Convergence of projected stochastic approximation algorithm

TL;DR

This paper proves the convergence of a projected stochastic approximation algorithm, strengthening the theoretical foundation for stochastic optimization methods like stochastic gradient descent.

Contribution

It fills a gap in convergence proofs for the Robbins-Monro algorithm with projections, extending applicability and relaxing previous assumptions.

Findings

Proves convergence of the projected stochastic approximation algorithm.

Uses the ODE method to analyze convergence to stationary points.

Provides a theoretical basis for stochastic optimization techniques.

Abstract

We study the Robbins-Monro stochastic approximation algorithm with projections on a hyperrectangle and prove its convergence. This work fills a gap in the convergence proof of the classic book by Kushner and Yin. Using the ODE method, we show that the algorithm converges to stationary points of a related projected ODE. Our results provide a better theoretical foundation for stochastic optimization techniques, including stochastic gradient descent and its proximal version. These results extend the algorithm's applicability and relax some assumptions of previous research.