Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

TL;DR

This paper analyzes the convergence behavior of stochastic gradient algorithms, providing new results that accommodate multiple and non-isolated minima, with implications for various recursive algorithms used across multiple fields.

Contribution

It introduces convergence rate bounds for stochastic gradient methods with complex minima, extending existing theory to more realistic scenarios.

Findings

Proves single limit-point convergence under broader conditions.

Derives tight bounds on convergence rates.

Applies results to algorithms in engineering, statistics, and machine learning.

Abstract

The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research.