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In this paper, we propose a new SVRG-style acceleated stochastic algorithmfor solving a family of non-convex optimization problems whose objectiveconsists of a sum of $n$ smooth functions and a non-smooth convex function. Ourmajor goal is to improve the convergence of SVRG-style stochastic algorithms tostationary points under a setting with a large condition number $c$ - the ratiobetween the smoothness constant and the negative curvature constant. Theproposed algorithm achieves the best known gradient complexity when $c\geq\Omega(n)$, which was achieved previously by a SAGA-style acceleratedstochastic algorithm. Compared with the SAGA-style accelerated stochasticalgorithm, the proposed algorithm is more practical due to its low memory costthat is inherited from previous SVRG-style algorithms. Compared with previousstudies on SVRG-style stochastic algorithms, our theory provides much strongerresults in terms of (i) reduced gradient complexity under a large conditionnumber; and (ii) that the convergence is proved for a sampled stagewiseaveraged solution that is selected from all stagewise averaged solutions withincreasing sampling probabilities instead of for a uniformly sampled solutionsacross all iterations.

Katalyst: Boosting Convex Katayusha for Non-Convex Problems with a Large Condition Number