[Paper Review] Accelerated Stochastic Mirror Descent Algorithms For Composite Non-strongly Convex Optimization

Le Thi Khanh Hien, Cuong V. Nguyen+3 more|arXiv (Cornell University)|May 23, 2016

Stochastic Gradient Optimization Techniques26 references17 citations

TL;DR

This paper proposes accelerated stochastic mirror descent algorithms for composite non-strongly convex optimization, avoiding the need for quadratic regularization. By extending Tseng's deterministic accelerated proximal gradient methods to randomized settings, the proposed algorithms achieve the optimal convergence rate of $O(1/k^2)$, with improved complexity through parameter tuning.

ABSTRACT

We consider the problem of minimizing the sum of the average function consisting of a large number of smooth convex component functions and a general convex function that can be non-differentiable. Although many methods have been proposed to solve the problem with the assumption that the sum is strongly convex, few methods support the non-strongly convex cases. Adding a small quadratic regularization is the common trick used to tackle non-strongly convex problems; however, it may worsen certain qualities of solutions or weaken the performance of the algorithms. Avoiding this trick, we extend the deterministic accelerated proximal gradient methods of Paul Tseng to randomized versions for solving the problem without the strongly convex assumption. Our algorithms achieve the optimal convergence rate $O( icefrac{1}{k^2})$. Tuning involved parameters helps our algorithms get better complexity compared with the deterministic accelerated proximal gradient methods. We also propose a scheme for non-smooth problem.

Motivation & Objective

• To address composite optimization problems where the smooth component is non-strongly convex and the regularizer is general and possibly non-differentiable.
• To eliminate the need for quadratic regularization, which can degrade solution quality or algorithm performance.
• To extend deterministic accelerated proximal gradient methods to randomized, stochastic settings under non-strongly convex assumptions.
• To achieve optimal convergence rates without relying on strong convexity.
• To develop a scheme applicable to non-smooth problems within the proposed framework.

Proposed method

• Adapts Paul Tseng's deterministic accelerated proximal gradient method to a stochastic, randomized setting for large-scale composite optimization.
• Introduces a randomized mirror descent framework that maintains acceleration without requiring strong convexity.
• Employs a stochastic approximation of the gradient of the smooth component function using a subset of components at each iteration.
• Uses a proximal operator to handle the non-smooth, general convex regularizer in the optimization subproblem.
• Tunes algorithm parameters to improve iteration complexity compared to deterministic counterparts.
• Proposes a specialized scheme for handling non-smooth components in the objective function.

Experimental results

Research questions

• RQ1 Can accelerated stochastic mirror descent achieve optimal convergence for non-strongly convex composite problems without quadratic regularization?
• RQ2 How does parameter tuning affect the iteration complexity of the proposed stochastic accelerated algorithm?
• RQ3 What is the convergence rate of the proposed randomized algorithm under non-strongly convex assumptions?
• RQ4 Can the framework be extended to handle non-smooth regularizers effectively?
• RQ5 How does the performance of the stochastic method compare to deterministic accelerated methods in terms of convergence speed and complexity?

Key findings

• The proposed algorithm achieves the optimal convergence rate of $O(1/k^2)$ for composite non-strongly convex problems.
• The method avoids quadratic regularization, preserving solution quality and avoiding performance degradation.
• Parameter tuning leads to improved iteration complexity compared to deterministic accelerated proximal gradient methods.
• The algorithm is applicable to non-smooth problems through a proposed scheme.
• The randomized variant maintains acceleration and convergence guarantees without strong convexity.
• The theoretical convergence rate matches the best-known rates for deterministic accelerated methods, but in a stochastic setting.