Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-\L{}ojasiewicz Condition

TL;DR

This paper demonstrates that the Polyak-jolashvilia inequality, an older condition, is weaker than recent conditions for linear convergence and applies it to analyze various gradient-based optimization methods in machine learning.

Contribution

It shows the PL inequality's broader applicability for proving linear convergence and extends the analysis to proximal-gradient methods for non-smooth problems.

Findings

Proves linear convergence under the PL inequality for multiple algorithms.

Provides simple convergence proofs for various machine learning models.

Extends analysis to non-smooth optimization with proximal-gradient methods.

Abstract

In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the \L{}ojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older Polyak-\L{}ojasiewicz (PL) inequality is actually weaker than the main conditions that have been explored to show linear convergence rates without strong convexity over the last 25 years. We also use the PL inequality to give new analyses of randomized and greedy coordinate descent methods, sign-based gradient descent methods, and stochastic gradient methods in the classic setting (with decreasing or constant step-sizes) as well as the variance-reduced setting. We further propose a generalization that applies to proximal-gradient methods for non-smooth optimization, leading to simple proofs of linear convergence of these methods. Along the way, we give simple convergence results for a wide variety of problems in machine learning: least squares, logistic regression, boosting, resilient backpropagation, L1-regularization, support vector machines, stochastic dual coordinate ascent, and stochastic variance-reduced gradient methods.