Linear convergence of first order methods for non-strongly convex optimization

TL;DR

This paper establishes linear convergence rates for first order methods in smooth convex optimization without requiring strong convexity, by introducing relaxed conditions applicable to various practical problems.

Contribution

It introduces relaxed strong convexity conditions that ensure linear convergence of first order methods in non-strongly convex settings, broadening applicability.

Findings

Linear convergence achieved under relaxed conditions

Applicable to linear systems, LP, and dual convex problems

Validates conditions with practical examples

Abstract

The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this paper, we derive linear convergence rates of several first order methods for solving smooth non-strongly convex constrained optimization problems, i.e. involving an objective function with a Lipschitz continuous gradient that satisfies some relaxed strong convexity condition. In particular, in the case of smooth constrained convex optimization, we provide several relaxations of the strong convexity conditions and prove that they are sufficient for getting linear convergence for several first order methods such as projected gradient, fast gradient and feasible descent methods. We also provide examples of functional classes that satisfy our proposed relaxations of strong convexity conditions. Finally, we show that the proposed relaxed strong convexity conditions cover important applications ranging from solving linear systems, Linear Programming, and dual formulations of linearly constrained convex problems.