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We study stochastic Cubic Newton methods for solving general possiblynon-convex minimization problems. We propose a new framework, which we call thehelper framework, that provides a unified view of the stochastic andvariance-reduced second-order algorithms equipped with global complexityguarantees. It can also be applied to learning with auxiliary information. Ourhelper framework offers the algorithm designer high flexibility forconstructing and analyzing the stochastic Cubic Newton methods, allowingarbitrary size batches, and the use of noisy and possibly biased estimates ofthe gradients and Hessians, incorporating both the variance reduction and thelazy Hessian updates. We recover the best-known complexities for the stochasticand variance-reduced Cubic Newton, under weak assumptions on the noise. Adirect consequence of our theory is the new lazy stochastic second-ordermethod, which significantly improves the arithmetic complexity for largedimension problems. We also establish complexity bounds for the classes ofgradient-dominated objectives, that include convex and strongly convexproblems. For Auxiliary Learning, we show that using a helper (auxiliaryfunction) can outperform training alone if a given similarity measure is small.

Unified Convergence Theory of Stochastic and Variance-Reduced Cubic Newton Methods