  We study stochastic Cubic Newton methods for solving general possibly
non-convex minimization problems. We propose a new framework, which we call the
helper framework, that provides a unified view of the stochastic and
variance-reduced second-order algorithms equipped with global complexity
guarantees. It can also be applied to learning with auxiliary information. Our
helper framework offers the algorithm designer high flexibility for
constructing and analyzing the stochastic Cubic Newton methods, allowing
arbitrary size batches, and the use of noisy and possibly biased estimates of
the gradients and Hessians, incorporating both the variance reduction and the
lazy Hessian updates. We recover the best-known complexities for the stochastic
and variance-reduced Cubic Newton, under weak assumptions on the noise. A
direct consequence of our theory is the new lazy stochastic second-order
method, which significantly improves the arithmetic complexity for large
dimension problems. We also establish complexity bounds for the classes of
gradient-dominated objectives, that include convex and strongly convex
problems. For Auxiliary Learning, we show that using a helper (auxiliary
function) can outperform training alone if a given similarity measure is small.
