  This work considers the non-convex finite sum minimization problem. There are
several algorithms for such problems, but existing methods often work poorly
when the problem is badly scaled and/or ill-conditioned, and a primary goal of
this work is to introduce methods that alleviate this issue. Thus, here we
include a preconditioner based on Hutchinson's approach to approximating the
diagonal of the Hessian, and couple it with several gradient-based methods to
give new scaled algorithms: Scaled SARAH and Scaled L-SVRG. Theoretical
complexity guarantees under smoothness assumptions are presented. We prove
linear convergence when both smoothness and the PL condition are assumed. Our
adaptively scaled methods use approximate partial second-order curvature
information and, therefore, can better mitigate the impact of badly scaled
problems. This improved practical performance is demonstrated in the numerical
experiments also presented in this work.
