Accelerated Randomized Mirror Descent Algorithms For Composite Non-strongly Convex Optimization

TL;DR

This paper introduces an accelerated randomized mirror descent algorithm for non-strongly convex composite optimization, avoiding regularization drawbacks and extending to inexact proximal computations and non-smooth components.

Contribution

It develops a novel accelerated randomized mirror descent method that handles non-strongly convex problems without regularization, including non-smooth components, extending existing deterministic methods.

Findings

Supports large-scale convex optimization with improved convergence.

Handles inexact proximal computations effectively.

Applicable to non-smooth convex component functions.

Abstract

We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-strongly convex case. Adding a small quadratic regularization is a common devise used to tackle non-strongly convex problems; however, it may cause loss of sparsity of solutions or weaken the performance of the algorithms. Avoiding this devise, we propose an accelerated randomized mirror descent method for solving this problem without the strongly convex assumption. Our method extends the deterministic accelerated proximal gradient methods of Paul Tseng and can be applied even when proximal points are computed inexactly. We also propose a scheme for solving the problem when the component functions are non-smooth.