Stochastic Heavy Ball with Polyak Step Size and Armijo Line Search: A General Convergence Analysis

2026-09-29

AI Summary

This study addresses the theoretical limitations in the convergence analysis of the stochastic heavy ball method combined with Polyak step sizes and Armijo line search by establishing a unified convergence framework. Methodologically, it introduces a modified Armijo rule and a momentum decoupling technique, providing general convergence proofs without requiring interpolation conditions. The contributions are threefold: first, the framework encompasses objectives ranging from strongly convex to nonconvex, achieving expected convergence; second, it strengthens almost sure convergence rates and last-iterate convergence guarantees in specific settings; third, under general assumptions, it proves that decaying variants converge exactly to optimal solutions or stationary points. Collectively, this work significantly extends the theoretical applicability boundaries of stochastic optimization algorithms.

Abstract

Polyak step size (PS) and Armijo line search (ALS) have received increasing attention in stochastic optimization, with encouraging empirical performance and theoretical guarantees. However, their convergence theory for stochastic heavy ball (SHB) methods remains limited. In this work, we develop a unified convergence analysis for SHB equipped with PS and ALS. To this end, we introduce a modified Armijo rule that closely parallels the Polyak step size, together with a decoupling analysis that isolates the historical dependence induced by momentum. For SHB with standard PS and ALS, we establish expected convergence for strongly convex, convex, and non-convex objectives without interpolation or restrictive conditions on the momentum parameter. Under interpolation or strong growth, we further strengthen the results to almost sure rates and last-iterate convergence. Moreover, for general settings beyond interpolation, we prove almost sure convergence to the exact optimum or to stationarity for SHB with diminishing variants of PS and ALS. These results provide a more comprehensive theoretical view of Polyak step size and Armijo line search for stochastic heavy ball methods.

Problem

• Stochastic Heavy Ball
• Polyak Step Size
• Armijo Line Search
• Convergence Analysis
• Stochastic Optimization

Innovation

• Stochastic Heavy Ball
• Polyak Step Size
• Armijo Line Search
• Unified Convergence Analysis
• Decoupling Analysis