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In this work, we generalized and unified recent two completely differentworks of Jascha \cite{sohl2014fast} and Lee \cite{lee2012proximal} respectivelyinto one by proposing the \textbf{prox}imal s\textbf{to}chastic\textbf{N}ewton-type gradient (PROXTONE) method for optimizing the sums of twoconvex functions: one is the average of a huge number of smooth convexfunctions, and the other is a non-smooth convex function. While a set ofrecently proposed proximal stochastic gradient methods, include MISO,Prox-SDCA, Prox-SVRG, and SAG, converge at linear rates, the PROXTONEincorporates second order information to obtain stronger convergence results,that it achieves a linear convergence rate not only in the value of theobjective function, but also in the \emph{solution}. The proof is simple andintuitive, and the results and technique can be served as a initiate for theresearch on the proximal stochastic methods that employ second orderinformation.

Proximal Stochastic Newton-type Gradient Descent Methods for Minimizing Regularized Finite Sums