Convergence of difference inclusions via a diameter criterion

TL;DR

This paper establishes convergence criteria for discrete difference inclusions using a diameter-based approach, applicable to various optimization algorithms including stochastic and momentum methods.

Contribution

It introduces a stratified descent framework and a diameter criterion that ensure convergence of first-order methods under diminishing step sizes.

Findings

Convergence is guaranteed when the inter-iterate diameter is controlled by a potential variation.

The framework applies to inexact, stochastic, and momentum-based methods for polynomially bounded objectives.

No continuous-time approximations are used in the convergence analysis.

Abstract

We study discrete dynamics governed by a difference inclusion whose increment is the sum of a selection from a set-valued map and a noise term. For any bounded realization, convergence follows once the inter-iterate diameter is controlled by the variation of a continuous potential. The limit point is then critical for a scaled outer limit of the update map. To certify this diameter criterion, we develop a stratified descent framework: we project iterates onto a suitable stratification and track a potential that decreases up to a summable error. Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order $1/k$. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures. Our arguments are entirely discrete, with no appeal to continuous-time approximations.