our
approach guarantees local convergence faster than with first-order oracle and
adapts to the problem's curvature. Interestingly, our method is not unbiased,
so our theory provides new intuition for designing new stochastic methods.
