Reproduce “Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition” on OpenResearch

About this paper

Hamed Karimi, Julie Nutini, Mark Schmidt

Researchers from the University of British Columbia demonstrated that the Polyak-Lojasiewicz (PL) inequality provides a fundamental and unifying condition for proving global linear convergence rates of various optimization algorithms. This work simplifies theoretical analyses for methods like gradient descent, coordinate descent, and proximal methods, applying to many machine learning models without requiring strong convexity.

Key insights

• The Polyak-Lojasiewicz inequality is a generally weaker (or equivalent) condition than many existing criteria for guaranteeing global linear convergence, providing a unifying theoretical framework.
• Employing the PL inequality allows for remarkably simpler and more direct proofs of linear convergence for a wide range of first-order optimization algorithms.
• The PL condition implies invexity, ensuring that algorithms converge to a global minimum even for certain non-convex functions, which is crucial for practical applications.

Summary by alphaXiv.