Finite Termination of a Generalized Perceptron Algorithm

TL;DR

This paper proves finite termination of a generalized perceptron algorithm for convex inequalities in Hilbert space under certain conditions, extending classical perceptron results.

Contribution

It establishes finite convergence for a subgradient method in convex systems, highlighting the importance of strict feasibility.

Findings

Finite termination is guaranteed under strict feasibility and bounded subgradients.

Without strict feasibility, convergence may fail or lead outside the feasible set.

The linear case aligns with the classical perceptron algorithm.

Abstract

Motivated by Ridgway's proof of the perceptron algorithm, we study a simple subgradient method for convex inequality systems in Hilbert space. Assuming strict feasibility and bounded subgradients, we establish finite termination for several natural step sizes. We also examine what can go wrong without strict feasibility: finite convergence may fail even for one function, and with several functions the method may converge to a point outside the feasible set. The linear setting recovers the classical perceptron algorithm.