Accelerated proximal incremental algorithm schemes for non-strongly convex functions

Abstract

There have been a number of recent advances in accelerated gradient and proximal schemes for optimization of convex finite sum problems. Defazio introduced a simple accelerated scheme for incremental stochastic proximal algorithms inspired by gradient based methods like SAGA. He was able to prove O(1/k) convergence for non-smooth function but only under the assumption of strong convexity of component terms. We introduce a slight modification of his scheme, called MP-SAGA for which we can prove O(1/k) convergence without strong convexity, but for smooth functions. Numerical results show that our method has better or comparable convergence to Defazio's scheme, even for non-strongly convex functions. As important special cases, we also derive an accelerated schemes for a multi–class formulation of SVM as well as clustering based on the SON regularization. Finally, we introduce a simplification of Point–SAGA, called SP–SAGA for problems such as SON with large number of variables and sparse relation between variables and objective terms.